East African Journal of

Biophysical and Computational

Sciences

( E A J B C S )

ISSN (Online): 2789-3618 and ISSN (Print): 2789-360X

COLLEGE OF NATURAL AND COMPUTATIONAL SCIENCES,

HAWASSA UNIVERSITY

June, 2026

Volume 7 Issue 1

East African Journal of Biophysical and

Computational Sciences

ISSN (Online): 2789-3618 and ISSN (Print): 2789-360X

East African Journal of

Biophysical and Computational

Sciences (EAJBCS)

ISSN (Online): 2789-3618 and ISSN (Print): 2789-360X

Volume 7 Issue 1

College of Natural and Computational Sciences

Hawassa University

June, 2026

Table of Contents

East African Journal of Biophysical

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

ARTICLE

ARTICLE INFO

Volume 7(1), 2026

https:

Bifurcation Analysis of Eco-Epidemiological

Mathematical Model with Saturated

Incidence Rate and General Holling Type

Response Function

//dx.doi.org/10.4314/eajbcs.v7i1.1S

ARTICLE HISTORY

Received: 16 November, 2025

Accepted: 30 April, 2026

Published Online: 10 June, 2026

Solomon Molla Alemu¹, Tesfaye Tefera Mamo², Mohammed Yiha

Dawed^3,∗�

CITATION

Alemu et.al (2026) Bifurcation Analysis

of Eco-Epidemiological Mathematical

Model with Saturated Incidence Rate

and General Holling Type Response

Function. East African Journal of

¹Addis Ababa Science and Technology University, Department of Mathematics, Addis Ababa, Ethiopia,

²Debre Berhan University, Department of Mathematics, Debre Berhan, Ethiopia,

³Hawassa University, Department of Mathematics, Hawassa, Ethiopia

Biophysical and Computational

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.1S.1-17

^∗Corresponding author: mohammedyiha@hu.edu.et

Abstract

OPEN ACCESS

This paper presents a bifurcation analysis of an Eco-epidemiological model with saturated incidence

rate and general Holling-Type functional responses. The model describes a predator–prey system in

which the prey population is infected by a communicable disease, and the predator feeds on both

susceptible and infected individuals. Fundamental properties of the system, including existence and

uniqueness, positivity, and boundedness of solutions, are established to ensure biological feasibility.

Equilibrium points are identified and their stability is examined. The basic reproduction number '

0

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

is derived to determine threshold conditions for disease persistence. Using Sotomayor’s theorem,

transcritical and Hopf bifurcations are rigorously verified. The results indicate that increasing the

inhibition rate stabilizes the system and promotes coexistence, whereas higher transmission rates

destabilize equilibria and generate sustained oscillations. Numerical simulations and bifurcation

diagrams validate the analytical findings, demonstrating transitions between stable steady states and

periodic dynamics.

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Eco-epidemiology, Saturated incidence rate, Bifurcation, General Holling Type, Emergent

carrying capacity

essential approach for investigating and understanding the transmission

1 Introduction

and control of infectious diseases.

Numerous researchers (e.g.,

Hugo and Simanjilo (2019) and Sieber et al. (2014)) have explored

predator–prey models incorporating disease dynamics, highlighting how

infections within the prey and/or predator populations can signiﬁcantly

inﬂuence the ecological interactions and system stability. The primary

focus of eco-epidemiological models revolves around how infections

impact species mortality, decrease reproduction rates, the nature of

contamination, changes in population size, the eradication or control

of epidemic outbreaks, the persistence and the overarching dynamics

of the diseased species (Sieber et al., 2014). Saifuddin et al. (2016)

demonstrated that, under an explicit carrying capacity, susceptible and

infected prey exhibit identical competitive abilities, whereas under an

emergent carrying capacity, infected prey compete less eﬀectively than

susceptible ones in the presence of disease. Biswas et al. (2015) examined

a modiﬁed Lotka-Volterra system that incorporates the prey infection

propagation term based on the mass action law, while Haldar et al.

(2021) focused on standard incidence within predator-prey interactions.

Liu et al. (1987) proposed an epidemiological model characterized by

a nonlinear incidence rate. Gumel and Moghadas (2003) formulated

In applied mathematics, mathematical modeling serves as an essential

tool for investigating real-world problems across diverse disciplines,

including biology, epidemiology, and ecology (Bezabih et al., 2021).

Numerous researchers have demonstrated that the dynamic interactions

between predator and prey populations can be eﬀectively analyzed

using the tools of mathematical ecology (Das, 2016; Demir, 2019).

Building upon the foundational works of Lotka (1925) and Volterra

(1927), various sophisticated predator–prey models have been developed

to describe complex ecological interactions under diﬀerent realistic

scenarios (Ghanbari, 2021; Sieber et al., 2014). Furthermore, Anderson

and May (1986) established a pioneering framework that integrates the

epidemiological models of Kermack and McKendrick (Brauer, 2005)

with classical Lotka–Volterra predator–prey dynamics. As a result,

recent decades have been marked by a growing body of research

devoted to analyzing the dynamical behavior of eco-epidemiological

models (Biswas et al., 2015).

Since conducting experiments is

often impractical or unethical, mathematical modeling has become an

Alemu et.al (2026)

1

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

a tritrophic dynamics that incorporating a distinct saturating incidence

term to more accurately capture complex transmission dynamics, while

Ruan and Wang (2003) extended this line of research by examining

an epidemic model that integrates essential system with a saturating

To address these gaps, the study proposes a novel eco-epidemiological

model integrating saturated incidence, generalized Holling-type

predation on susceptible and infected prey, and an emergent carrying

capacity framework with distinct competition eﬀects. This integrated

approach strengthens theoretical understanding of system stability,

persistence, and complex population oscillations.

incidence term to investigate the overall system behavior.

Their

approach is deemed more justiﬁable because it considers behavioral

changes and the crowding eﬀect among infected individuals, thus

preventing the contact rate from becoming unbounded by selecting

appropriate parameters (Maiti et al., 2019). Hu et al. (2017) analyzed

a discrete-time eco-epidemiological framework, focusing on the system

dynamic behavior under a Holling type-II incidence function in place

of the bilinear incidence rate. Following these inﬂuential studies have

incorporated disease transmission into prey and/or predator populations

under various incidence mechanisms, including mass action, standard

incidence, and nonlinear forms. Among these, saturated incidence

rates have attracted considerable attention because they incorporate

behavioral changes and crowding eﬀects, thereby preventing unrealistic

unbounded transmission when the infected population becomes large.

Such formulations provide a more biologically realistic representation of

disease spread.

The remainder of this paper is organized as follows: Section 2 presents

the mathematical formulation of the model; Section 3 establishes

fundamental dynamical properties; Section 4 is devoted to stability

and bifurcation analysis; Section 5 provides numerical simulations

that support the analytical ﬁndings; Finally, the concluding section

summarizes the main results and discusses their ecological implications.

2 Mathematical Model

In this section, we investigate the eco-epidemiological dynamics to

explore the inﬂuence of a saturated incidence function on the sustainable

coexistence of two interacting species within the same ecosystem. Let ꢀ(C)

and ꢁ(C) denote the prey and predator densities at time C, respectively.

The model is formulated based on the following biological assumptions:

From an ecological perspective, predator–prey dynamics are strongly

inﬂuenced by the prey’s response to predation, while the predator

population, in turn, directly or indirectly regulates the prey population

(Panja, 2020). In order to accurately characterize the responsiveness

of predation rates to variations in prey biomass across diﬀerent

population densities, ecologically realistic functional responses have been

formulated that explicitly incorporate prey behavioral patterns. The

following functional responses are developed: Beddington-DeAngelis (Li

& Takeuchi, 2011), Crowley-Martin (Maiti et al., 2019), General Holling

type (Dawed et al., 2020), Michaelis-Menten type (HT-II), Holling type

III, Holling type IV (which came later) (Holling, 1959). Holling responses

are commonly categorized into speciﬁc forms (Type I–IV), each with

distinct ecological characteristics. However, in this study, the use of

the term “General Holling-Type functional responses” is intentionally

and methodologically justiﬁed. We mean either of these forms or

combinations of them:

The total prey population is divided into two compartments

ꢀ(C) = ꢀ (C) + ꢀ (C)

1

2

where ꢀ (C) and ꢀ (C) represent the susceptible and infected prey

1

2

populations, respectively.

The researchers assume that the lifespan of infected prey is shorter than

that of susceptible prey (Haldar et al., 2021). The susceptible prey

population ꢀ (C) follows logistic growth in the absence of predation

and disease. ¹Furthermore, both susceptible and infected prey share

limited environmental resources. However, they do not possess identical

competitive abilities. To capture this ecological feature, we incorporate

distinct competition coeﬃcients representing emergent carrying capacity:

0G

1 + G

0G²

1 + G²

0G

5(G) = 0G, ,(G) =

,

ℎ(G) =

,

A(G) =

,

1

denotes intra-speciﬁc competition among susceptible prey, while 1

2

1

1 + 1G + 2G²

represents inter-speciﬁc competition between susceptible and infected

prey (Ghanbari, 2021; Sieber et al., 2014). Thus, the logistic growth of

susceptible prey is given by

where, 0 is attack rate, 1 is a half saturation constant and 2 is the

measure of the predator tolerance to the prey to attack. Haque and

Venturino (2007) studied an eco-epidemic model in which the predator

population is infected and predation follows a ratio-dependent functional

response. Moreover, Kooi et al. (2011) also have discussed on tritrophics

food web eco-epidemiological system with predator infection, where the

infection transmitted among predators follow a hybrid response function

as Holling type-IV functional response and Beddington–DeAngelis type

functional response (Li & Takeuchi, 2011). Capasso and Serio (1978)

introduced an interaction term to account for the saturation eﬀect in

large infectious populations. Consequently, incorporating saturation

in disease transmission (Cai & Li, 2010) becomes particularly relevant

in eco-epidemiological models when the number of infectives is high.

Real-world predation involves complex mechanisms (Wayesa et al., 2024,

2025) such as prey refuge, handling time, predator interference, and

adaptive feeding, which can be captured using general Holling-type

functional responses. However, most eco-epidemiological models rely on

simpliﬁed predation terms and standard disease transmission functions,

with limited attention given to combining generalized predation

dynamics and saturated incidence. Key research gaps include:

3ꢀ

3C

1

= A_ꢀꢀ₁(1 − 1 ꢀ₁− 1 ꢀ ) .

1

2

1

The disease spreads among prey solely through direct contact. Infected

prey do not recover or acquire immunity; instead, they are removed from

the system through predation, disease-induced mortality at rate ꢀ, and

natural death at rate ꢁ .

1

We assume that susceptible prey become infected according to a

nonlinear saturated incidence function

ꢂꢀ ꢀ

2

1 +¹Bꢀ

2

as proposed in Maiti et al. (2019). Here, ꢂꢀ represents the force

2

1

of infection rate, while

accounts for behavioral changes and

1 + Bꢀ

2

crowding eﬀects among infected individuals. This formulation prevents

the transmission rate from becoming unbounded for large infected

populations (Ruan & Wang, 2003).

• Lack of systematic analysis of the combined eﬀects of saturated

disease transmission and general Holling-type responses on system

stability.

Ecologically, infected prey is generally more vulnerable to predation due

to its weakened physiological condition. To capture this phenomenon, we

incorporate distinct general Holling-Type functional responses, Φ_ꢀ(ꢀ )

and Φ_ꢀ(ꢀ ), which represent the predator’s consumption of susce¹ptib¹le

• Limited exploration of how these nonlinear mechanisms drive

qualitative changes such as transcritical and Hopf bifurcations.

2

• Few models incorporating emergent carrying capacity with

unequal competition between susceptible and infected prey.

and infected prey biomass, respectively.

Accordingly, the susceptible prey dynamics are given by

• The absence of a rigorous analytical framework linking the

basic reproduction number, stability switching, and bifurcation

dynamics Wang et al. (2016) under such generalized conditions.

ꢂꢀ ꢀ

3ꢀ

3C

1

2

1

= A_ꢀꢀ₁(1 − 1 ꢀ₁− 1 ꢀ ) −

− Φ_ꢀ(ꢀ )ꢁ,

1

2

1

1 + Bꢀ

2

Alemu et.al (2026)

2

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

while the infected prey dynamics are described by

3.1 Positivity of the solution

ꢂꢀ ꢀ

3ꢀ

3C

2

1 +¹Bꢀ

2

Let us denote R³= {((, ꢃ, %) ∈ R³: ( > 0, ꢃ ≥ 0, % > 0}, the positive

=

− (ꢁ₁+ ꢀ)ꢀ₂− Φ (ꢀ )ꢁ.

2

ꢀ

2

octants of the so⁺lution of our model system (3).

2

The model assumes a specialist predator population ꢁ(C) that feeds

on both susceptible and infected prey, with predation governed by

general Holling-type functional responses. Accordingly, the predator’s

population dynamics are formulated based on these generalized

predation interactions.

T heorem 1. The non-negative octant in R³is remain positive under the

dynamics for the model (3).

Proof. We want to verify

3ꢁ

3C

= ꢂ Φ (ꢀ )ꢁ + ꢂ Φ (ꢀ )ꢁ − ꢁ ꢁ,

1

2

ꢀ

2

1

(()) > 0,

ꢃ()) > 0,

%()) > 0,

for all , ) ≥ 0,

where ꢂ and ꢂ denote the conversion eﬃciencies of susceptible and

1

2

Rewrite the system (3) in the form

infected prey into predator biomass, respectively, and ꢁ represents the

2

natural mortality rate of the predator.

ꢀ

ꢁ

ꢄꢃ

1 + ꢅꢃ

2

ꢆ # (()%

3(

3)

1

(

= ( 1 − ( − ꢃꢃ −

−

= (& ((, ꢃ, %),

1

The descriptions of state variables and parameters are provided in Table 1.

All parameters are assumed to be positive. Hence, based on the above

assumptions, the governing eco-epidemiological model takes the form

(

ꢂ

ꢃ

ꢄ(

ꢆ # (ꢃ)%

3ꢃ

3)

ꢃ

ꢅ

= ꢃ

− ꢇ −

= ꢃ& ((, ꢃ, %),

2

1 + ꢅꢃ

ꢀ

ꢁ

ꢄ



ꢂꢀ ꢀ

3%

3)

3ꢀ

3C

1

2

1



= % #₍(() + #_ꢃ(ꢃ) − ꢈ = %& ((, ꢃ, %).

= A_ꢀ

ꢀ

1 − 1 ꢀ₁− 1 ꢀ

−

− Φ_ꢀ(ꢀ )ꢁ,

3

1

2

1

1 + Bꢀ

2





From the above expression and the initial conditions (4), we have:

ꢂꢀ ꢀ

3ꢀ

3C

1 +¹Bꢀ

2

(1)

(2)

=

− (ꢁ₁+ ꢀ)ꢀ₂− Φ (ꢀ )ꢁ,

2

ꢀ

2



2



3ꢁ

3C



= ꢂ Φ (ꢀ )ꢁ + ꢂ Φ (ꢀ )ꢁ − ꢁ ꢁ,

1

2

ꢀ

2

1



ꢂ

ꢃ

¹

)

(()) = (₀exp

& ((, ꢃ, %)3D

,

1

with initial conditions

0

ꢂ

ꢃ

¹

ꢀ (0) = ꢀ⁰> 0, ꢀ (0) = ꢀ⁰≥ 0, ꢁ(0) = ꢁ⁰> 0.

)

1

2

1

2

ꢃ()) = ꢃ₀exp

& ((, ꢃ, %)3D

2

,

0

ꢂ

¹

)

%()) = % exp

& ((, ꢃ, %)3D

3

.

2.1 Non-Dimensionalization

0

Non-Dimensionalization simplify and make the equations easier to

interpret. The transformation equations could be:

As, the initial conditions (4) and the exponential form are positive, thus,

1

B

1

all the state variables (()), ꢃ()) and %()) are positive ∀) ≥ 0. Therefore,

ꢀ

=

(, ꢀ

A_ꢀ

=

₁ꢃ, ꢁ = ₁%, C =

), Φ_ꢀ

1

2

1

A_ꢀ

1

every solutions of the mathematical model 3 are positive.

1

A_ꢀ

(ꢀ ) = ¹#₍((), and Φ_ꢀ(ꢀ ) = ¹#_ꢃ(ꢃ).

1

2

ꢂ

2

1

3.2 Bounded behavior of trajectories

Thus, the scaled form of the dynamical system is

ꢀ

ꢁ

ꢄ(ꢃ

1 + ꢅꢃ

 3(



T heorem 2. All possible solution of the dynamical system (3) are consistently

= ( 1 − ( − ꢃꢃ

−

− ꢆ # (()%,

1

(



3)

bounded in R³and enter in the invariant zone

+





ꢄ(ꢃ

3ꢃ

(3)

=

− ꢇꢃ − ꢆ # (ꢃ)%,

2

ꢃ



3)

1 + ꢅꢃ

= #₍(()% + #_ꢃ(ꢃ)% − ꢈ%,



nꢀ

ꢁ



Σ = (()), ꢃ()), %()) ∈ R³: 0 < ( ≤ max{( , 1},

0

3%

3)

+



ꢆ

ꢇꢇ



2

(1 + ꢈ − <)

0 < ꢉ ≤ max

, ꢉ

(5)

0

ꢂ

1

B

1

ꢁ₁+ ꢀ

4(ꢈ − <)

where ꢃ = ², ꢆ

=

, ꢆ

2

=

, ꢄ =

, ꢅ =

, ꢇ =

,

1

ꢂ

1 A

1

A_ꢀ

1

2

ꢀ

1

ꢁ

2

where ꢉ()) = (()) + ꢃ()) + ꢆ %()), 0 < # (ꢃ) ≤ <.

1

ꢃ

ꢈ =

, and

A_ꢀ

1

((0) = ( > 0, ꢃ(0) = ꢃ ≥ 0, %(0) = % > 0.

(4)

0

Proof. As established in Proposition (1), the solutions (()), ꢃ()), and %())

of system (3) remain positive for all ) ≥ 0. Considering the ﬁrst equation

of the model (3), it follows that

3 Mathematical Model Analysis

ꢀ

ꢁ

ꢄ(ꢃ

1 + ꢅꢃ

3(

3)

= ( 1 − ( − ꢃꢃ

−

− ꢆ # (()% ≤ ((1 − ()

1

(

The analysis of mathematical models in eco-epidemiology provides

valuable insights into disease transmission dynamics, host–pathogen

interactions, and the ecological feedback mechanisms within the

system. Such analysis helps to explore system behavior, identify critical

parameters, and examine aspects like stability, bifurcation, and possible

outcomes, including disease outbreaks. Furthermore, properties such as

the existence and uniqueness of solutions, positivity, boundedness, as

well as permanence, persistence, and numerical simulations of the model

system 3, will be studied.

This directly leads to

ꢈ

ꢉ₋₁

ꢀ

ꢁ

1

(

0

(()) ≤ 1 +

− 1 4⁻⁾

=

.

(

₀+ (1 − ( )4⁻⁾

(

0

Therefore,

lim sup (()) ≤ max{( , 1}.

0

)→∞

Alemu et.al (2026)

3

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Table 1: The state variables and parameters description

Ecological Meaning

Variables/Parameters

Dimension

ꢀ

ꢁ

Susceptible prey density

Per Area

Per time

Area

1

2

Infected prey density

Predator density

Response functions

Φ_ꢀ(resp.Φ_ꢀ

A_ꢀ

)

1

2

Natural propagation rate of susceptible prey

Intra-speciﬁc competition coeﬃcient among susceptible prey

Inter-speciﬁc competition coeﬃcient between susceptible and infected prey

Transmission rate

1

2

ꢂ

B

Area

Per time

Area coverage

No unit

Per time

Inhibition rate

ꢂ

Proportion of susceptible prey into predator

Proportion of infected prey into predator

Natural death rates of infected prey/predator

Disease induced mortality rate

1

2

ꢁ /ꢁ

1

2

ꢀ

Thus, ((C) is bounded. To show other state variables ꢃ()) and %()) are

Proof. Let the right parts of the dynamical system (3) be denoted by ꢄ =

bounded we consider

( 5 , 5 , 5 ). Since 5 , 5 and 5 are continuous function, ꢄ = ( 5 , 5 , 5 )

2

3

1

is¹continuous function²in seve³ral variables, that is, ꢄ ∈ ꢂ¹(R³).¹Th²us,³ꢄ

satisfy the Lipschitz condition with respect to G in ꢅ. Hence, t⁺he solution

of system (3) exists. The locally Lipschitz condition of ꢄ is veriﬁed using

ꢉ = ( + ꢃ + ꢆ %

1

By diﬀerentiating ꢉ with respect to time ), we obtain

% 5₈

, 8, 9 = 1, 2, 3 to be continuously bounded within the domain ꢅ

3ꢉ

3)

3(

3)

3ꢃ

3)

3%

%G₉

=

+

+ ꢆ

¹3)

3(

3)

(Bezabih et al., 2021). We note that ꢄ ∈ ꢂ¹(R³, !8?) in ꢅ and 5

=

,

ꢀ

ꢁ

ꢄ(ꢃ

1 + ꢅꢃ

1

+

= ( 1 − ( − ꢃꢃ

−

− ꢆ # (()%

1

(

% 5₈

3ꢃ

3%

3)

5

=

and 5

=

. To show

, 8, 9 = 1, 2, 3 to be continuously

ꢂ

ꢃ

2

3

3)

%G₉

ꢄ (ꢃ

1 + ꢅꢃ

2

+

− ꢇꢃ − ꢆ # (ꢃ)%

2

ꢃ

bounded. Now we get

ꢄ

ꢅ

+ ꢆ #₍(()% + #_ꢃ(ꢃ)% − ꢈ%

1

ꢀ

ꢁ

= ( 1 − ( − ꢃ(ꢃ − ꢇꢃ − ꢆ # (ꢃ)% + ꢆ # (ꢃ)% − ꢆ ꢈ%

2

1

ꢃ

ꢄ

ꢅ

% 5

ꢄꢃ

1 + ꢅꢃ

1

= 1 − 2( − ꢃꢃ −

− ꢆ #⁰(()% ≤ 1,

ꢀ

ꢁ

1

(

%(

% 5

≤ ( 1 − ( − ꢃ(ꢃ − ꢇꢃ + ꢆ # (ꢃ)% − ꢆ ꢈ%

1

ꢃ

ꢂ

ꢃ

ꢄ(

1

= (1 + ꢈ)( − (²− ꢃ(ꢃ − (ꢇ − ꢈ)ꢃ + ꢆ # (ꢃ)% − ꢈ(( + ꢃ + ꢆ %)

= −ꢃ( −

= −( ꢃ +

,

2

1

ꢃ

%ꢃ

(1 + ꢅꢃ)

ꢊ

ꢂ

ꢃ

≤ (1 + ꢈ)( − (²+ ꢆ # (ꢃ)% − ꢈꢉ

1

3 5

1

3ꢃ

ꢄ(

ꢃ

This implies

= ( ꢃ +

< ∞, as ( and ꢃ are bounded,

2

(1 + ꢅꢃ)

ꢊ

3 5

1

3%

3 5

1

ꢊ

The general Holling Type response function #_ꢃ(ꢃ) is bounded, say, #_ꢃ(ꢃ) ≤

= −ꢆ # (() implies

= −ꢆ # (() = ꢆ # (() < ꢆ # ,

1

(

3%

<, 0 < < < ꢈ ≤ ꢇ. Then after simpliﬁcation we arrive

as # (() ≤ # ∈ R,

1

(

2

3ꢉ

3)

(1 + ꢈ − <)

% 5

2

ꢄꢃ

ꢄ

ꢅ

≤ (1 + ꢈ − <)( − (²− (ꢈ − <)ꢉ ≤

− (ꢈ − <)ꢉ.

=

≤

,

4

%(

% 5

1 + ꢅꢃ

ꢄ(

2

We can thus conclude that

− ꢇ − ꢆ #⁰(ꢃ)% ≤

< ∞,

2

ꢃ

2

%ꢃ

(1 + ꢅꢃ)

ꢂ

ꢃ

2

(1 + ꢈ − <)

ꢉ()) ≤

(1 + ꢈ − <)

− ꢉ 4^{−(ꢈ−<))}.

as ( and ꢃ are bounded,

−

0

4(ꢈ − <)

ꢊ

3 5

2

3%

3 5

2

3%

ꢊ

= −ꢆ # (ꢃ) =⇒

= −ꢆ # (ꢃ) = ꢆ # (ꢃ) < ꢆ # ,

2

ꢃ

As a result, we ﬁnd that

as # (ꢃ) ≤ # ∈ R,

= #₍⁰(()% < ∞,

2

ꢃ

ꢆ

ꢇ

2

(1 + ꢈ − <)

% 5

3

lim sup ꢉ()) ≤ max

, ꢉ

.

0

4(ꢈ − <)

)→∞

%(

% 5

3

= #⁰_ꢃ(ꢃ)% < ∞,

Hence, ꢉ()) remains bounded for all ) ≥ 0, which implies that the other

%ꢃ

% 5

state variables are also bounded. Consequently, all solutions of system (3)

3

are uniformly bounded on [0, ∞).

= # (() + # (ꢃ) − ꢈ < # (() + # (ꢃ) ≤ #₁+ # < ∞.

2

(

ꢃ

(

ꢃ

%%

As these all are continuous and bounded, ꢄ satisfy the locally Lipschitz

condition. Therefore, the unique solution of the system (3) is veriﬁed as

3.3 Existence & Uniqueness

it is explained in Allen et al. (2007) and Hale (2009).

T heorem 3. Let ꢄ = ( 5 , 5 , 5 ). If ꢄ satisﬁes the Lipschitz condition and has

continuous ﬁrst partial deriv²atives with respect to G in a domain ꢅ, then ꢄ(), G)

1

3

3.4 Equilibrium points

is locally Lipschitz in G. Consequently, for any initial point () , G ) ∈ ꢅ, there

0

exists a unique solution G(), ) , G ) of the system

0

The ﬁxed points of the dynamical system (3) are the roots of a nonlinear

system of equations.

3G

3)

= ꢄ(), G), G() ) = G ,

0

ꢀ

ꢁ

ꢄ(ꢃ

1 + ꢅꢃ

( 1 − ( − ꢃꢃ

−

− ꢆ # (()% = 0,

(6)

1

(

which passes through () , G ).

0

Alemu et.al (2026)

4

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

ꢄ(ꢃ

1 + ꢅꢃ

2ꢅꢇ

ꢄ

− ꢇꢃ − ꢆ # (ꢃ)% = 0,

(7)

(8)

• If ꢇ > ꢄ (i.e., Δ > 0) and

+ ꢄ + ꢃ > ꢅ (i.e Δ > 0), then (11) has

2

3

2

ꢃ

no positive root meaning that there is no feasible equilibrium point

#₍(()% + #_ꢃ(ꢃ)% − ꢈ% = 0.

ꢆ².

Hence, the extinction ﬁxed point is ꢆ⁰(0, 0, 0), the axial ﬁxed point is

ꢆ¹(1, 0, 0).

• If ꢇ < ꢄ (i.e., Δ < 0), then there exists a unique equilibrium point

3

The predator free equilibrium point ꢆ²is obtained by the intersection

ꢆ².

ꢂ

ꢃ

3(

3)

point of the zero growth isocline of susceptible

= 0 and the zero

2ꢅꢇ

ꢄ

ꢂ

ꢃ

• If ꢇ > ꢄ (i.e., Δ > 0) and

+ ꢄ + ꢃ < ꢅ (i.e., Δ < 0), then

3ꢃ

3)

3

2

growth isocline of infected species

= 0 where % = 0. That is,

equation (11) has two positive roots, consequently two equilibrium

points ꢆ₁², and ꢆ₂².

ꢀ

ꢁ

ꢄ(^∗ꢃ^∗

1 + ꢅꢃ^∗

∗

(

1 − (^∗− ꢃꢃ^∗

−

ꢄ(^∗ꢃ^∗

1 + ꢅꢃ^∗

= 0,

(9)

− ꢇꢃ^∗= 0,

(10)

Disease free equilibrium point

The infection free ﬁxed point of the form ꢆ³((^∗, 0, %^∗) is solution of

From equation (10), we get

non-linear system

ꢇꢅ

ꢃ^∗

ꢇ

(^∗=

+

ꢀ

ꢁ

ꢄ

∗

(

1 − (^∗− ꢆ # ((^∗)%^∗= 0 and # ((^∗)%^∗− ꢈ%^∗= 0.

1

(

Substitute this equation in (9), after simpliﬁcation we arrived

ꢀ

ꢁ

Δ ꢃ^∗2+ Δ ꢃ^∗+ Δ = 0,

(11)

∗

(

1 − (^∗

1

2

3

This gives #₍((^∗) = ꢈ and %^∗=

.

ꢂ

ꢃ

ꢇꢅ

ꢄ

2ꢅꢇ

ꢄ

ꢇ

ꢈꢆ

where Δ = ꢅ ꢃ +

> 0, Δ

=

+ ꢄ + ꢃ − ꢅ and Δ

=

− 1.

1

2

3

ꢄ

The positive roots ꢃ^∗in the quadratic equation above is possible provided

Table 2 provides an explanation of the illness free equilibrium point’s

existence criteria. where ℎ is half saturation constant and F denote

predator attack rate.

that the discriminant of an equation is positive, that is, Δ²₂− 4Δ Δ > 0

and follow from Descartes’ rule of sign. We have the following r¹esults:

2

Table 2: Existence conditions of the disease free ﬁxed point.

HT

HT-I

HT-II

HT-III

ℎ²ꢈ

HT-IV

p

q

2

4($−ꢈ)+ 4²(ꢈ−$) −4ꢈℎ4

ꢈ

ℎꢈ

∗

(

$−ꢈ

2ꢈ

√

$

$ − ꢈ

Conditions

ꢈ < $

ℎꢈ < $ − ꢈ and $ > ꢈ

ℎ²ꢈ < $ − ꢈ and $ > ꢈ

(ꢈ − $)4 > 2 ꢈℎ

The basic reproduction number, '

Thus,

0

V (ꢃ) ≈ ꢇꢃ + ꢆ #⁰(0)%^∗ꢃ.

2

ꢃ

According to Layek (2015), the basic reproduction number is the average

number of new infections from a single sick individual in a community

that is completely susceptible over the course of the infectious period. It is

used to predict whether the epidemic will spread or die out (Omar et al.,

2024). To compute the basic reproduction number, we consider only the

infected compartment of system (3)

The linearized equation becomes

ꢋ

ꢌ

ꢄ

ꢅ

3ꢃ

3)

=

ꢄ(^∗− ꢇ + ꢆ #⁰(0)%^∗ꢃ.

2

ꢃ

ꢄ(ꢃ

3ꢃ

=

− ꢇꢃ − ꢆ # (ꢃ)%.

(12)

2

ꢃ

3)

1 + ꢅꢃ

Hence, the new infection rate is

Following the next-generation matrix approach, we write

ꢄ = ꢄ(^∗,

3ꢃ

3)

= ℱ (ꢃ) − V (ꢃ),

and the total removal rate is

where the new infection and the transition (removal) terms represent

+ = ꢇ + ꢆ #⁰(0)%^∗.

2

ꢃ

ꢄ(ꢃ

1 + ꢅꢃ

ℱ (ꢃ) =

, and V (ꢃ) = ꢇꢃ + ꢆ # (ꢃ)%.

2

ꢃ

By the next-generation method,

Since ' measures the invasion of infection when ꢃ is small, we linearize

the syst⁰em around ꢃ = 0. Using Taylor expansion,

'

= ꢄ+⁻¹

.

0

ꢄ(ꢃ

1 + ꢅꢃ

= ꢄ(ꢃ(1 − ꢅꢃ + $(ꢃ²)).

Therefore,

ꢄ(^∗₀

keeping only ﬁrst-order terms gives ℱ (ꢃ) ≈ ꢄ(ꢃ. Similarly, expanding

'

=

.

(13)

#_ꢃ(ꢃ) near ꢃ = 0, #_ꢃ(ꢃ) ≈ #⁰(0)ꢃ.

0

ꢇ + ꢆ # (0)%^∗

2

ꢃ

Alemu et.al (2026)

5

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Coexistence Equilibrium Point

4.1 Local stability analysis

T heorem 4. The system admits a coexistence equilibrium ꢆ^∗((^∗, ꢃ^∗, %^∗) if the

4.1.1 Stability nature near ꢆ⁰(0, 0, 0)

following conditions hold:

ꢄ(^∗

1 + ꢅꢃ^∗

(^∗+ ꢃꢃ^∗< 1,

> ꢇ, #₍((^∗) + #_ꢃ(ꢃ^∗) = ꢈ,

'

> 1

0

.

1

0

©

-

ª

®

ꢇ(ꢆ⁰) =

^-0 −ꢇ

-

Proof. To determine the coexistence equilibrium point ꢆ^∗((^∗, ꢃ^∗, %^∗), we

0

−ꢈ

«

¬

set

3(

3)

3ꢃ

3)

3%

3)

= 0,

= 0.

Thus, ꢊ = 1 > 0, ꢊ = −ꢇ < 0, and ꢊ = −ꢈ < 0. Hence, the trivial ﬁxed

point ꢆ⁰is unstable². Biologically, this³indicates that total extinction of the

1

Thus the equilibrium point satisﬁes the algebraic equations

populations is impossible.

ꢄ(^∗ꢃ^∗

(^∗(1 − (^∗− ꢃꢃ^∗) −

− ꢆ # ((^∗)%^∗= 0,

(3.9)

1

(

1 + ꢅꢃ^∗

4.1.2 System behavior near ꢆ¹(1, 0, 0)

ꢄ(^∗ꢃ^∗

− ꢇꢃ^∗− ꢆ # (ꢃ^∗)%^∗= 0,

(3.10)

(3.11)

2

ꢃ

1 + ꢅꢃ^∗

#₍((^∗)%^∗+ #_ꢃ(ꢃ^∗)%^∗− ꢈ%^∗= 0.

ꢄ(^∗

1 + ꢅꢃ^∗

−1 −ꢃ − ꢄ −ꢆ #₍(1)

ꢄ − ꢇ

¹0

©

-

ª

®

From the ﬁrst, second and third equations we have (^∗+ꢃꢃ^∗< 1,

>

ꢇ(ꢆ¹) =

0

ꢇ, and #₍((^∗) + #_ꢃ(ꢃ^∗) = ꢈ, respectively.

0

#₍(1) − ꢈ

«

¬

Note that

ꢈꢆ ꢄ #⁻¹(ꢈ)

1

(

'

=

ꢀ

ꢁ .

0

ꢇꢈꢆ₁+ ꢆ #⁰(0)#⁻¹(ꢈ) 1 − #₍⁻¹(ꢈ)

The eigenvalues are ꢊ = −1, ꢊ = ꢄ − ꢇ, and ꢊ = #₍(1) − ꢈ. Thus,

1

2

the axial ﬁxed point is locally asymptotically sta³ble whenever ꢄ < ꢇ

and #₍(1) < ꢈ. This has a biological implication that susceptible prey

population survive alone whenever no disease in the environment and

without predator whenever the conditions holds.

2

ꢃ

(

ꢄ(^∗

1 + ꢅꢃ^∗

ꢄ(^∗

ꢇ(1 + ꢅꢃ^∗)

> ꢇ ⇐⇒

> 1.

At equilibrium (using (^∗= #₍⁻¹(ꢈ) we obtain

4.1.3 System behavior near consumer free fixed point ꢆ²((^∗, ꢃ^∗, 0)

ꢄ(^∗

ꢇ(1 + ꢅꢃ^∗)

= ' .

0

T heorem 5. The consumer-free ﬁxed point ꢆ²((^∗, ꢃ^∗, 0) is locally

asymptotically stable if the following conditions hold

ꢄ(^∗

> ꢇ ⇐⇒

'

> 1.

0

1 + ꢅꢃ^∗

(i) #₍((^∗) + #_ꢃ(ꢃ^∗) < ꢈ,

(ii) ꢇ + 2(^∗+ ꢃꢃ^∗+

ꢄꢃ^∗

ꢄ(^∗

< 1 +

,

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

ꢂ

ꢃ

ꢄꢃ^∗

4 Stability and Bifurcation

Analysis

(iii)

1 − 2(^∗− ꢃꢃ^∗−

1 + ꢅꢃ^∗

ꢂ

ꢃ

ꢄ(^∗

×

+

− ꢇ

(1 + ꢅꢃ^∗)²

ꢂ

ꢃ

ꢄ(^∗

ꢄꢃ^∗

1 + ꢅꢃ^∗

By examining sign of the derivative matrix’s eigenvalues, we can

determine the stability of a ﬁxed points as in Dawed et al. (2020). The

system (3) has a stable ﬁxed point ꢆ^∗((^∗, ꢃ^∗, %^∗) if all characteristic roots of

the Jacobian matrix, ꢇ(ꢆ^∗),

ꢃ(^∗+

> 0.

(1 + ꢅꢃ^∗)²

ꢇ₁₁ꢇ₁₂ꢇ₁₃

^-ꢇ₂₁ꢇ₂₂ꢇ₂₃

ꢇ₃₁ꢇ₃₂ꢇ₃₃

©

ª

®

-

ꢇ(ꢆ^∗) =

(14)

Proof. The community matrix of the model (3) at ꢆ²is given by

-

«

¬

have negative real part where

ꢄꢃ^∗

ꢄ(^∗

(1 + ꢅꢃ^∗)²

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

ꢇ

= 1 − 2(^∗− ꢃꢃ^∗−

− ꢆ #⁰((^∗)%^∗, ꢇ = −ꢃ(^∗−

,

1 − 2(^∗− ꢃꢃ^∗−

−ꢃ(^∗−

−ꢆ # ((^∗)

11

1

12

1 + ꢅꢃ^∗

©

-

ª

®

(

1

(

(1 + ꢅꢃ^∗)

2

ꢄꢃ^∗

ꢄ(^∗

2

ꢄꢃ^∗

ꢄ(^∗

− ꢇ − ꢆ #⁰(ꢃ^∗)%^∗,

ꢇ(ꢆ ) =

ꢇ

= −ꢆ # ((^∗), ꢇ

=

, ꢇ

22

=

13

1

21

2

− ꢇ

−ꢆ # (ꢃ^∗)

(

2

ꢃ

1 + ꢅꢃ^∗

ꢃ

(1 + ꢅꢃ^∗)²

1 + ꢅꢃ^∗

2

(1 + ꢅꢃ^∗)

= −ꢆ # (ꢃ^∗), ꢇ = #⁰((^∗)%^∗, ꢇ = #⁰(ꢃ^∗)%^∗, and ꢇ = #₍((^∗) +

(

ꢃ

0

# ((^∗) + # (ꢃ^∗) − ꢈ

«

¬

23

2

31

32

33

ꢃ

(

ꢃ

#_ꢃ(ꢃ^∗) − ꢈ.

The associated auxiliary equation is 34C(ꢇ(ꢆ²) − ꢊꢃ ) = 0, that is

3

The corresponding characteristic equation is det(ꢇ(ꢆ^∗) − ꢊꢃ ) = 0, that is

3

ꢊ

∗

ꢋ − ꢊ

ꢌ

−ꢆ # (( )

1

(

ꢊ

^ꢊꢇ − ꢊ

ꢇ₁₂

ꢇ₂₂− ꢊ

ꢇ₃₂

ꢇ₁₃

ꢇ₂₃

∗

ꢊ

11

ꢇ₂₁

ꢇ₃₁

ꢌ

ꢋ − ꢊ −ꢆ # (ꢃ ) = 0,

2

ꢃ

ꢊ

= 0

(15)

ꢊ

0

ꢋ − ꢊ

3

ꢇ₃₃− ꢊ^ꢊ

ꢊ

Alemu et.al (2026)

6

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

ꢄꢃ^∗

ꢄ(^∗

(1 + ꢅꢃ^∗)²

where, ꢋ = 1 − 2(^∗− ꢃꢃ^∗−

, ꢋ

=

− ꢇ, ꢋ = #₍((^∗) +

we write

1

2

3

1 + ꢅꢃ^∗

ꢄ(^∗

ꢄꢃ^∗

ꢊ

= (ꢇ + ꢆ #⁰(0)%^∗)('₀− 1).

1

2

# (ꢃ^∗) − ꢈ, ꢌ = −ꢃ(^∗−

< 0 and ꢌ

=

> 0. Thus, one

ꢃ

1

2

ꢃ

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

eigenvalue at ꢆ²((^∗, ꢃ^∗, 0) is ꢊ = ꢋ = # ((^∗) + # (ꢃ^∗) − ꢈ. Now, ꢊ is

1

3

1

(

ꢃ

Thus, if ' < 1, then ꢊ < 0 and the infection cannot invade. The others

two eigenvalues are computed from the matrix

0

1

negative if #₍((^∗) + #_ꢃ(ꢃ^∗) < ꢈ. The rest two eigenvalues are found from

the matrix

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

(1 + ꢅꢃ^∗)²

1 − 2(^∗− ꢆ #⁰((^∗)%^∗−ꢆ # ((^∗)

1 − 2(^∗− ꢃꢃ^∗−

−ꢃ(^∗−

1

(

©

-

ª

®

©

-

ª

®

ꢇ =

2

#₍⁰((^∗)%^∗

0

¯

ꢇ(ꢆ ) =

.

«

¬

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

− ꢇ

(1 + ꢅꢃ^∗)²

«

¬

By the Routh-Hurwitz stability rule, th₀e two eigenvalues o₀f ꢇ possess

negative real parts provided that ꢆ ꢈ# ((^∗)%^∗> 0 (i.e., #₍((^∗) > 0)

¯

Using the Routh–Hurwitz criterion, the two eigenvalues of ꢇ are negative

1

(

and #₍⁰((^∗)(^∗2− (2ꢈ + #₍⁰((^∗))(^∗+ ꢈ < 0. Thus, if conditions (1)–(3) are

satisﬁed, we conclude that the model system (3) is locally asymptotically

stable at the disease-free ﬁxed point ꢆ³. The predator eating eﬃciency is

so high whenever conditions (1)–(3) are satisﬁed. The predator will only

eat healthy prey because there is no infected prey present.

in their real parts provided that

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

ꢇ + 2(^∗+ ꢃꢃ^∗+

< 1 +

and

(1 + ꢅꢃ^∗)²

ꢂ

ꢃ ꢂ

ꢃ

∗

ꢄꢃ

ꢄ(

∗

1 − 2( − ꢃꢃ −

− ꢇ +

∗

∗ 2

1 + ꢅꢃ

(1 + ꢅꢃ )

ꢂ

ꢃ

∗

ꢄ(

ꢄꢃ

∗

ꢃ( +

> 0.

∗

∗ 2

1 + ꢅꢃ

(1 + ꢅꢃ )

Therefore, we infer that the model system (3) is locally asymptotically

stable at the predator free equilibrium point ꢆ²as long as the conditions

4.1.5 Global stability analysis using the Bendixson-Dulac

theorem

(i), (ii), and (iii) hold.

T heorem 7. If the Infection Free Equilibrium point ((^∗, 0, %^∗) is locally

4.1.4 Local Stability Near the Disease-Free Equilibrium Point

asymptotically stable in the positive (% - plane region, then it is also globally

#₍(()

#₍((^∗)

T heorem 6. Local asymptotic stability of the infection-free ﬁxed point

asymptotically stable in the same region if

≥

.

ꢆ³((^∗, 0, %^∗) of system (3) is ensured if the following criteria are satisﬁed:

∗

(

1. ' < 1,

0

2. #₍⁰((^∗) > 0,

3. #₍⁰((^∗)((^∗)²− 2ꢈ + #₍⁰((^∗) (^∗+ ꢈ < 0.

Proof. Consequently, the system can be reduced to the following

two-dimensional subsystem

ꢄ

ꢅ

3(

= ((1 − () − ꢆ # (()%,

(17)

(18)

1

(

3)

3%

3)

Proof.

= #₍(()% − ꢈ%.

5₁

−ꢃ(^∗− ꢄ(^∗−ꢆ # ((^∗)

1

(

©

-

ª

®

1

(%

3

Consider ℎ((, %)

=

as a Dulac positive function in the positive

0

5₂

#⁰_ꢃ(0)%^∗

0

ꢇ(ꢆ ) =

(16)

-

®

quadrant. Also, deﬁne the following functions

#⁰((^∗)%^∗

5₃

«

¬

(

ℎ ((, %) = ((1 − () − ꢆ # (()%,

(19)

(20)

1

(

ℎ ((, %) = # (()% − ꢈ%.

2

where, 5₁= 1 − 2(^∗− ꢆ #⁰((^∗)%^∗, 5₂= ꢄ(^∗− ꢇ − ꢆ #⁰(0)%^∗and

(

1

2

(

ꢃ

5₃= #₍((^∗) − ꢈ = 0.

Then,

The associated auxiliary equation of ꢇ(ꢆ³) is

ꢂ

ꢃ

(#⁰(() − #₍(()

ꢊ

%

1

%

(

∗

ꢊ

)((, %) =

(ℎ ℎ ) +

(ℎ ℎ ) = −

+ ꢆ

.

5 − ꢊ

−ꢃ( − ꢄ(

−ꢆ # (( )

1

2

1

(

ꢊ

2

%(

%%

(

ꢊ

0

5 − ꢊ

2

0

= 0

Hence, )((, %) is a negative function of its arguments if (#₍⁰(() −

ꢊ

#₍(() ≥ 0. Note that by mean value thorem (#⁰(() − #₍(() ≥ 0 and

0

∗

# (( )%

(

∗

# (0)%

ꢃ

5 − ꢊ

3

(

#₍(()

#₍((^∗)

≥

are equivalent. Since )((, %) does not change sign and

∗

(

is not identically zero in the positive quadrant of the (%-plane, by the

Bendixson - Dulac criterion the infection free equilibrium point is globally

asymptotically stable in the region

Since the matrix is block triangular with respect to the infected variable,

one eigenvalue is

ꢊ

= 5₂= ꢄ(^∗− ꢇ − ꢆ #⁰(0)%^∗.

1

2

ꢃ

ꢆ

ꢇ

#₍(()

#₍((^∗)

ꢅ = ((, %) ∈ '²

:

≥

, % > 0

.

Using the basic reproduction number

+

∗

(

ꢄ(^∗₀

'

=

,

0

ꢇ + ꢆ # (0)%^∗

Moreover, the system has no limit cycle in the region.

7

2

ꢃ

Alemu et.al (2026)

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

4.1.6 System stability conditions near endemic equilibrium point

Proof. (1). Consider the community matrix of model (3) evaluated at

ꢆ¹(1, 0, 0):

−1 −(ꢃ + ꢄ) −ꢆ # (1)

T heorem 8. Local asymptotic stability of the endemic equilibrium point

©

-

ª

®

ꢇ(ꢆ¹) =

.

0

ꢄ − ꢇ

¹0⁽

ꢆ^∗((^∗, ꢃ^∗, %^∗) holds provided that the following conditions are met:

0

#₍(1) − ꢈ

«

¬

The eigenvalues of ꢇ(ꢆ¹) are ꢊ = −1 < 0, ꢊ = ꢄ − ꢇ, and ꢊ = # (1)− ꢈ.

(8) ꢅ + ꢈ < 0,

1

2

3

(

Therefore, ꢆ¹is locally asymptotically stable provided that ꢄ < ꢇ and

#₍(1) < ꢈ hold. Substituting either ꢇ = ꢄ or ꢈ = #₍(1) into ꢇ(ꢆ¹) yields a

zero eigenvalue in the characteristic equation.

(88) @ @₄− ꢆ@₅+ ꢈ@₂+ ꢅ@ < 0,

1

3

(888) (ꢅ + ꢈ)(ꢅꢈ + @₂+ @₃+ ꢆ@ ) + @ @₄− ꢆ@₅+ ꢈ@₂+ ꢅ@ > 0,

4

1

3

where the parameters ꢅ, ꢆ, ꢄ, and ꢈ are deﬁned in the proof.

With ꢇ

=

ꢇ^[1], the eigenvectors + and ,, associated with the zero

eigenvalue of the Jacobian ꢇ^[1](ꢆ¹, ꢇ^[1]) and its transpose, respectively,

are

Proof. The positive ﬁxed point ꢆ^∗((^∗, ꢃ^∗, %^∗) of the dynamics (3) is locally

asymptotically stable if all the characteristic roots of the Jacobian matrix,

ꢇ, has negative real parts, where

−(ꢃ + ꢄ)0

0

©

-

ª

®

© ª

0

1

+ =

,

, =

,

- ®

0

«

¬

« ¬

)

where 0, 1 ≠ 0. The derivative of the vector ﬁeld ꢄ((, ꢃ, %) = (ꢄ , ꢄ , ꢄ )

1

2

3

ꢅ

ꢆ

−ꢃ(^∗− ꢄ(^∗ꢄ²−ꢆ #₍((^∗)

1

with respect to ꢇ is

©

-

ª

®

ꢇ(ꢆ^∗) =

.

ꢈ

−ꢆ #_ꢃ(ꢃ^∗)

2

0

©

ª

®

© ª

ꢄ (-, ꢇ) = −ꢃ

,

ꢄ_ꢇ(ꢆ¹, ꢇ^[1]) = 0 ,

#⁰((^∗)%^∗

#⁰_ꢃ(ꢃ^∗)%^∗

0

-

- ®

ꢇ

«

¬

(

0

«

¬

« ¬

implying

1

,⁾ꢄ_ꢇ(ꢆ¹, ꢇ^[1]) = 0.

where, ꢅ = 1 − 2(^∗− ꢃꢃ^∗− ꢄꢃ^∗ꢄ − ꢆ #⁰((^∗)%^∗, ꢆ = ꢄꢃ^∗ꢄ, ꢄ =

and

1

1 + ꢅꢃ^∗

(

ꢈ = ꢄ(^∗ꢄ²− ꢇ − ꢆ #⁰(ꢃ^∗)%^∗.

Hence, the ﬁrst condition of Sotomayor’s theorem Pirayesh et al., 2016 for

a transcritical bifurcation is met.

2

ꢃ

The characteristic equation is

Next, we compute

ꢊ

ꢅ − ꢊ

−ꢃ(^∗− ꢄ(^∗ꢄ²^{−ꢆ #}₍⁽⁽^∗⁾ꢊ

0

−1

0

−0

0

1

ꢊ

©

-

ª

®

©

-

ª

®

ꢅꢄ_ꢇ(ꢆ¹, ꢇ^[1]) =

,

ꢅꢄ_ꢇ(ꢆ¹, ꢇ^[1])+ =

,

∗

ꢊ

ꢆ

ꢈ − ꢊ

#⁰_ꢃ(ꢃ^∗)%^∗

−ꢆ #_ꢃ(ꢃ )

2

= 0

ꢊ

«

¬

«

¬

0

#₍((^∗)%^∗

−ꢊ

so that

,⁾[ꢅꢄ_ꢇ(ꢆ¹, ꢇ^[1])+] = −01 ≠ 0.

This implies

Finally, the second derivative of ꢄ along + is

ꢊ³+ : ꢊ²+ : ꢊ + : = 0

(21)

1

2

3

−2E₁²+ 2ꢄꢅE₂²+ 2(−ꢃ − ꢄ)E E

1

2

©

-

ª

®

ꢅ²ꢄ(ꢆ¹, ꢇ^[1])(+, +) =

−2ꢄꢅE₂²+ 2ꢄE E

,

1

2

0

where, : = −(ꢅ + ꢈ), : = ꢅꢈ + @ + @ + ꢆ@ , : = −(@ @

−

1

2

3

4

3

4

ꢆ@ + ꢈ@ + ꢅ@ ), @ = ꢆ # (ꢃ^∗)#⁰((^∗)%^∗, @ = ꢆ # ((^∗)#⁰(¹(^∗)%^∗,

«

¬

5

2

3

1

2

1

ꢃ

(

giving

@

= ꢆ # (ꢃ^∗)#⁰(ꢃ^∗)%^∗, @ = ꢃ(^∗+ ꢄ(^∗ꢄ²and @ = ꢆ # ((^∗)#⁰(ꢃ^∗⁽)%^∗.

3

2

4

5

1

ꢃ

(

ꢃ

,⁾[ꢅ²ꢄ(ꢆ¹, ꢇ^[1])(+, +)] = −21ꢇ(ꢅE₂²− E E ) ≠ 0 whenever ꢅE ≠ E .

Consequentially, if ꢅ +ꢈ < 0, then : > 0. If @ @ −ꢆ@ +ꢈ@ +ꢅ@ < 0,

then : > 0. Moreover, : : − : > 0 if (ꢅ +¹ꢈ)(ꢅꢈ + @ +²@ + ꢆ@ ) +

1

4

5

3

1

2

1

3

1

2

3

2

3

@ @₄− ꢆ@₅+ ꢈ@₂+ ꢅ@ > 0. According to the Routh–Hurwitz criter⁴ion,

the model system 3 is l³ocally asymptotically stable at the endemic ﬁxed

1

Therefore, by Sotomayor’s theorem Pirayesh et al., 2016; Yu et al., 2020,

point ꢆ^∗= ((^∗, ꢃ^∗, %^∗) if the corresponding conditions are satisﬁed.

the model (3) demonstrate transcritical bifurcation at ꢇ = ꢇ^[1]near the

axial ﬁxed point ꢆ¹(1, 0, 0) provided that ꢅE ≠ E .

2

1

4.2 Local bifurcation analysis

Now, let us examine the bifurcation at ꢈ = ꢈ^[2]= #₍(1). The eigenvectors

+ and ,, associated to the zero eigenvalues of the matrices ꢇ^[2](ꢆ¹, ꢈ^[2]

)

and its transpose respectively, can be written as

Bifurcations analysis helps to predict and understand transitions in

the behavior of dynamical system as parameters value change. Local

bifurcation refers, change in the qualitative behavior of dynamical system

near ﬁxed point as a system’s parameters are varied.

ꢄ

ꢅ

)

+ = (E

E

E ) = −ꢆ # (1)

0

2

and , = (0

0

3)

1

2

3

1

(

where 2 and 3 are nonzero real numbers. It follows that

T heorem 9. (Transcritical Bifurcation)

0

−%

0

©

-

ª

®

© ª

0

ꢄ (-, ꢈ) =

,

ꢄ_ꢇ(ꢆ¹, ꢈ^[2]) =

.

- ®

ꢈ

0

1. The diseased model (3) demonstrate a transcritical bifurcation at parameter

«

¬

« ¬

values ꢇ = ꢇ^[1]= ꢄ or ꢈ = ꢈ^[2]= #₍(1) in the neighborhood of the

This implies that, ,⁾ꢄ_ꢈ(ꢆ¹, ꢈ^[2]) = 0. Moreover,

equilibrium point ꢆ¹(1, 0, 0).

0

∗

ˆ

©

-

0⁽

2

ª

®

2. When the parameter ꢈ attains the bifurcation threshold ꢈ = #₍(( ) +

ꢅꢄ_ꢈ(ꢆ¹, ꢈ^[2]) =

,

#_ꢃ(ꢃ^∗), the system (3) near the equilibrium ꢆ²((^∗, ꢃ^∗, 0) exhibits

−1

«

¬

(i). No saddle-node bifurcation occurs but

(ii). A transcritical bifurcation is observed.

0

−ꢆ # (1)2

0

1

©

-

ª ©

® -

ª

®

©

ª

®

DF_ꢈ(ꢆ¹, ꢈ^[2])+ =

=

.

-

−1

−2

«

¬ «

¬

«

¬

Alemu et.al (2026)

8

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

¤

Hence, ,⁾[ꢅꢄ_ꢈ(ꢆ¹, ꢈ^[2])+] = −23 ≠ 0. Furthermore,

3ꢊ

3ꢅ

ꢅ₂+ ꢊꢅ

¹. Thus, at ꢅ = ꢅ^∗it is reduced to

This implies that,

= −

ꢅ₁+ 2ꢊ

−2E₁²− 2ꢆ #⁰(1)E E

1

0

1

3

(

ꢈ

ꢉ

ꢈ

ꢉ

√

2

©

-

ª

®

ꢅ²ꢄ(ꢆ¹, ꢈ^[2])(+, +) =

¤

3ꢊ

3ꢅ

ꢅ₂+ ꢊꢅ

−ꢅ

ꢅ

1

2

2#⁰(1)E E

= −

=

+ 8

2ꢅ

.

√

ꢅ

2

ꢅ₁+ 2ꢊ

2ꢅ

1

3

«

¬

∗

2

(

ꢅ=ꢅ

ꢊ=8

which implies that,

ꢉ

√

¤

3ꢊ₉

ꢅ

1

ꢅ

2

Hence, '4

= −

≠ 0.

,⁾[ꢅ²ꢄ(ꢆ¹, ꢈ^[2])(+, +)] = 2#⁰(1)E E = −232²ꢆ ꢈ#⁰(1) ≠ 0.

3ꢅ

2ꢅ

∗

2

ꢅ=ꢅ

1

3

1

(

Hence, the transversality condition

Hence, based on the Sotomayor’s theorem as in Pirayesh et al. (2016) and

ꢂ

ꢃ

Yu et al. (2020) the model exhibit transcritical bifurcation at ꢈ = ꢈ^[2]

#₍(1) near to the axial equilibria ꢆ¹(1, 0, 0).

=

3ꢊ₉

<

≠ 0, 9 = 2, 3,

3ꢅ

∗

ꢅ=ꢅ

See the proof of (2) in the appendix A.

is satisﬁed, which conﬁrms the occurrence of a Hopf bifurcation at ꢅ = ꢅ^∗.

Furthermore, it can be demonstrated that there exists a threshold value of

the parameter ꢄ at which the present model also demonstrate a stability

switch via Hopf bifurcation.

T heorem 10 (Hopf Bifurcation). The system undergoes a Hopf bifurcation

near to the equilibrium point ꢆ²((^∗, ꢃ^∗, 0) at the parameter value ꢅ = ꢅ^∗,

provided the following criteria are hold:

1. At ꢅ = ꢅ^∗, we have ꢅ = 0 and ꢅ > 0, which guarantees the existence

1

2

T heorem 11. If the bifurcation parameter is given by

ꢆ ꢈꢄ ( − ꢆ #⁰(0)((1 − ()

of a pair of purely imaginary eigenvalues, and

2. The transversality condition is satisﬁed, i.e.,

ꢂ

ꢃ

3ꢊ₉

1

2

<

≠ 0, 9 = 2, 3,

ꢃ

3ꢅ

ꢇ^[0]

=

,

∗

ꢅ=ꢅ

ꢆ ꢈ

1

where ꢊ₉denote the eigenvalues of the auxiliary equation

then the model system (3) at the infection-free ﬁxed point ꢆ³((^∗, 0, %^∗) does not

exhibit a saddle-node bifurcation. Instead, the transcritical bifurcation of the

system is observed. See the proof in the appendix B.

ꢊ²+ ꢅ ꢊ + ꢅ = 0

1

2

associated with ꢆ². Here, ꢅ = −(ꢋ + ꢋ ) and ꢅ = ꢋ ꢋ − ꢌ ꢌ ,

2

1

2

1

2

with ꢋ₈and ꢌ₈(8 = 1, 2) deﬁ¹ned in th¹e proof part.

5 Computational Analysis

Proof. The auxiliary equation of the model (3) at ꢆ²is

In order to validate the theoretical results, the researchers numerically

explore the dynamic behavior of their model using the ode45 solver in

MATLAB. Owing to the unavailability of empirical data, a biologically

feasible and representative set of parameter values is adopted for the

ꢊ

∗

ꢊ

ꢋ − ꢊ

ꢌ

−ꢆ # (( )

1

(

ꢊ

∗

ꢌ

ꢋ − ꢊ −ꢆ # (ꢃ ) = 0,

2

ꢃ

ꢊ

purpose of numerical simulations. - = {ꢁ = 0.001, ꢁ = 0.007, ꢄ = 2.5,

1

2

ꢊ

ꢆ

1

= 0.04, ꢆ = 0.02, ꢇ = 0.04, ꢃ = 1.3, ꢈ = 0.01, ꢅ = 0.03, ℎ = 0.064,

2

0

ꢋ − ꢊ

3

and 4 = 0.05}. Moreover, for simulation purposes, we consider four

representative models selected from the sixteen possible combinations of

Holling-type functional responses. Speciﬁcally:

ꢄꢃ^∗

ꢄ(^∗

where, ꢋ = 1 − 2(^∗− ꢃꢃ^∗−

, ꢋ

2

=

− ꢇ, ꢋ = #₍((^∗) +

1

3

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

ꢄ(^∗

ꢄꢃ^∗

# (ꢃ^∗) − ꢈ, ꢌ = −ꢃ(^∗

−

< 0 and ꢌ

=

> 0. After

1 + ꢅꢃ^∗

1

2

ꢃ

(1 + ꢅꢃ^∗)²

• Model 1: Represents the disease model with(HT-I–HT-II).

• Model 2: Corresponds to the combination (HT-II–HT-III).

• Model 3: Deﬁned by the combination (HT-III–HT-II).

• Model 4: Represents the combination (HT-IV–HT-III).

simpliﬁcation

(ꢋ₃− ꢊ)(ꢊ²+ ꢅ ꢊ + ꢅ ) = 0

(22)

1

2

where

ꢂ

ꢃ

ꢄꢃ^∗

ꢄ(^∗

ꢅ

= −(ꢋ₁+ ꢋ ) = − 1 − 2(^∗− ꢃꢃ^∗−

+

− ꢇ

,

1

2

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

ꢂ

ꢃ ꢂ

ꢃ

ꢄꢃ^∗

ꢄ(^∗

= ꢋ ꢋ₂− ꢌ ꢌ

=

1 − 2(^∗− ꢃꢃ^∗−

− ꢇ +

1

2

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

These four sample models are selected as representative cases among the

sixteen Holling-type response function combinations to capture diﬀerent

nonlinear interaction mechanisms and to compare their qualitative

impacts on the disease transmission dynamics. When the consumer

species is absent in the dynamics 3, then the model is dominated by prey

population. The time-series plots in Figure (1) show that both ((C) and

ꢃ(C) converge smoothly to their ﬁxed point values ꢆ²((^∗, ꢃ^∗, 0), indicating

local asymptotic stability. The infected prey population increases initially

due to infection transmission, then stabilizes at a moderate level, while

the susceptible population decreases and reaches a steady state.

ꢂ

ꢃ

ꢄ(^∗

ꢄꢃ^∗

ꢃ(^∗+

.

1 + ꢅꢃ^∗

(1 + ꢅꢃ^∗)²

Eq’n (22) has pure imaginary roots if ꢅ = 0 and ꢅ > 0 from which the

1

2

√

threshold value ꢅ = ꢅ^∗. Thus, when ꢅ = ꢅ^∗, ꢊ = ꢋ , ꢊ = 8 ꢅ , and

1

3

2

√

ꢊ

= −8 ꢅ . Diﬀerentiating equation (22) with respect to ꢅ, we have

3

2

3ꢊ

3ꢅ

3ꢊ 3ꢅ

1

2

3ꢅ

2ꢊ

+ ꢊ

+ ꢅ

+

= 0.

3ꢅ

¹3ꢅ

Alemu et.al (2026)

9

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

(a)

(b)

(c)

(d)

Figure 1: Time series plot of the system (3), where the parametric values ꢃ = 0.3; ꢄ = 0.8; ꢅ = 0.2; ꢆ = 0.4; ꢆ = 0.3; ꢇ = 0.25; G8 = 0.5; 0₁= 0.6; 0₂= 0.5; 1₁= 0.4;

1

2

1₂= 0.3; 2₁= 0.2; and initial condition (0.70, 0.120, 0.3).

When host is absent in the model system (3), the dynamics reduce to

a predator-prey subsystem involving ( and %. In Figure (2) the time

series plots show both populations converging to the host-free ﬁxed point

ꢆ³((^∗, 0, %^∗). Consumers grow up initially fueled by prey availability,

then stabilize as prey density decreases. Additionally, the phase diagrams

conﬁrm that trajectories approach the subspace ꢃ = 0. The infection-free

ﬁxed point is locally asymptotically stable under the parameter sets

considered, consistent with Theorem 6.

predation in prey dynamics and the stabilizing inﬂuence of functional

response saturation.

The bifurcation diagrams in Figures 4 conﬁrm the predicted transcritical

bifurcations in system (3) (Theorem 9). As the parameter ꢇ cross its

thresholds, equilibria exchange stability, with the infected equilibrium ꢃ^∗

smoothly transitioning from stable to unstable.

Figure 3 shows the system dynamics begin with periodic oscillation and

through time it goes to a locally asymptotically stable endemic ﬁxed point,

where the computational laboratory is performed for some possible

Holling Type response function combinations of the mathematical

Eco-Epidemiology model for the diseased-model (3).

Ecologically, small changes in disease-induced mortality ꢇ can shift

the system between disease-free and infected states, or from predator

extinction to coexistence, reﬂecting the inﬂuence of nonlinear functional

responses #₍and #_ꢃ.

Models 1 and 2 show bifurcations at lower thresholds, indicating higher

sensitivity and faster infection spread under simpler responses. In

contrast, Models 3 and 4, show stronger saturation or complex predation

terms, display delayed transitions, highlighting greater ecological

resilience.

Overall, the simulations show that both predator-free and disease-free

equilibria are stable, while nonlinear functional responses mainly aﬀect

the rate and amplitude of transient dynamics rather than the ﬁnal steady

state. These results emphasize the regulatory roles of infection and

Alemu et.al (2026)

10

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Figure 2: Time series plot of the system (3), where the parametric values ꢃ = 0.10; ꢄ = 0.4; ꢅ = 0.10; ꢇ = 0.30; ꢆ = 0.50; ꢆ = 0.70; ꢈ = 0.40; 0 = 0.65; 1 = 0.3;

1

2

ℎ = 0.20; ℎ₁= 0.30; ℎ₂= 0.25; and initial condition (0.80, 0.1, 0.3).

(a)

(b)

Figure 3: Time seires plot of the model system 3, where the parameter values ꢄ = 1.2, ꢆ = 0.5, ꢆ = 0.3, ꢇ = 0.3, ꢃ = 0.1, ꢈ = 0.4, 0 = 0.6, 1 = 0.3, 2 = 0.2, ꢅ = 0.03,

1

2

ℎ₁= 0.3, ℎ₂= 0.25 and for the, initial condition (0.8, 0.1, 0.3).

Alemu et.al (2026)

11

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Figure 4: Bifurcation diagrams for Models 1–4 with respect to ꢇ = ꢄ near the equilibrium ꢆ¹(1, 0, 0). The horizontal axis represents the bifurcation parameter ꢇ, and the

vertical axis represents the infected equilibrium ꢃ^∗. Solid lines denote stable equilibria, while dashed lines denote unstable equilibria and the bifurication value is ꢇ = 0.4.

5.1 Impact of the inhibition rate, ꢅ

such as crowding, limited contact, or behavioral avoidance. Ecologically,

ꢅ represents density-dependent inhibition in the infection process due

to immunity, crowding, or behavioural avoidance among prey. As ꢅ

increases, the eﬀective contact rate between susceptible and infected

prey decreases, reducing infection pressure. This reduction weakens

the oscillatory feedback between prey and predator populations, thereby

promoting stability in the coexistence equilibrium.

The inhibition rate ꢅ appears in the infection term which regulates the rate

at which susceptible prey become infected. From Figure(5), the parameter

ꢅ controls the saturation level of the infection process for small values of ꢅ,

the incidence rate is almost linear in ꢃ, leading to rapid spread of infection;

for large ꢅ, the infection saturates quickly, representing inhibitory eﬀects

Figure 5: Time series showing the impact of the inhibition rate ꢅ on system stability for HT-II–HT-III, where parameters value ꢅ = 0.1, 0.3, 0.5, ꢃ = 0.5; ꢄ = 0.8; ꢆ = 0.2;

1

ꢆ

2

= 0.15; ꢇ = 0.6; ꢈ = 0.5; 0 = 0.6; 1 = 0.3; 2 = 0.2; and initial condition (0.7, 0.2, 0.1).

5.2 Impact of the transmission rate, ꢄ

transmission rates ꢄ increasing the infected prey population rise up, and

making the nature of stability of the coexistence equilibrium is becomes

more periodic and take long time to stable. Moreover, for diﬀerent

Holling Type response functions combination, the patterns of stability of

endemic equilibrium point are identical.

As illustrated in Figure 6, the transmission parameter ꢄ has also have a

signiﬁcation eﬀect on the dynamical behaviour of population ꢃ , and as

Alemu et.al (2026)

12

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Figure 6: Time series plot of the dynamical system ( 3) for diﬀerent values of transmission rate ꢄ = 0.4, 0.6, 0.8 for Mode 1-4, where other parameter values ꢃ = 0.5,

ꢅ = 0.3, ꢆ = 0.2, ꢆ = 0.15, ꢇ = 0.6, ꢈ = 0.5; 0 = 0.6; 1 = 0.3; 2 = 0.2 and initial condition (0.7, 0.2, 0.1).

1

2

(a)

(b)

Figure 7: Time series plot(periodic solution) and phase diagram (limit cycle) of the model system HT-II–HT-III, where the parameter values ꢃ = 0.4, ꢄ = 5.0, ꢇ = 0.6,

= 1.2, ꢆ = 1.0, 0₁= 1.5, 0₂= 1.3, 1 = 0.4, ℎ = 0.2, ꢈ = 0.3, ꢅ = 2.0, and initial condition (0.6, 0.3, 0.1).

ꢆ

1

2

Figure 7 indicates existence of Hopf bifurcation which veriﬁes Theorem

4.6. Ecologically, ꢅ measures the strength of inhibitory (saturation)

eﬀects regulating predation or disease transmission. For ꢅ < ꢅ^∗, the

populations coexist at a stable steady state. When ꢅ > ꢅ^∗, the equilibrium

loses stability and sustained oscillations emerge due to feedback between

infection spread and predation pressure. Increased infection enhances

predator growth, which subsequently suppresses the host population,

leading to predator decline and eventual host recovery. This recurring

mechanism generates persistent population cycles, reﬂecting realistic

eco-epidemiological ﬂuctuations observed in natural ecosystems.

6 Result

In this section, we concisely summarize the main analytical and numerical

ﬁndings obtained in Sections 4 and 5 for the eco-epidemiological

model (3). The local and global stability conditions of all equilibrium

points, corresponding to diﬀerent combinations of Holling-Type

functional responses, are presented in Table (3). These results establish

the parametric regimes under which the system exhibits disease-free,

endemic, predator-free, or coexistence states.

The bifurcation analysis results of the eco-epidemiological dynamics 3

near equilibrium points are summarized in table (4).

Alemu et.al (2026)

13

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Table 3: Stability analysis result of Equilibrium points of the model system 3: Note; LAS ≡ locally asymptotically stable, GAS ≡ globally asymptotically stable

Equilibria

Stability conditions

Stability status

ꢆ⁰

ꢆ¹

ꢆ²

ꢆ³

ꢆ^∗

Always

Unstable

LAS

ꢄ < ꢇ and #₍(1) = ꢇ

Conditions stated in Theorem (5) (i)–(iii)

LAS

Conditions stated in Theorem (6) (1)–(3)

#₍(() ≥ exp

LAS

∫

3D

(

)

0

GAS

Conditions stated in Theorem (8)(i)–(iii)

LAS

The collective results depicted in Figures 4 clearly demonstrate how

variations in the key bifurcation parameter ꢇ regulate the coexistence

and persistence of prey, infected prey, and predator populations. The

transcritical bifurcation marks a critical threshold where the system shifts

from a disease-free to an endemic equilibrium, reﬂecting a change in

ecological stability and disease prevalence.

From an ecological perspective, increasing the recovery rate (ꢈ) helps

drive the system toward a disease-free state. Comparing Models 1–4,

introducing nonlinear saturation in infection and predation stabilizes

the system by postponing bifurcations. This highlights the importance

of using realistic functional responses in eco-epidemiological models

to capture key biological feedbacks and better understand ecosystem

resilience under disease pressure.

Table 4: Local bifurcation analysis result of Equilibrium points of the model 3: TB ≡ Transcritical bifurcation, HB ≡ Hopf bifurcation

E.P

ꢆ¹

ꢆ²

T hreshold value

ꢄ = ꢇ

Stability condition

Bifurcation

ꢅE ≠ E

2

TB

HB

1

#₍(1) = ꢈ

always

ꢈ = #₍(( ) + #_ꢃ(ꢃ )

#⁰((^∗)+ + #⁰(ꢃ^∗)+ ≠ 0

∗

ˆ

1

2

(

ꢃ

ꢈ

ꢉ

3ꢊ₉

ꢅ = ꢅ^∗

ꢅ

1

= 0, ꢅ > 0 and '4

≠ 0

2

3ꢅ

∗

ꢅ=ꢅ

0

ꢆ

1

ꢈꢄ (−ꢆ # (0)((1−()

ꢆ³

ꢇ^[0]

=

−(2ꢄꢅ(^∗+ ꢆ #⁰⁰(0)%^∗)+²Υ + 2(ꢄ+ + Υ − ꢆ #⁰(0)+ + )Υ ≠ 0

TB

2

ꢃ

2

1

2

3

ꢆ

1

ꢈ

2

ꢃ

The inhibition (saturation) parameter ꢅ plays a critical regulatory

role. When ꢅ is below the critical threshold (ꢅ^∗

2), the system

results.

=

settles into a stable coexistence of susceptible prey, infected prey, and

predators. However, once ꢅ exceeds this value, a Hopf bifurcation occurs:

the equilibrium becomes unstable and a stable limit cycle emerges.

Biologically, this leads to recurring oscillations driven by feedback

between disease transmission and predation. Increased susceptible

prey boosts infection and predator growth; predators then reduce prey

populations, which in turn lowers predator numbers, allowing prey to

recover and restarting the cycle.

The system exhibits oscillatory behavior for lower values of the inhibition

rate (ꢅ), whereas higher inhibition rates promote stability. Hopf

bifurcation analysis, taking ꢅ as the bifurcation parameter, revealed

that increasing inhibition enhances system stability. Furthermore,

when the predation rates ($ , $ ) exceed a critical threshold, the

predator-free equilibrium becomes²unstable, and a stable disease-free

coexistence of prey and predator emerges. The bifurcation analysis

indicates that disease transmission and predator–prey interactions jointly

determine ecosystem stability. Managing infection parameters such

as the transmission rate can prevent oscillatory outbreaks and species

extinction. Hence, controlling ecological feedbacks through parameter

tuning plays a crucial role in maintaining biodiversity and long-term

coexistence within predator–prey systems. Overall, the theoretical and

numerical investigations are carried out under saturating incidence rates

demonstrate the biological consistency of the proposed model. The

results provide valuable insights into the interplay between infection,

predation eﬃciency, and population stability in eco-epidemiological

systems. The primary contribution of this work lies in providing a

comprehensive bifurcation analysis under these combined nonlinear

mechanisms. We rigorously establish threshold dynamics through

the basic reproduction number and employ bifurcation theory to

demonstrate the occurrence of transcritical and Hopf bifurcations.

The results reveal how inhibition and transmission parameters govern

transitions between disease-free equilibria, endemic coexistence, and

sustained oscillatory outbreaks.

1

Overall, the Hopf bifurcation shows how changes in inhibitory eﬀects

can shift the ecosystem from stable coexistence to sustained oscillations,

underscoring the delicate balance between disease dynamics and

predator–prey interactions.

7 Conclusion

In this study, the researchers have investigated an eco-epidemiological

mathematical model in which

a

prey population is infected by

microparasites, while predators feed on both susceptible and infected

prey following a general Holling-type functional response. The model

was developed to explore how disease transmission and predation

eﬃciency aﬀect the overall community structure and population

dynamics. An emergent carrying capacity was introduced to reﬂect the

fact that infected prey, having reduced ﬁtness, are more easily captured

by predators. The stability and bifurcation conditions were derived

for diﬀerent equilibrium points, including trivial, axial, predator-free,

disease-free, and endemic states. Analytical and numerical analyses

showed strong agreement between theoretical predictions and simulation

Future studies can extend this work by incorporating time delays

representing disease incubation or predator gestation periods, which

may lead to more complex dynamical behaviors such as chaos or

multiple attractors. Furthermore, integrating stochastic eﬀects, seasonal

Alemu et.al (2026)

14

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

variations, or optimal control strategies may enhance the model

applicability to real-world ecological management and conservation

policies.

Haldar, S., Khatua, A., Das, K., & Kar, T. K. (2021). Modeling and analysis

of a predator–prey type eco-epidemic system with time delay.

Modeling Earth Systems and Environment, 7, 1753–1768.

Hale, J. K. (2009). Ordinary diﬀerential equations. Courier Corporation.

Haque, M., & Venturino, E. (2007). An ecoepidemiological model with

disease in predator: The ratio‐dependent case. Mathematical

methods in the Applied Sciences, 30(14), 1791–1809.

Data Availability Statement

Holling, C. S. (1959). The components of predation as revealed by a study

of small-mammal predation of the european pine sawﬂy1. The

canadian entomologist, 91(5), 293–320.

The data supporting the ﬁndings of this study are available from the

authors upon reasonable request.

Hu, Z., Teng, Z., Zhang, T., Zhou, Q.,

&

Chen, X. (2017).

Globally asymptotically stable analysis in a discrete time

eco-epidemiological system. Chaos, Solitons & Fractals, 99,

20–31.

Conflicts of interest

Hugo, A., & Simanjilo, E. (2019). Analysis of an eco-epidemiological

model under optimal control measures for infected prey.

Applications and Applied Mathematics: An International Journal

(AAM), 14(1), 8.

The authors declare that they have no conﬂicts of interest relevant to this

study.

Kooi, B. W., van Voorn, G. A., & pada Das, K. (2011). Stabilization and

complex dynamics in a predator–prey model with predator

suﬀering from an infectious disease. Ecological Complexity, 8(1),

113–122.

Layek, G. C. (2015). An introduction to dynamical systems and chaos

(Vol. 449). Springer.

Author Contributions

All have equal contribution.

Li, H., & Takeuchi, Y. (2011). Dynamics of the density dependent

predator–prey system with beddington–deangelis functional

response. Journal of Mathematical Analysis and Applications,

374(2), 644–654.

Liu, W. M., Hethcote, H. W., & Levin, S. A. (1987). Dynamical behavior of

epidemiological models with nonlinear incidence rates. Journal

of mathematical biology, 25, 359–380.

Funding

This research received no speciﬁc grant from any funding agency.

Lotka, A. J. (1925). Elements of physical biology. Williams; Wilkins.

Maiti, A. P., Jana, C., & Maiti, D. K. (2019). A delayed eco-epidemiological

model with nonlinear incidence rate and crowley–martin

functional response for infected prey and predator. Nonlinear

Dynamics, 98, 1137–1167.

Omar, F. M., Sohaly, M. A., & El-Metwally, H. (2024). Lyapunov

functions and global stability analysis for epidemic model with

n-infectious. Indian Journal of Physics, 98(5), 1913–1922.

Panja, P. (2020). Prey–predator–scavenger model with monod–haldane

type functional response. Rendiconti del Circolo Matematico di

Palermo Series 2, 69(3), 1205–1219.

Pirayesh, B., Pazirandeh, A., & Akbari, M. (2016). Local bifurcation

analysis in nuclear reactor dynamics by sotomayor’s theorem.

Annals of Nuclear Energy, 94, 716–731.

Ruan, S., & Wang, W. (2003). Dynamical behavior of an epidemic model

with a nonlinear incidence rate. Journal of diﬀerential equations,

188(1), 135–163.

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

Appendix A

2

∗

2

ˆ

let the Jacobian matrix the model(3) at the predator free equilibrium point ꢆ (( , ꢃ , 0) denote by ꢇ(ꢆ ) = (ꢀ₈₉)_3×3

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

(1 + ꢅꢃ^∗)²

1 − 2(^∗− ꢃꢃ^∗−

−ꢃ(^∗−

−ꢆ # ((^∗)

1

(

©

ª

-

®

-

®

ꢄꢃ^∗

1 + ꢅꢃ^∗

ꢄ(^∗

2

ꢇ(ꢆ ) =

.

− ꢇ

−ꢆ # (ꢃ^∗)

2

ꢃ

(1 + ꢅꢃ^∗)²

-

®

-

®

0

#₍((^∗) + #_ꢃ(ꢃ^∗) − ꢈ

«

¬

From the condition at which ꢇ(ꢆ²) has zero eigenvalue, that is, ꢊ = ꢋ = # ((^∗) + # (ꢃ^∗) − ꢈ = 0 the bifurcation value is

1

3

(

ꢃ

∗

ˆ

ꢈ = #₍(( ) + #_ꢃ(ꢃ ).

2

)

ˆ

2

ˆ

Now we compute the Jacobian matrix ꢇ(ꢆ ) = (ꢀ )_3×3at ꢈ = ꢈ which is same as above ꢇ(ꢆ ) except ꢀ = 0. The eigenvectors of ꢇ(ꢆ , ꢈ) and ꢇ (ꢆ , ꢈ),

33

89

corresponding to the zero eigenvalue are, respectively

Ψ e

+

0

1

e

1

©

-

ª

®

©

-

ª

®

© ª

+

0

+ =

=

and , =

- ®

2

Ψ e

+

3

f

2

«

¬

«

¬

« ¬

ˆ

21

ˆ

21

ˆ

ꢀ

ꢀ₂₃− ꢀ

ꢀ

ˆ

ꢀ

ˆ

ꢀ

ꢀ₂₂− ꢀ

ꢀ

12

ˆ

22

ˆ

11

12

ˆ

where, Ψ

=

¹³, Ψ

=

²¹, moreover e and f are nonzero real numbers. From our model system (3), we have:

1

2

ˆ

ꢀ₁₃− ꢀ

11 23

0

©

-

ª

®

© ª

2

ˆ

0

ꢄ (-, ꢈ) =

ꢈ

=⇒ ꢄ (ꢆ , ꢈ) =

- ®

ꢈ

−%

0

«

¬

« ¬

)

2

ˆ

Thus, , ꢄ_ꢈ(ꢆ , ꢈ) = 0. Applying Sotomayor’s theorem (Pirayesh et al., 2016) for local bifurcation, the saddle node bifurcation does not occur near to

the equilibrium point ꢆ²((^∗, ꢃ^∗, 0). For Bogdanov– Takens bifurcation, there must be two equilibria : saddle and non-saddle. Therefore, BT bifurcation

cannot appear here also.

We noted that the ﬁrst condition , ꢄ_ꢈ(ꢆ , ꢈ) = 0 of Sotomayor’s theorem for the existence of transcritical bifurcation is satisﬁed. Now,

)

2

ˆ

0

Ψ e

0

1

e

©

-

ª

®

©

-

ª ©

® -

ª

®

©

-

ª

®

2

ˆ

ꢅꢄ (ꢆ , ꢈ) =

=⇒ ꢅꢄ (ꢆ , ꢈ)+ =

=

ꢈ

−1

−1 Ψ e

−Ψ e

2

«

¬

«

¬ «

¬

«

¬

)

2

ˆ

So, we have , [ꢅꢄ (ꢆ , ꢈ)+] = −Ψ ef ≠ 0. Moreover,

2

ꢈ

ꢂ

ꢃ

ꢀ

ꢁ

2ꢄꢅ(^∗

ꢄ

−2+²

+

− 2 ꢃ +

+ + + ꢆ #⁰((^∗)+ +

2

©

ª

®

1

2

1

3

2

1

2

(

(1 + ꢅꢃ^∗)³

(1 + ꢅꢃ^∗)²

-

®

ꢂ

ꢃ

−2ꢄꢅ(^∗

ꢄ

2

ˆ

+ + − ꢆ #⁰(ꢃ^∗)+ +

ꢅ ꢄ(ꢆ , ꢈ)(+, +) =

2

+

+ 2

1

2

3

2

ꢃ

-

(1 + ꢅꢃ^∗)³

(1 + ꢅꢃ^∗)²

®

-

®

2+ (#⁰((^∗)+ + #⁰(ꢃ^∗)+ )

3

1

2

«

¬

(

ꢃ

0

(

0

ꢃ

)

2

∗

ˆ

Thus, we have , [ꢅ ꢄ(ꢆ , ꢈ)(+, +)] = 2+ f(# (( )+ + # (ꢃ )+ ) ≠ 0. Therefore, by Sotomayor’s theorem, transcritical bifurcation occurs near to the

3

1

2

predator-free stationary point ꢆ²((^∗, ꢃ^∗, 0).

Appendix B

The Jacobian matrix of the system (3) at the infection free equilibrium point ꢆ³((^∗, 0, %^∗) denote by ꢇ(ꢆ³) = (ꢅ₈₉)_3×3as

1 − 2(^∗− ꢆ #⁰((^∗)%^∗

−ꢃ(^∗− ꢄ(^∗

−ꢆ # ((^∗)

1

(

©

«

ª

-

®

ꢇ(ꢆ³) =

0

ꢄ(^∗− ꢇ − ꢆ #⁰(0)%^∗

0

.

2

ꢃ

-

®

#₍⁰((^∗)%^∗

#⁰_ꢃ(0)%^∗

#₍((^∗) − ꢈ

¬

Thus, ꢇ(ꢆ³) has zero eigenvalue, while, ꢊ = 5₂= ꢄ (^∗− ꢇ − ꢆ #⁰(0)%^∗= 0. and the model bifurcate when

1

2

ꢃ

ꢆ ꢈꢄ(^∗− ꢆ #⁰(0)(^∗(1 − (^∗)

1

2

ꢃ

ꢇ^[0]

=

.

ꢆ ꢈ

1

Alemu et.al (2026)

16

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 1-17

To perform the Jacobian matrix ꢇ^[0](ꢆ³) = (ꢅ )_3×3at ꢇ = ꢇ^[0]which is same as above ꢇ(ꢆ³) except ꢅ = 0. The eigenvectors of ꢇ^[0](ꢆ³, ꢇ^[0]) and

22

89

(ꢇ^[0](ꢆ³, ꢇ^[0])) , corresponding to the zero eigenvalue are, respectively

)

Υ

©

ª

-

®

ꢅ

ꢅ₃₁− ꢅ

ꢅ

32 13

+

0

13

12

11 33

1

©

-

ª

®

© ª

Υ

+

Γ

+ =

=

and , =

- ®

2

ꢅ₃₃− ꢅ

-

®

+

3

0

«

¬

« ¬

ꢅ

ꢅ₃₂− ꢅ

ꢅ

11

12 13

Υ

ꢅ₁₂ꢅ₃₃− ꢅ

ꢅ

32 13

)

«

¬

where,Υ and Γ are nonzero real numbers. From our model system (3), use derivative we get:

0

−ꢃ

0

©

-

ª

®

© ª

=⇒ ꢄ (ꢆ³, ꢇ^[0]) =

0

ꢄ (-, ꢇ) =

- ®

ꢇ

0

«

¬

« ¬

Thus, applying Sotomayor’s theorem ﬁrst condition ,⁾ꢄ_ꢇ(ꢆ³, ꢇ^[0]

)

=

0. Hence, the dynamical system (3) saddle node bifurcation does not

demonstrate at the disease free equilibrium point ꢆ³((^∗, 0, %^∗).

Now we try to perform the other conditions of Sotomayor’s theorem for the existence of transcritical bifurcation. Thus,

0

−1

0

©

-

ª

®

ꢅꢄ (-, ꢇ) =

ꢇ

«

¬

Υ

0

©

ª

®

0

−1

0

ꢅ

ꢅ₃₁− ꢅ

ꢅ

13

11 33

-

©

-

ª

®

ꢅ

ꢅ₃₃− ꢅ

ꢅ

32 13

©

-

ª

®

Υ

11

12

13 31

=⇒ ꢅꢄ_ꢇ(ꢆ³, ꢇ^[0])+ =

=

Υ

ꢅ₃₃− ꢅ

- ꢅ¹²ꢅ − ꢅ³²ꢅ¹³

ꢅ₃₃− ꢅ

11 32

12 13

«

¬

Υ

0

«

¬

ꢅ

− ꢅ

ꢅ

)

«

12 33

32 13

¬

ꢅ

ꢅ₃₃− ꢅ

ꢅ

11

12

13 31

Hence, we arrived that ,⁾[ꢅꢄ_ꢇ(ꢆ³, ꢇ^[0])+] =

ΥΓ ≠ 0. In addition,

ꢅ₃₃− ꢅ

32 13

ꢀ

ꢁ

(−2 − ꢆ #⁰⁰((^∗)%^∗)+²+ 2ꢄꢅ(^∗+₂²− 2 (ꢃ + ꢄ1)+ + + ꢆ #⁰((^∗)+ +

1

2

1

3

2

1

©

(

ª

-

®

-

®

ꢅ²ꢄ(ꢆ³, ꢇ^[0])(+, +) =

(−2ꢄꢅ(^∗− ꢆ #⁰⁰(0)%^∗)+²+ 2(ꢄ+ + − ꢆ #⁰(0)+ + )

2

1

2

3

2

ꢃ

-

®

#⁰⁰((^∗)%^∗+₁²+ #⁰⁰(0)%^∗+₂²+ 2+ (#⁰((^∗)+ + #⁰(0)+ )

3

1

2

«

¬

(

ꢃ

(

ꢃ

Thus, we have

,⁾[ꢅ²ꢄ(ꢆ³, ꢇ^[0])(+, +)] = −(2ꢄꢅ(^∗+ ꢆ #⁰⁰(0)%^∗)+²Υ + 2(ꢄ+ + Υ − ꢆ #⁰(0)+ + )Υ ≠ 0.

2

1

2

3

2

ꢃ

Therefore, according to Sotomayor’s theorem (Pirayesh et al., 2016), our dynamical system (3) experience transcritical bifurcation at the disease-free

ﬁxed point ꢆ³((^∗, 0, %^∗).

Alemu et.al (2026)

17

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

ARTICLE

Computational Study of MHD Blood Flow

through Bifurcated Artery Using

Caputo-Fabrizio Fractional Derivative,

Thermal Radiation, and Magnetic Field for

Tumor Therapies

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.2S

Isah Abdullahi¹, Dauda Gulibur Yakubu^1,∗, Muhammad

ARTICLE HISTORY

Shamsuddeen Dauda², Mahmood Abdulhameed³,Saidu Abubakar

Kadas⁴,Mohammed Abdulhameed⁵, and Garba Tahiru Adamu⁶

Received: March 10, 2026

Accepted: 23 May, 2026

Published Online: 10 June, 2026

CITATION

¹Department of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria

²Department of Biological Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria

³Department of Electrical Electronic Engineering, Abubakar Tafawa Balewa University, Bauchi, Nigeria

⁴Department of Obstetrics Gynaecology, ATBU, Teaching Hospital, Bauchi, Nigeria

⁵School of Science and Technology, The Federal Polytechnic Bauchi, Nigeria

Abdullahi et.al (2026). Computational

Study of MHD Blood Flow through

Bifurcated Artery Using Caputo-Fabrizio

Fractional Derivative, Thermal

Radiation, and Magnetic Field for Tumor

Therapies. East African Journal of

Biophysical and Computational

⁶Department of Mathematical Sciences, Bauchi State University, Gadau, Bauchi, Nigeria

^∗Corresponding author: dgyakubu@atbu.edu.ng

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.2S.18-26

Abstract

OPEN ACCESS

This study investigates the impact of heat sources, thermal radiation, and chemical reactions on the

magnetohydrodynamic blood flow through a bifurcated artery in the presence of a slanted magnetic

field. Using Laplace transform and the method of undetermined coefficients, the constitutive equations

for the mathematical model of Caputo-Fabrizio fractional derivative order were solved. Blood flow

velocity, temperature distribution, and concentration were found to have analytical expressions. The

effects of certain physical parameters on blood velocity, temperature and concentration are graphically

represented, and these representations accurately depict the flow disturbances. We discovered that

the bifurcation apex of the artery with a symmetrical divider has steady blood flow. This may lead

to significant shear stresses on either side of the bifurcation. Near the apex, when the flow is

substantially different, obstruction may result from the formation of boundary layers on the inner walls

of the bifurcation. Sluggish flow also occurred along the outer walls of the bifurcation. It has also

been discovered that the temperature distribution, concentration, and arterial blood flow velocity are

significantly influenced by the fractional order parameter, the slanted magnetic fields, the heat source,

and the chemical reaction parameter. This study offers significant benefits for medical applications in

biomechanical engineering, biomedical engineering, and medicine.

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Chemical reaction; Heat source; MHD Blood low; Slanted magnetic ield; Thermal

radiation

(Shit & Majee, 2015). Understanding many facets (aspects) of the

medical sciences, such as homeostasis, treating cancerous tumors, and

administering medication using magnetic particles, depends on the

study of biomagnetic ﬂuid dynamics (Shaw & Murthy, 2010). Blood’s

hemoglobin molecules are regarded as biomagnetic ﬂuids with magnetic

properties. Blood can also be considered as a Newtonian ﬂuid if it ﬂows

through the bigger arteries at a high shear rate. These arteries are thought

to be homogenous whose ﬂow behavior can be described by a Newtonian

model (see, Caro et al., 2011; MacDonald, 1979). Numerous researchers

have looked into various possibilities for studying physiological ﬂuids

1 Introduction

Bio-magnetic ﬂuid dynamics (BFD), which is the study of bio-ﬂuid ﬂow in

the presence of magnetic ﬁeld, is a rapidly developing subject of study in

ﬂuid mechanics (Tzirtzilakis, 2005). This ﬁeld of study is of tremendous

importance to the ﬁeld of medical science and has the possibility

to be utilized in a diversity of domains, including the delivery of

medications via the utilization of magnetized particles, the management

of severe bleeding, and the assistance in dealing with malignant cancers

Abdullahi et.al (2026)

18

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

using porous media (see, Bhatti & Lu, 2019; Dash et al., 1996; Ramesh &

Devakar, 2015; Shit & Roy, 2015) developed blood ﬂow model via porous

medium. Based on Darcy’s law, Bhargava et al. (2007) and Ghasemi et al.

(2015) investigated the pulsatile ﬂow and mass transfer of an electrically

conducting Newtonian bio-ﬂuid via a channel comprising porous media

using blood as the porous medium ﬂuid. Bhatti et al. (2018) developed a

mathematical model to investigate heat transfer, mass transfer, and blood

ﬂow in a porous medium channel while accounting for the integrated

Darcy-Brinkman-Forchheimer model. Blood behaves non-Newtonian

even in larger arteries at low shear rates, as demonstrated by Liepsch

(1986). When blood ﬂows through arteries at a low shear rate, it can be

treated as Cassons ﬂuid (Srivastava & Srivastava, 1984). Many researches

have supported the Casson ﬂuid model for blood ﬂow via tiny arteries at

low shear rates (see,Hayat et al., 2016; Nagarani et al., 2006; Venkatesan

et al., 2013 ). Many authors (see, Abdulhameed et al., 2017; Misra & Shit,

2009; Mondal & Shit, 2017; Yakubu et al., 2020; Zeeshan et al., 2017 )

have regarded blood as a non-Newtonian ﬂuid, because of its electrical

conductivity, displays magneto hydrodynamic behavior.

Therefore, it wasn’t until the last few decades that a signiﬁcant number

of scholars started to highlight the fact that diﬀerential equations and

fractional derivatives have numerous applications in a variety of domains

(see, Abdulhameed et al., 2023; Imoro et al., 2024). These days, fractional

derivative order diﬀerential equation problems are the most eﬀective

and successful ways to model the nonlinear processes that emerge in

many domains of applied study, including biology, chemistry, ecology,

engineering, and many other application areas. Several mathematical

models have shown that they oﬀer a more realistic depiction of the

phenomenon under research. Examples of these models include those

employed in biomedical engineering, viscoelastic mechanics, boundary

layers, electromagnetic, and porous media. Bansi et al. (2018) investigated

a fractional blood ﬂow model in the oscillatory artery with magnetic ﬁeld

and heat radiation eﬀects. With the aid of fractional time derivative,

(Yakubu et al., 2021) examined the eﬀects of pressure gradient, body

acceleration, and magnetic ﬁeld on blood ﬂow through artery. The

eﬀects of blood ﬂow parameters, Caputo’s time fractional derivatives, and

the external magnetic ﬁeld on the cylindrical domain were studied by

(Shah et al., 2016). Ali et al. (2017) solved a fractional order model for

Cassons ﬂuid ﬂow using the Hankel transform and Laplace transform

techniques to determine the exact solutions. He and collaborators (2019)

used the fractional order Caputo derivative to investigate the complexity

of blood in arteries under various forces. In the ﬁeld of medicine, magneto

hydrodynamic ﬂow plays a crucial role. It is considered for the reduction

of bleeding from wounds and for the treatment of malignant tumors.

Kumar et al. (2021) employed a chemical reaction, heat source, and

inclined magnetic ﬁeld to cure malignancies.

Many authors considered the examination of the heat and mass

transfer occurrences generated from these processes to be a highly

relevant element with respect to modeling physiological processes

(Prasad et al., 2025) and industrial processes (Sademaki et al., 2026).

Electromagnetohydrodynamics is the study of ﬂuids whose motion is

constantly aﬀected by externally applied magnetic ﬁeld and electric

ﬁeld. In order to comprehend the impact of magnetohydrodynamic

(MHD) and electrohydrodynamic (EHD) forces on the ﬂow of normal

ﬂuids, including blood, several studies have mostly concentrated on

the theoretical, computational, and experimental aspects of these forces.

Cell-based therapies, medication delivery, and biological processes are

just a few of the ﬁelds where the application of (EHD) has shown notable

advancement. Additional force components, primarily the Lorentz and

Coulomb forces, are incorporated into momentum equations and have a

direct eﬀect on ﬂuid velocity. Magnetohydrodynamics or MHD has been

used in a wide variety of biomedical applications (Vardanyan, 1973).

The fractional order time derivative of MHD blood ﬂow via a bifurcated

artery in the presence of a slanted magnetic ﬁeld, as well as the

coupling impact of heat transfer and blood ﬂow concentration, are

described here using Newtonian ﬂuid. The goal is to investigate

how magneto-hydrodynamic blood ﬂow through a bifurcated artery is

aﬀected by thermal radiation and a slanted magnetic ﬁeld during tumor

treatments. The Laplace transform and the indeterminate coeﬃcients

approach were used to ﬁnd the exact solutions, which were then

simulated to produce graphical outputs and the implications of several

important parameters on the outcomes were explored. The study was

motivated by the fact that there is currently very little information

available on the ﬂow in arterial bifurcation since the phenomena is

currently not stringent to mathematical analysis or precise experimental

measurement. The present investigation shows that the vast number of

variables involved are the main challenge in both situations.

The heat transmission and magnetohydrodynamic (MHD) blood ﬂow in

a restricted artery were studied by Majee and Shit (2017). Akbar and Butt

(2017) considered ferromagnetic blood ﬂow in a restricted, smaller artery

with a porous wall. The radiant heat transfer that takes place in the blood

vessels must also be considered while treating hyperthermia. Oncology

professionals are familiar with the medical practice of using heat therapy

to cancer patients. Chinyoka and Makinde (2014) investigated the eﬀects

of magnetic ﬁelds and heat radiation on arterial blood ﬂow. Sinha and

Shit (2015) investigated the magnetic hydrodynamic blood ﬂow in the

presence of thermal radiation. Tabi et al. (2017) studied the combined

eﬀects of magnetic ﬁelds and external radiation on blood ﬂow in the major

blood arteries. Yakubu et al. (2022) examined blood ﬂow of Oldroyd-B

ﬂuids in order to investigate the erratic ﬂow, with magnetic ﬁeld applied

perpendicular to the ﬂow direction. Heat transfer processes were studied

in the peristaltic ﬂow of blood with variable viscosity particle-liquid

suspensions by Bhatti et al. (2016). Blood ﬂow is greatly aﬀected when

the human body is exposed to a vibratory environment, as occurs when

operating machines or traveling in spacecraft. When the human body

undergoes body acceleration, a number of health problems might arise,

such as an elevated heart rate and vision loss. In the study of the impact of

body acceleration, a number of researches have produced mathematical

simulations of oscillatory blood ﬂow (see, Chaturani & Palanisamy, 1990;

Ghasemi et al., 2016; Sud & Sekhon, 1984 ). Bhatti and Lu (2019)

investigated the propagation of a hydro elastic single wave in a channel

with uniform ﬂow. Blood ﬂow characteristics have been discovered

to promote blood velocity in a vibratory environment using fractional

order derivative diﬀerential equation problems. Fractional diﬀerential

equations are the most used method for modeling natural phenomena.

This is due to the fact that equations oﬀer the possibility for a system to

either retain memory or to be hereditary with the properties of its history,

similar to how dynamic systems work (Syed et al., 2026).

2 Methods

2.1 Physical Structure and Mathematical

formulation

Blood considered in this study, is Newtonian, incompressible,

homogeneous, sticky ﬂuid that ﬂows from the trunk to the branches.

A mass stream’s rate at any cross-section that is perpendicular to its

direction is equal to m = 2bv, where b is the stream’s radius and v is

its mean speed. The mass stream’s speed at any cross section of the

extended channel is equal to m/2, and the bifurcating divider (internal

apical curve) has no eﬀect on this (see Figure 1). The magnetic ﬁeld is

applied to the ﬂow at an angle (φ) since the evaluated magnetic Reynolds

number is low. Therefore, it is believed that the magnetic and electric

ﬁelds produced by blood ﬂow are insigniﬁcant, the angle of bifurcation

is set to zero (Θ = 0), that is, the blood ﬂow region is bifurcated into two

streams that ﬂow parallel to the principal artery (the trunk). Figure 2

demonstrates the smooth muscle ﬁbers of the three concentric layers that

make up the walls of a typical elastic artery. These ﬁbers are controlled

by the sympathetic nervous system to contract or relax.

Fractional order derivatives have been applied in many ﬁelds of study,

including the complicated dynamics and rheological properties of

diﬀerent kinds of ﬂuids. The behavior of ﬂuid ﬂow is well depicted by

substituting fractional-order derivatives for the ordinary time derivative

in the constitutive equations (see, Atangana and Baleanu, 2016; Caputo

and Fabrizio, 2015; Samko et al., 1993). The concept of fractional calculus

was initially proposed by L’Hôpital in 1695, more than four centuries ago.

Abdullahi et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

where D is the diﬀusion coeﬃcient and G = −k₁(C − C_∞) represents

chemical reaction rate in the ﬂuid ﬂow. It is important to mention that the

eﬀect of an electric ﬁeld in the concentration equation was also ignored.

θ = `^−z˙

,

u = `^−z˙

,

v = `^−z˙

,

C = `^−z˙at y = −1,

1

(5)

and θ → 0, u → 0, v → 0, C → 0 at y = 1.

However, by using the proper normalizing factors, the governing

equations (1)–(4) can be converted to dimensionless form. We present

the non-dimensional parameters as follows:

Figure 1: Physical ﬂow diagram of the bifurcated artery with zero angle of

bifurcation

x¯

b

y¯

b

u¯

v¯

dp¯/dx¯

u_HSηm/2b³ρ

x = , y = , u =

,

v =

, h(x, t) =

mu_HS/2bρ

ꢂ

ꢃ

ꢂ

ꢃ

3

2

3

2

¯

θ

C

2b

mηu

ρ

2b

mηu

ρ

¯

η

ρ

¯

t

k

ρ/η

ꢀ

ꢁ

k= ^ꢀ

and

^ꢁ, C =

, θ =

, τ =

,

t =

,

2

b

ρ/η

HS

−16δT₀³

¯

∂T

∂y¯

q¯ =

(6)

3k⁰

Applying (5) and (6) to eqns. (1)- (4) and removing the bars we obtain:

Figure 2: The structure of an artery walls (Transverse section through an artery)

∂u

∂t

∂²u

∂y²

ꢄ

u

k

2.2 Fundamental flow equations and their

solutions

+ h =

+ gβθ + gβ⁰C − M²sin²φ −

(7)

(8)

ꢅ

ꢄ

ꢅ

∂θ

=

∂t

1

∂²θ

∂y²

S

τp_r

+ R

+

θ

τp_r

Blood ﬂows through a porous media as two-limit layers when it is

subjected to magnetic ﬁelds and heat, with the assumptions made in the

numerical deﬁnition guiding its movement. In the stream ﬁeld headings

of x and y at time t , let u and v be the speed parts, η and ρ denote

blood density and thickness, respectively. Blood pressure is represented

by p , warm conductivity (K_T), C_pis the speciﬁc heat capacity at steady

strain, hotness is represented by Q , temperature is represented by T ,

the volumetric development boundary is represented by β , the angle of

the slanted (inclined) magnetic ﬁeld is represented by φ, and porosity

parameter is represented by K . With these, we have the equations

provided by (see, Ali et al., 2017; He et al., 2019; Kumar et al., 2021), with

some additional terms as follows:

∂u

∂x

∂v

∂y

+

= 0

(9)

∂C

∂t

1 ∂²C

S_C∂y²

=

− ωC

(10)

where

σB²

⁰, p_r=

16δT₀³

3k⁰τ

Qb²

k_T

τ

k₁b²

τ

ρC

k_T

M²

=

^ρ, R =

, S =

, S_C

=

, ω =

ρ

bD

2

∂u

∂t

1 ∂p

ρ ∂x

η ∂²u

cB_αsin φ

v

k

+

=

+ gβ(T − T_∞) + gβ⁰(C − C_∞) −

v −

2

ρ

∂y

are the magnetic ﬁeld parameter, Prandtl number, thermal radiation

parameter, heat source parameter, Schmidt number, chemical reaction

parameter and θ is the temperature conveyance. Now, using the

Caputo-Fabrizio fractional derivative as stated in (Caputo and Fabrizio

2015), we consider the time fractional momentum equations as:

(1)

∂²T

Q

ρC_o

∂q

∂y

¯

k_T

∂T

∂t

=

+

(T − T_o) −

(2)

(3)

∂y²

ρC_o

ꢄ

ꢅ

1

∂u(y, τ)

∂τ

α(t − τ)

t

^CFD_t^αu(y, t) =

exp

−

dτ, 0 < α < 1

0

1 − α

∂u

∂x

∂v

∂y

= 0

n

o

su(y, s) − u(y, 0)

L

^CFD_t^αu(y, t)

=

(1 − α)s + α

∂

∂t

(11)

where,

is the material time derivative. On the other hand, the

dimensionless concentration equation for medication (concentration)

delivery in magneto hydrodynamic blood ﬂow through permeable

bifurcated artery is provided by,

u(y, 0) =

(12)

∂²C

∂y²

The Caputo-Fabrizio derivative corresponding to equations (7), (8) and

(10) are as follows:

∂C

∂ f

= D

+ G

(4)

Abdullahi et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

(

)

A₇+ A cosh A₂y + B sinh A₂y + A₈cos A₆y + A₉sin A₆y +

F =

.

∂²u

∂y²

ꢄ

u

k

+ A₁₀cosh Λy + A₁₁sinh Λy

^CFD_t^αu(y, t) + h =

+ gβθ + gβ⁰C − M²sin²φ −

(13)

(14)

(15)

(30)

ꢅ

1

τp_r

∂²θ

∂y²

S

τp_r

^CFD_t^αθ(y, t) =

+ R

+ (

)θ

We now have blood velocity in the axial direction, using equation (30) and

equation (19) as

∂C

∂t

1 ∂²C

S_C∂y²

^CFD_t^αC (y, t) =

=

− ωC.

!

A₇+ A cosh A₂y¯ + B sinh A₂y¯ + A₈cos A₆y¯ +

1

u¯(y, s) =

s + λ²

A₉sin A₆y¯ + A₁₀cosh Λy¯ + A₁₁sinh Λy¯

Applying Laplace transform to equations (13)-to-(15), and using the

boundary condition in equation (12) we have;

(31)

Equation (26) and equation (21) together provide the usual direction of

blood velocity in the bifurcated artery, which is given by

su(y, s)

(1 − α)s + α

∂²u¯

∂y¯²

u¯

k

0

2

¯

+ h =

+ gβθ + gβ C − M sin φ −

(16)

(17)

(18)

1

v¯(y, s) = A₁

(32)

s + λ²

ꢄ

ꢅ

2

¯

su(y, s)

(1 − α)s + α

1

τp_r

∂ θ

S

τp_r

¯

)θ

=

+ R

+ (

∂y¯²

According to equations (28) and (20), the temperature distribution in the

bifurcated artery is given as follows:

2

¯

su(y, s)

(1 − α)s + α

1 ∂ C

¯

=

− ωC.

S_C∂y¯²

ꢄꢄ

ꢅ

cos A₆y¯

2 cos A₆

sin A₆y¯

1

¯

θ(y, s) =

−

(33)

2 sin A₆s + λ²

2.3 Exact solutions

Equations (22) and (29) provide the drug’s concentration in the ﬂowing

blood in the carotid artery as follows:

Here we assume the following as the arbitrary solutions of equations (9),

(16), (17) and (18),

1

¯

u¯ = F(y)

ꢄ

ꢅ

,

(19)

(20)

(21)

(22)

s + λ²

cosh A₉y¯

2 cosh A₉

sinh A₉y¯

1

¯

C(y, s) =

−

.

2 sinh A₉s + λ²

1

¯

θ = H(y)

,

s + λ²

1

¯

v¯ = G(y)

,

(34)

s + λ²

1

Equations (31) through (34) yield the inverse Laplace transform.

Using Mathcad software, we simulated the given solutions using

Gaver-Stehfest’s algorithm, and the results are shown graphically in the

next section.

¯

C = I(y)

,

s + λ²

then the boundary conditions in eqns. (5) reduce to;

H = 1, I = 1, F = 1, at y = −1,

H → 0, I → 0, F → 0, at y = 1.

3 Results and Discussion

(23)

To get the ﬂow information, we simulated the solutions of equations

(31), (33), and (34) using Mathcad software. The inﬂuence of the

fractional-parameter (α) on velocity, temperature and blood concentration

are displayed graphically and discussed. Axial ﬂuid velocity, temperature

distribution, and concentration are explored as functions of several

dimensionless factors, including: slanted (inclined) magnetic ﬁeld

parameter (M), radiation parameter (R), fractional parameter (α), heat

source parameter (S), and Schmidt number (S_C). In all the dimensionless

parameter calculations, we vary the value of the fractional parameter (α),

but we maintain other values constant, such as, t = 1, S_C= 0.5, ω = 0.5,

S = 1, P_r= 2, K = 2, R = 0.5, h = 0.5, β = 0.5, φ = 30^◦.

As a result, the following are the simpliﬁed governing equations of

motions with arbitrary solutions:

2

¯

d F

0

¯

− A₂F = A₃− gβH − gβ I,

(24)

dy²

2

¯

d H

2

¯

+ A₆H = 0,

(25)

(26)

(27)

dy²

G = A₁(constant),

2

¯

d I

dy²

2

¯

− A₉I = 0.

3.1 Velocity Profile

Equation (23)’s boundary conditions are used to solve equations (24)

through (27) and the following solutions are obtained:

Consequently, the magnetic ﬁeld always has a greater inﬂuence on the

blood velocity proﬁle. The application of the magnetic ﬁeld to the system,

as shown in Figure 3, increases the Lorentz force, a resistive force that

primarily restricts the ﬂow of ﬂuid Bunonyo and Ebiwareme (2023) and

Vardanyan (1973). For fractional order (α = 0.4), as the magnetic ﬁeld

parameter’s strength increases, as seen in Figure 3(a), the blood velocity

reduces sharply, whereas it declines gradually for (α = 1) as shown in

Figure 3b.

ꢄ

ꢅ

cos A₆y¯

2 cos A₆

sin A₆y¯

2 sin A₆

¯

H =

−

(28)

(29)

cosh A₉y¯

2 cosh A₉

sinh A₉y¯

2 sinh A₉

¯

I =

−

Abdullahi et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

However, for the fractional order parameter in Figure 6a and 6b, the ﬂow

velocity vanishes between angles of (80^◦to 85^◦).

Because several of the related parameters’ values have changed, the

graphs in Figure 6c and 6d behave very diﬀerently from one another.

Using Figure 6c as an example, y = 0.004, p = 4, and the fractional

parameter (α

=

0.2), whereas Figure 6d also includes the fractional

parameter (α = 0.4) and y = 0.004, p = 3.

(a)

(b)

Figure 3: Axial velocity proﬁle for diﬀerent values of magnetic parameter: (a)

α = 0.4 (b) α = 1, ω = 0.5, S = 1, P = 2, K = 2, R = 0.5, t = 1, h = 0.5,

r

β = 0.5, φ = 30^◦, Sc = 0.5.

For therapeutic purposes and treatment procedures related to

atherosclerosis, bone fractures, controlled tissue damage, and malignant

tumors, to mention a few, a regulated magnetic ﬁeld can therefore be a

useful tool (Imoro et al., 2024). For both the fractional parameter (α = 0.4)

and the integer order model blood ﬂow (α = 1), Figure 4 shows the

variation in blood ﬂow at diﬀerent heat source parameter (S) values. It is

clear that an increase in the heat source has an impact on blood velocity

and the fractional ﬂuid parameter (α = 0.4) (see Figure 4a). As seen in

Figure 4b, the axial velocity does, however, drop symmetrically as the

heat source parameter increases.

(a)

(b)

(c)

(d)

Figure 6: Axial velocity proﬁle for various angles of inclination of the magnetic ﬁeld:

(a) α = 0.4 (b) α = 1 (c) y = 0.004, p = 4 and α = 0.2 (d) y = 0.004, p = 3, α = 0.4,

ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5.

r

(a)

(b)

Figure 4: Proﬁle of axial velocity for various heat source parameter: (a) α = 0.4 (b)

The blood ﬂow velocity proﬁle at two independent times, t

=

0.01

α = 1, ω = 0.5, M = 0.5, P = 2, K = 2, R = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

r

and 0.5, is shown in Figure 7 with ﬁve diﬀerent values of the fractional

parameter (α = 0.2, 0.4, 0.6, 0.8, and 1). It has been observed that the

fractional parameter (α) plays a critical role in regulating blood velocity.

The fractional derivative ﬂuid velocity initially moves faster than the

integer order ﬂuid model when the time is relatively small (t = 0.01).

However, for a longer period of time (t = 0.5), the reverse behavior

is seen, that is, ﬂuids with integer order have a faster velocity than

those with fractional order parameters. Naturally, this results from the

system’s stability, which can improve over longer timescales. For both

fractional order derivative ﬂuid models and integer order derivative ﬂuid

models, it is often observed that blood velocity increases with increasing

time t. Figure 7a shows the evolution of the primary velocity proﬁle,

showing how the ﬂow develops into fully formed Poiseuille ﬂow, which

is distinguished by the typical parabolic proﬁle. Figure 7b clearly depicts

the velocity proﬁle at the fork section for various values of the fractional

parameter. It is signiﬁcant to note that when (α = 1), a zone of sluggish

ﬂow appears along the outside wall and gets worse as time goes on, as

was previously observed by Gade et al. (2026).

Sc = 0.5.

Figure 5 displays the velocity distribution based on diﬀerent thermal

radiation parameters (R). The blood velocity increases as the radiation

parameter (R) increases, as indicated by both the fractional parameter

(α

=

0.4) and the classical order parameter (α

=

1). Remarkably,

comparable results for a related ﬂuid model were discussed in Tabi et al.

(2017). According to Yakubu et al. (2022), heat radiation possesses the

capability to modify the eﬀective viscosity of ﬂuids, hence potentially

causing an indirect inﬂuence on the velocity proﬁle.

(a)

(b)

Figure 5: Proﬁle of axial velocity for various heat source parameter: (a) α = 0.4 (b)

α = 1, ω = 0.5, M = 0.5, P = 2, K = 2, R = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

r

Sc = 0.5.

(a)

(b)

The applied magnetic ﬁeld parameter for various tilted values is

displayed in Figure 6. Blood ﬂow is reduced over the aﬀected area when

the applied magnetic ﬁeld’s angle of inclination is increased for both the

fractional order parameter and the classical order parameter (α = 1).

Figure 7: Axial velocity proﬁle for diﬀerent values of α at: (a) t = 0.01 (b) t = 0.5

ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5

r

Abdullahi et.al (2026)

22

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

3.2 Temperature profile

It’s noteworthy to see in Figure 9b that the temperature at the middle line

of the channel decreases as the heat source’s values rise. The temperature

exhibits oscillating behavior for diﬀerent amounts of the heat source in

Figures 9c and 9d. As the values of the heat source rise, the temperature

is maximum at the center, decreases, and ﬁnally approaches zero at the

artery walls. The diﬀerent values of the fractional parameter really cause

a shift in the temperature distribution, as shown in Figure 10. It illustrates

how temperature increases with increasing fractional parameter . This

implies that the fractional order ﬂuid model’s temperature distribution is

more faster and higher over a longer period of time, which is what causes

the variation shown in Figures 10a and 10b, as mentioned earlier. The

temperature gradually decreases toward the artery’s axis in Figures 10c

and 10d, eventually tending to align with the axis.

Temperature proﬁles for diﬀerent radiation parameters (R), fractional

parameters , and heat source parameters (S) are shown in Figures 8–??.

The temperature change for diﬀerent values of the radiation parameter

R, as shown in Figure 8. It is evident that when the thermal radiation

increases, temperature increases for both fractional and integer order

derivatives. The temperature varies near the center line for both the

integer order and the fractional order derivative, as shown in Figure 8a

and 8b. Consequently, it is more visible in the graphs of Figure 8b.

(a)

(b)

Figure 8: Temperature proﬁle for diﬀerent values of radiation parameter: (a) α = 0.4

(b) α = 1, ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦,

(a)

(b)

r

Sc = 0.5

During hyperthermia, the temperature distribution is very important. It

is commonly recognized that hyperthermia results from a breakdown

in thermoregulation, which takes place when the body absorbs heat

from outside sources like radiation or a body temperature that is being

generated or absorbed. When a person has hyperthermia, the blood’s

internal temperature increases without damaging the tissues around

the blood vessel. We have not taken into account the temperature

exchange at the artery wall to account for this, meaning that the wall’s

temperature is zero. In light of this, the blood temperature in the current

model is low at the artery wall and high at the midline for classical

ﬂuid. Numerous theoretical and experimental studies for Newtonian

and non-Newtonian ﬂuids of integer order reported similar phenomena,

for example in (Ramesh & Devakar, 2015). Similar to radiation, the heat

source (S) another crucial factor, has a large impact on the bloodstream’s

temperature distribution. More mitochondria per cell increase the

thermal activity involved with the heat production process, as seen

in Figure 9, which raises the system’s temperature. The heat source

improves the temperature distribution and supplies more heat to the

blood ﬂow system even though the wall temperature must remain zero in

order to meet the boundary conditions. The temperature distribution at

the channel walls, which decreases and becomes more ﬂattened toward

the channel’s center line when the heat source is increased as shown in

Figure 9a, which is ampliﬁed to maintain a constant ﬂow rate, as seen in

Figure 9a.

(c)

(d)

Figure 10: Temperature proﬁle for diﬀerent values of α at: (a) t = 0.05, y = 0.004,

p = 1 and α = 1, t = 0.1 (b) t = 0.25, ω = 0.5, S = 1, P = 2, K = 2, M = 0.5, t = 1,

r

h = 0.5, β = 0.5, φ = 30^◦, Sc = 0.5, y = 0.004, p = 3, α = 0.4. (c) y = 0.004, p = 2

and α = 1, t = 0.1 (d) y = 0.004, p = 10, α = 0.01, t = 0.1.

3.3 Concentration profile

The concentration proﬁle for diﬀerent values of fractional order

parameter (α) , Schmidt number (Cs) , and chemical process (ω) is

shown in Figures 11 to 13. There is a relationship between the blood

concentration and the quantity of blood cells ﬂoating in the plasma. Red

blood cells (RBCs) are important blood cells because of their size and

density in the bloodstream. RBCs assembled at the center of the vessel,

where there is a greater concentration of solutes, due to their revolving

nature. However, because the oﬀ-axis zone is an area predominantly

represented by cells that carry plasma, the solute concentration there

decreases to a minimum. This observation is displayed in all of the

concentration graphs in this section. The fractional model ﬂuid in Figure

11 reaches a greater concentration more quickly than the integer order

ﬂuid (Imoro et al., 2024).

(a)

(b)

(c)

(d)

(a)

(b)

Figure 9: Temperature proﬁle for diﬀerent values of heat source parameter: (a)

α = 0.4 (b) α = 1, ω = 0.5, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5,

Figure 11: Concentration proﬁle for diﬀerent values of α at: (a) t = 0.1 (b) t = 0.5

r

β = 0.5, φ = 30^◦, Sc = 0.5

Sc = 0.5, S = 1, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5, φ = 30^◦

r

Abdullahi et.al (2026)

23

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 18-26

This is because a fractional order derivative that restricts ﬂuid ﬂow

is included in the model. The Schmidt number exhibits the opposite

pattern. The blood cells show an additional force of the temperature

gradient in the presence of the Schmidt number, as seen in Figure 12,

which further increases the concentration. Therefore, lower Schmidt

number values, for example in industrial applications, physically

represent hydrogen gas as the species diﬀusing (Sademaki et al., 2026).

4 Conclusion

Currently, a fractional-order model of the magneto hydrodynamic blood

ﬂow via a bifurcated artery under the inﬂuence of thermal radiation,

a slanted magnetic ﬁeld, and a heat source during tumor treatment is

being developed. The Laplace transform and the combined methods

of indeterminate coeﬃcients were used to solve the mathematical

models. The fractional order parameter has a major eﬀect on the

blood velocity proﬁles, concentration, and temperature distribution.

It has been noted that ﬂuids with fractional order can occasionally

move faster than those with integer order. Fractional model ﬂuid

ﬂow is slower than integer-order ﬂuid ﬂow over longer dimensionless

durations. The impact of ﬂuid velocity is demonstrated by the fact

that the rate of increase in ﬂuid velocity is slower at larger levels of

the magnetic ﬁeld parameter. As the chemical reaction parameter rises,

the blood ﬂow concentration falls. The blood concentration rises as the

Schmidt number rises. As the fractional parameter and the heat source

increase, the blood ﬂow’s dimensionless temperature rises, which also

aﬀects the radiation parameter. We noted that the outcomes will be

intriguing to comprehend and evaluate throughout cancer therapy using

hyperthermia. Additionally, it will be useful in understanding the drug

particle concentration phenomena for applications and administrations

involving drug delivery. Our research’s ﬁndings should serve as a

foundation for the study of increasingly sophisticated blood ﬂow models

and also serve as a basis for in vitro and in vivo testing, particularly

in the application areas like medicine, biomedical engineering, biology,

pathology, and other related domains.

(a)

(b)

Figure 12: Concentration proﬁle for diﬀerent values of Schmidt number: (a) α = 0.4

(b) α = 1, ω = 0.5, S = 1, R = 0.5, P = 2, K = 2, M = 0.5, t = 1, h = 0.5, β = 0.5,

r

φ = 30^◦

Figure 12 illustrates how the species’ chemical molecular diﬀusivity

decreases dramatically with increasing Schmidt number (Sc) , making

it easier for the species to enter the ﬂow ﬁeld and raising the mass

transfer function. Higher Schmidt number compounds can enhance

mass transfer and dispersion properties in the bloodstream, especially

for pharmaceutical diﬀusion in pulse blood ﬂow. The amplitude of the

blood ﬂow concentration is larger for the integer order derivative. As

demonstrated in Figure 13a and 13b, this phenomena is clearly seen along

the ﬂow axis (0 ≤ y ≤ 0.5) and slowly declines in the region (0 ≤ y ≤ 1)

for both fractional and integer order derivatives, respectively. As can be

observed from all of the graphs in Figure 13, the blood ﬂow decreases

along the distensible tube’s length where the graphs begin to ﬂuctuate

because of the size of the chemical reaction parameter’s peak value

(pressure gradient) (see, Abdul-Wahab & Al-Saif, 2024). Furthermore,

it has been demonstrated that the variation is more pronounced in the

larger section of the artery wall, permitting the ﬂow to pass without

producing a perceptible pressure gradient. Nevertheless, the substantial

pressure gradient is usually required to maintain a consistent ﬂow rate as

it passes through the constrictions in the artery.

Acknowledgments

This work was supported by Tertiary Education Trust Fund (TETFund)

Ref. No. TETF/ DR&D/CE /UNI /BAUCHI /IBR /2025/ VOL.1.

Therefore, the authors gratefully acknowledged the ﬁnancial support of

the TETFUND. The authors would like to express their sincere gratitude

to the handily editor and the reviewers for their helpful and informative

comments, which have enhanced the manuscript.

Declaration of Generative AI

The authors declare that they do not used generative AI in the scientiﬁc

writing.

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Appendix

1

k

s

τp_r

Rτp_r+ 1

A₁= 1, A₂= M²sin²φ −

−

,

A₃= h(s + λ²), A₄=

,

(1 − α)s + α

ꢄ

ꢅ

p

τp_r

Rτp_r+ 1

S

τp_r

S

A₃

A

A₃= h(s + λ²), A₄=

,

A₅=

−

(s + λ²), A₆= A₄A₅, A₇=

,

(1 − α)s + α

q

−gβ

gβ

A₉= Sc(ω + s((1 − α)s + α)), A₁₀

=

,

A₁₁

=

2(A²₉− A₂) cosh A₉

2(A²₉− A₂) sinh A₉

Abdullahi et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 27-33

ART ICLE

Prevalence and Determinant Factors of

Malaria Infection among Patients Attending

Gimbichu Primary Hospital, Soro District,

Central Ethiopia, Ethiopia

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.3S

Melese Birmeka ^1,∗�, Gebremedhin Gebrezgabiher², Tekleweyni

Asayehegn³, and Mohammed Kasso⁴

ARTICLE HISTORY

Received: 12 June, 2025

¹Department of Biology, Hawassa University, Hawassa, P. O. Box 05,

Accepted: 21 January, 2026

Published Online: 10 June, 2026

²Department of Veterinary Medicine, College of Veterinary Medicine and Animal Sciences, Samara

University, P. O. Box 132, Samara, Afar, Ethiopia

³Department of Aquatic Sciences, Fisheries and Aquaculture, Hawassa University, Hawassa, P. O. Box 05,

⁴Department of Biology, Hawassa University, Hawassa, P. O. Box 05.

CITATION

Birmeka M. et.al (2026). Prevalence and

Determinant Factors of Malaria Infection

among Patients Attending Gimbichu

Primary Hospital, Soro District, Central

Ethiopia, Ethiopia. East African Journal

of Biophysical and Computational

^∗Corresponding author: melesebirmeka@yahoo.com

Abstract

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.3S.27-33

In the world, particularly in Ethiopia, malaria has a great influence on human health and economy.

This study intended to determine the prevalence, trends and associated risk factors of malaria patients

visiting Gimbichu Primary Hospital, Ethiopia. To assess the trend and parasitological examination,

a hospital-based cross-sectional study was carried out. To determine factors that significantly

associated with infection, a bivariate and multivariable logistic regression analyses were performed

with statistical significance set at p < 0.05. The findings of the study revealed the overall malaria

prevalence of 72.4% among suspected patients. The study also revealed that males (AOR = 3.5, 95%

CI; 1.5 - 3.8, p<0.001), individuals under five years (AOR = 2.8, 95% CI: 1.13 - 2.2), 5-20 years (AOR

= 1.75, 95% CI: 1.1 - 1.91) and 21-45 years (AOR = 1.65, 95% CI: 1.01 - 1.49) were at higher risk.

Additionally, study participants living close to mosquito breeding sites (AOR = 2.54, 95% CI: 2.53 -

4.14), rural (AOR = 2.13, 95% CI: 1.01 - 2.6), houses with thatch roof (AOR = 1.43, 95% CI: 1.01-2.30),

not using bed nets (AOR = 1.51, 95% CI: 2.01 - 4.1), homes with wall openings (AOR = 1.6, 95%

CI: 1.13 - 2.57), monthly income of less than 1,000 Ethiopian Birr (AOR = 2.93, 95% CI: 1.3 - 4.6),

and pregnant women (AOR = 1.6, 95% CI: 1.13 - 2.57) had maximum risk for malaria infection. The

analysis from the retrospective data showed the overall decreasing trend in malaria infection rates,

despite the fluctuations recorded between 2015 - 2021. The study indicates that malaria is persistent

and a significant public health challenge which is driven by a complex interrelationship of demographic,

social and environmental factors. Plasmodium vivax infection is the most prevalent species known to

cause malaria in the study area. These findings necessitate targeted interventions focusing on housing

improvements, economic support, and vector control measures.

OPEN ACCESS

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Determinants; Malaria; Prevalence; Soro District, Trend

both mother and the child. Similarly, children under ﬁve are at higher

risk because their immune systems are not fully developed, with a child

dying of malaria every 45 seconds worldwide.

1 Introduction

Malaria is a contagious disease that is caused by parasitic proatozoa

(Ferede et al., 2013). It has great impact on world population health and

economy (Baird, 2013). It is dominantly caused by Plasmodium falciparum

and Plasmodium vivax. Out of the two species P. falciparum is the stronger

pathogenic species that causes most deaths by malaria diseases at world

scale, accounting for more than 90% of the world malaria mortality

(Baird, 2013; Ferede et al., 2013). The malaria disease is a severe disease

particularly in children and pregnant women. Pregnant women are more

susceptible due to decreased immunity during pregnancy, endangering

As an infectious vector-borne disease, malaria continues to be a key public

health challenge in the country, with transmission patterns varying across

regions depending on climatic conditions, rainfall, and altitude. In

Ethiopia, malaria is known to be dominantly caused by P. falciparum (60%)

and P. vivax (40%) (FMOH (Federal Ministry of Health), 2018). In the

country, about 75% of areas located below 2,000 meters above sea level are

susceptible to malaria epidemics and the persistent risk of transmission

(Girum et al., 2019). Annually, approximately 4,782,000 reported cases

Birmeka M., et.al (2026)

27

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 27-33

and related deaths, with morbidity and mortality increasing markedly

during epidemic periods were recorded in the country (Alemayehu et al.,

2014). Of this, the large-scale epidemics tend to occur every ﬁve to eight

years although smaller, localized outbreaks are reported annually (Tsige

et al., 2011). An estimated 68% of Ethiopians, a country with a population

of over 100 million people, are at risk of contracting malaria (WHO (World

Health Organization), 2016).

Exclusion criteria: Malaria suspected patients who were not to give

consent for participation in this study.

Sampling and Sample Size Determination All patients suspected of

having malaria were consecutively selected during their visits to the

outpatient department of Gimbichu Primary Hospital till the compulsory

sample size was achieved. The sample size was estimated using Daniel’s

formula (Daniel, 2004).

z²(1 − p)

In Ethiopia, malaria transmission shows a considerable variation across

seasons, years, and geographic settings. The high impact of malaria is

particularly pronounced in rural areas (Donnelly et al., 2005), largely

due to proximity to mosquito breeding sites, limited coverage of control

interventions, widespread poverty, low literacy levels, land-use practices

and poor housing conditions (Stratton et al., 2008). Exceptionally, a

yearly based transmission observation is evident in the southwestern

lowland regions bordering neighboring countries (Zhou et al., 2016).

The communities with lower socioeconomic status are known to be

disproportionately aﬀected (WHO (World Health Organization), 2012),

because it limits access to medical care and preventative measures like

indoor spraying, bed nets treatment and eﬃcient antimalarial therapy

(Yamamoto et al., 2010). The drug-resistant strains of P. falciparum and

P. vivax have also emerged and spread, posing a signiﬁcant challenge to

the control of malaria which contributed to the recent increase in malaria

cases in the nation (Yarcho, 2010).

N =

(1)

d²

Where - p = 50%, because of the absence of previous malaria prevalence

studies in the area, - d = margin of error at 5% and - z = 1.96 at 95% CI

Consequently, the sample size was determined to be 384.

2.5 Data Collection

The structured pretested questionnaires were used to collect information

on socio-demographic and economic status of study participants. Blood

sample collection was done by ﬁnger prick by healthcare professional,

and on the same slide both thick and thin blood smears were prepared.

Throughout the data collection process, continuous monitoring and

supervision were maintained. The activities performed by laboratory

technicians, interviewers and nurses were closely overseen. Additionally,

retrospective data spanning for seven years (2015 -2021) was retrieved

from hospital registration records.

The Government of Ethiopian has made signiﬁcant progresses since

2005 in malaria control interventions such as diagnostic testing, rapid

case treatment, and prevention strategies for pregnant women through

intermittent preventive therapy. High eﬀorts also implemented on the

distribution of IRS and ITNs. However, the widespread emergence of

drug resistance in parasites and insecticide resistance in vectors have

obstructed eﬀorts of malaria eradication (Abeku et al., 2015; Tafese

et al., 2018), particularly in the Hadiya Zone of central Ethiopia. This

situation underscores the need for continuous evaluation and monitoring

of malaria control interventions to address existing gaps.

2.6 Data Analysis

Following a completeness check, the data was analyzed by using SPSS

version 24. Logistic regression studies were performed to identify the

relationship between a few possible risk variables and malaria infection.

To determine the existence and strength of a connection, AOR at 95% CI

were calculated; if p < 0.05, statistical signiﬁcance was proclaimed.

2 Materials and Methods

2.1 Study Area

2.7 Ethical Consideration

The Institutional Research Ethics Review Committee of CNCS of Hawassa

University examined and approved the study proposal and ethical

clearance was received (Ref.no. IRB/279/13). Additional, permission

was also granted by the Hadiya Zone Health Department and the Soro

District Primary Hospital. Conﬁdentiality and privacy were strictly

upheld, and participation in the study was entirely voluntary. After

awareness made on the objectives of the study, participants gave their

consent participation. Conﬁdentiality was also maintained.

The study was conducted at Gimbichu Primary Hospital which provides

care for the Soro District in Hadiya Zone, central Ethiopia region. The

district is about 264 kilometers south of the nation’s capital, Addis Ababa.

Soro District is home to a substantial population of 233,015 people, nearly

evenly split between genders (115,825 men and 117,190 women), giving

the hospital a wide and diverse community to serve.

2.2 Study Design and Period

3 Results

An institution-based cross-sectional study was carried out between

October 2022 and January 2023.

3.1 Characteristics of Study Participants

in Retrospective Study of Malaria in

Soro District, 2015 - 2021

2.3 Study Population

All individuals who presented to Gimbichu Primary Hospital with

suspected malaria during the data collection period and satisﬁed the

eligibility requirements were involved in the study population

A total of 65,211 clients were registered in the laboratory logbooks of

Gimbichu Primary Hospital. Of these, 36,132(55.4%) were males and

29,089(44.6%) were females. Between 2015 and 2021, 65,211 blood ﬁlms

were microscopically examined. The majority of the cases were males

accounting 17,813(49.3%). Although malaria prevalence ﬂuctuated from

2016 to 2021, there was an overall decreasing trend. Over the seven year

period, a considerable malaria cases were recorded in the age 15 - 24 years

old (18,718 cases, 28.7%), followed by 5 - 14 years old (15,131 cases, 23.2%).

The lowest number of cases was over 54 years (8,740 cases, 13.4%) (Table

1).

2.4 Eligiblity Criteria

Inclusion criteria: Malaria suspected patients who were consented to

participate in the study.

Birmeka M., et.al (2026)

28

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 27-33

Table 1: The social and demographic characteristics of microscopically examined suspected patients in Soro District, 2015 - 2021.

Socio-demographic

variables

Category

Malaria+ve(%) COR(95%CI)

P-value AOR(95%CI)

P-value

Female

Male

106(62.4)

172(80.4)

1

1.67(1.3 - 2.65) < 0.001

3.5(1.5 - 3.8) < 0.001

Age

<5

27(34)

161(79)

50(86)

40(90)

1.6(2.13 - 3.1)

0.01

2.8(1.13 - 2.2)

0.024

5 - 20

21 - 45

>45

1.86(1.14 - 2.16) < 0.001

1.75(1.1 - 1.91) < 0.001

1.75(1.01 - 2.79) 0.03

1

1.65(1.01 - 1.49) 0.03

1

Residence

Urban

Rural

110(66.7)

168(76.7)

1

1.43(1.14 - 2.31) < 0.001

2.13(1.01 - 2.6)

0.01

Income level of < 1000

90(81.8)

65(71.4)

83(70.3)

40(61.5)

1.55(1.06 - 4.62) < 0.001

2.93(1.3 - 4.6)

0.86(0.21 - 5.2)

2.75(0.14 - 4.3)

1

0.004

0.31

0.22

Household

100-3000

3000-5000

>5000

0.56(0.22 - 5.6)

1.63(0.11 - 2.6)

1

0.324

0.234

Use of bed nets

Yes

No

112(67.1)

166(76.5)

1

1.54(2.03 - 5.12) < 0.001

1.51(2.01 - 4.1) < 0.001

IRS in the past twelve

month’s

yes

No

70(69.3)

208(73.5)

1

1.9(1.23 - 2.96) < 0.001

1.89(1.3 - 2.76) < 0.001

Mosquito Breeding Site Yes

145(72.8)

133(71.9)

1.54(2.03 - 5.16) < 0.001

2.54(2.53 - 4.14) < 0.001

near to home

No

Home wall opening

Yes

No

67(62.6)

211(76.2)

1.89(1.23 - 2.96) < 0.001

1

1.6(1.13 - 2.57)

1

0.01

House Roof Type

Corrugated iron sheet 191(71.8)

Thatch 87(73.7)

1

1.63(1 - 2.2)

0.02

1.43(1 - 2.30)

0.001

*For each respective characteristic, the percentage calculated is from total examined

3.7 Discussion

falciparum transmission is more common in lowland areas. Conversely,

the predominance of P. vivax observed in this study is agree with reports

from Wolkite Health Center (Degeﬁe, 2017) , Dilla District (Ehsetu &

Besha, 2015), Hallaba District (Girum, 2014) , and East Shewa Zone (Firew

& Andrew, 2018) (Firew and Andrew, 2017). This similarity may be

related to comparable altitudinal conditions or the relapsing nature of P.

vivax, particularly during cooler seasons.

This study was conducted to examine trends in malaria infection over

time, estimate its prevalence and identify factors associated with malaria

infection among patients attending Gimbichu Primary Hospital in Soro

District, South-central Ethiopia.

A study by Deressa et al., 2006 reported that malaria has been one

of the major causes of mortality, hospital admissions and outpatient

visits in Ethiopian health facilities for a long time. In line with these

ﬁndings, the present analysis showed that malaria cases peaked in 2015,

accounting for 74.5% of all reported cases, while the lowest prevalence

was observed in 2019 (10.4%). From the year 2015- 2021, overall trend

analysis demonstrated the ﬂuctuations in malaria incidence, with an

overall declining pattern across the seven-year period.

The present study showed as the infection malaria was signiﬁcantly

more common among males than females which seem males being 3.5

times more likely to be infected. This result diﬀers from case reported

by Graves et al., 2009 , although it is consistent with other Ethiopian

studies (Abebe et al., 2012; Girum, 2014). More than half proportion of

the malaria infection was observed among individuals aged 5–20 years,

followed by those aged 21–40 years. This may be attributed to increased

outdoor activities such as farming and other productive work, which

elevate exposure to mosquito bites. Additionally, malaria prevalence

was assumed to be higher among individuals residing in rural areas

when compared with those living in urban settings. This observation

aligns with ﬁndings from Dilla District (Ehsetu & Besha, 2015). Such

higher burden in rural areas may be due to lower levels of awareness,

substandard housing conditions, limited resources and reduced access to

eﬀective malaria control measures.

Results from the current study revealed an overall malaria prevalence of

72.4% among suspected patients. This prevalence diﬀers from reports

of similar studies conducted in other parts of Ethiopia either higher

or lower. For instance, the prevalence rate observed in this study was

higher than those reported from Dilla District by Ehsetu and Besha, 2015

, Kola Diba District by Abebe et al., 2012, Wolkite Health Center by

Degeﬁe, 2017, Arba Minch Hospital by Belayneh, 2014, Sibu Sira District

by Girum, 2014 and the East Shewa Zone of Oromia Region by Firew

and Andrew, 2018. In contrast, it was lower than the prevalence reported

from Hallaba District by Girum (2014). These diﬀerences might be due to

variations in study period, season, altitude, local communities’ awareness

and diﬀerences in malaria prevention and control strategies.

The record of the higher prevalence of malaria cases among pregnant

women when compared with non-pregnant women probably due to

immunity reduction associated with pregnancy. Low household income

may also signiﬁcantly associate with increased malaria infection. For

instance, individuals with lower income levels may face greater malaria

risk due to limited access to preventive tools and healthcare services,

inadequate housing that permits mosquito entry, and compromised

health and nutrition status (Dejene, 2014). These ﬁndings are also

consistent with studies from Muleba District in the Kagera region

of Tanzania, which reported a similar association between family

employment status and malaria prevalence among children under ﬁve

(Mushashu, 2012). WHO (World Health Organization), 2012 also pointed

out as economically disadvantaged households may have limited access

to healthcare facilities and insuﬃcient resources to aﬀord vector control

In the study area, P. vivax was the predominant Plasmodium species

(86.1%), followed by P. falciparum (8.3%) and mixed infections of P.

falciparum and P. vivax (5.6%). Hence, the result of present study

contradicts with the national estimates which indicate as P. falciparum

accounts for approximately 60% of malaria incidences while P. vivax

accounts for the remaining 40% in Ethiopia (Eliyas, 2014). Similar

predominance of P. falciparum has been documented in studies conducted

in Ayire District (Eliyas, 2014) and Arba Minch Hospital (Belayneh, 2014).

Such inconsistencies may be explained by topographical diﬀerences, as P.

Birmeka M., et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 27-33

interventions including insecticide-treated nets (ITNs), indoor residual

spraying (IRS) and antimalarial medications.

Acknowledgments

The authors are grateful for laboratory technical staﬀ of Gimbichu

Primary Hospital and Hawassa University for technical support.

The result of present study regarding the use of indoor residual spraying

within the past 12 months was signiﬁcantly associated with malaria

infection. In similar way, Sintasath et al., 2005 also indicated as

IRS remains a cornerstone of the national malaria control strategy,

particularly for epidemic prevention and mitigation. Their study also has

shown substantially reduces in malaria morbidity and mortality. These

ﬁndings are also consistent with reports from the Jiga area in northwest

Ethiopia (Seble, 2014) , although they contrast with ﬁndings from Muleba

District in Tanzania (Mushashu, 2012).

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4 Conclusions

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ART ICLE

Feeding Habits and Trace Metal

Concentrations in Organs of the Nile Catfish,

Synodontis schall (Bloch & Schneider) (Pisces:

Mochokidae), in Lake Abaya, Ethiopia

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.4S

Elias Dadebo ¹�, Daniel WM-Bekele², Abnet Woldesenbet ¹�, Tamirat

ARTICLE HISTORY

Handago³, Tekleweyni Asayehegn¹, Tiruken Aziz¹, and Teshome Belay

Received: 14 April, 2026

4,∗

Accepted: 26 May, 2026

�

Published Online: 10 June, 2026

¹Department of Aquatic Sciences, Fisheries and Aquaculture, Hawassa University, P. O. Box 5, Hawassa,

Ethiopia

CITATION

²Biology Department, Environmental Toxicology Program, Hawassa University, P. O. Box 5, Hawassa,

Ethiopia

Dadebo et.al (2026). Feeding Habits

and Trace Metal Concentrations in

Organs of the Nile Catfish, Synodontis

schall (Bloch & Schneider) (Pisces:

Mochokidae), in Lake Abaya, Ethiopia .

East African Journal of Biophysical and

Computational Sciences Volume 7(1),

2026. .https://dx.doi.org/10.4314/eajbcs.

v7i1.4S.34-42

³Department of Biology, Wachemo University, PO Box 667, Hosaena, Ethiopia

⁴Department of Animal Sciences, Dilla University, PO Box 419, Dilla, Ethiopia.

^∗Corresponding author: teshimeansc@gmail.com

Abstract

This study investigated the feeding habits and trace metal concentrations in different organs of the

Nile catfish, Synodontis schall, in Lake Abaya, Ethiopia. Stomach content analysis was conducted

using frequency of occurrence and volumetric analysis. The results of the study indicated that S. schall

is an omnivore with polyphagous feeding habits; dominant food categories included phytoplankton,

detritus, insects, zooplankton, and macrophytes. Seasonal shifts were observed: phytoplankton was

the primary food source during the dry season, whereas zooplankton predominated during the wet

season. Ontogenetic dietary shifts were also noted with juveniles consuming mainly phytoplankton and

zooplankton, while adults mainly fed insects, detritus and phytoplankton. Trace metal analysis identified

copper (Cu), cadmium (Cd), nickel (Ni), zinc (Zn), and manganese (Mn) in liver, kidney and muscle tissues,

while lead (Pb) and cobalt (Co) were not detected. Metal concentrations in the liver were ranked as Cu >

Zn > Mn > Ni > Cd, while in muscle and kidney tissues, the order was Zn > Cu > Mn > Ni > Cd. Significant

difference (p < 0.05) in mean concentrations of Cu, Cd, and Zn were noted among tissues. All detected

heavy metals were within the FAO and EU safety limits, suggesting that S. schall from Lake Abaya is safe

for human consumption.

OPEN ACCESS

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Feeding habits; Lake Abaya; Omnivory; S. schall; Trace metals

River and its tributaries in the west, and in the Wabishebele River

in the southeast (Golubtsov & Habteselassie, 2010; Golubtsov et al.,

1995). Generally, S. schall is classiﬁed as an omnivore and benthic

ﬁsh species, and its diet covers a wide spectrum of food ranging

from plankton to invertebrates and plants (Lalèyè et al., 2006).

This dietary ﬂexibility, combined with a high tolerance for adverse

environmental conditions, allows the species to remain abundant

1 Introduction

The genus Synodontis is widely distributed across African

freshwaters ranging from the Nile basin, Chad, Niger, and much

of the West African region (Paugy et al., 2003) (Cuvier, 1816,). In

Ethiopia, the Nile catﬁsh Synodontis schall (Bloch and Schneider,

1801) is found in Lakes Abaya and Chamo in the south, the Baro

Dadebo et.al (2026)

34

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

in most African fresh waters (Lowe-McConnell, 1987).

Lake Abaya has a length of 79 km, a width of 29 km, and a surface

area of 1,160 km2 (Baxter, 2002). It has a maximum depth of 13 m

and is located at an elevation of 1,268 m (Baxter, 2002; Grove et al.,

1975). Lake Abaya is a home to 21 diﬀerent ﬁsh species that have

economic and ecological roles (Golubtsov & Habteselassie, 2010).

In Lake Abaya, S. schall is abundant in both littoral and pelagic

environments, likely due to low predation and minimal ﬁshing

pressure (Dadebo et al., 2012) . While the species is among the

most favored edible ﬁshes in some African countries (Lalèyè et al.,

2006), it currently holds low commercial importance in Lake Abaya.

Although, it remains ecologically indispensable as a primary prey

species for the commercially signiﬁcant catﬁsh, Bagrus docmac

(Forsskål, 1775) (Anja & Mengistou, 2001). Previous studies across

Africa have highlighted the species’ opportunistic feeding nature

(Adeyemi, 2010; Akombo et al., 2014; Arame et al., 2021; Dadebo

et al., 2012). Yongo et al. (2019) reviewed the feeding habits of

some Synodontis species in African freshwaters, and reported that

the genus feeds on a variety of food items, including vegetable

materials, insects, mollusks, detritus, macrophytes, ﬁsh scales,

and plankton. In Quémé River, the most frequent food items in

the stomachs of S. schall were macrophytes, algae, crustaceans,

rotifers, and mollusks (Lalèyè et al., 2006). Ofori-Danson (1992)

reported that the frequent food items of S. schall in the Kpong

head pond were benthic macroinvertebrates. Adeosun et al. (2017)

indicated the importance of insects, rotifers, crustaceans, ﬁsh parts

and phytoplankton in the diet of S. schall.

Figure 1: Geographic location of Lake Abaya (Source: adopted from Shishitu, 2024)

Beyond ecological dynamics, the health of ﬁsh populations is

increasingly threatened by the accumulation of trace metals from

natural and anthropogenic sources (Ali & Khan, 2018). Because ﬁsh

occupy various trophic levels, they can accumulate toxic substances

in vital organs and muscle tissues, posing risks not only to aquatic

biota but also to human consumers through trophic transfer (Garai

et al., 2021). Given the benthic feeding habits of S. schall, it is

particularly susceptible to metals associated with lake sediments.

2.2 Sampling and Measurements

A total of 849 S. schall specimens were collected during the dry

season (January to February, 2020) and the wet season (June to July,

2020) (wet season). Sampling was conducted at both littoral and

pelagic sites of the lake using a beach seine (25 m long and three

meters wide with a mesh size of 0.6 cm) in the shallow littoral

area and Nordic survey multi-mesh monoﬁlament nylon gillnets

(Appelberg et al., 1995) at the pelagic area of the lake.

Despite its ecological importance, there is lack of information

regarding the biology and ecology of S. schall in Ethiopia. To the

knowledge of the researchers, there is no published data regarding

the feeding habits and heavy metal load in the organs of S. schall

speciﬁcally within Lake Abaya. Therefore, the aim of this study

was to investigate the dietary patterns and concentrations of trace

metals in diﬀerent organs of this species. Such information is

vital for future management of the ﬁsh stock for assessing the

environmental health of the Lake Abaya ecosystem.

The multi-mesh gillnets consisted of twelve randomly distributed

panels of the mesh sizes 5, 6.25, 8, 10, 12.5, 15.5, 19.5, 25, 29, 35, 43,

and 53 mm (bar mesh). Each panel was 2.5 m long, and hence the

total length of each net was 30 m. The gillnets were set between

three to ﬁve meters depths in the open water, about 1.5 km inward

from the littoral sampling station. The gillnets were set early in the

morning around 7.00 a.m. local time and pulled around 3.00 p.m.

in the afternoon.

2 Materials and Methods

For each specimen, fork lengths (FL) and standard length (SL) were

measured to the nearest mm using a measuring board. Total weight

(TW) was measured to the nearest 0.1g using a SCALTEC digital

balance (model 23565, USA). The stomach of each ﬁsh was split

open, and the contents were collected and preserved in 5% formalin

solution and transported to Hawassa University Fishery Laboratory

for further analysis.

2.1 Study Area

Lake Abaya is the second largest lake in Ethiopia and

◦

0

00

◦

0

00

geographically located between 5 55 9 and 6 35 30 N latitude,

◦

0

00

◦

0

00

and 37 36 90 and 38 03 45 E longitude in the southern part of

the Ethiopian Rift Valley, East of the Guge Mountains (Figure 1)

(Shishitu, 2024) . The lake is fed on its northern shore by the

Bilate River, which rises on the southern slope of mount Gurage

(Golubtsov & Habteselassie, 2010). Other rivers that drain in to

the lake include, Gelana, Milate, Gidabo, Harre, Baso, and Amesa.

The only outﬂow of the lake is through the lower reaches of Kulfo

River directly below an alluvial pan at an elevation of 1,190 m.

Arba Minch town lies on its southwestern shore and the southern

shores are part of the Nechisar National Park (Teﬀera et al., 2019).

2.3 Stomach Content Analysis

The stomach contents of each specimen were examined visually

to identify macroscopic food items, whereas

a

dissecting

microscope (Leica, MS5, magniﬁcation- 40x) and a compound

microscope (Leica DME, magniﬁcation- 1000x) were used to

identify microscopic food items. Quantitative analysis of the diet

Dadebo et.al (2026)

35

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

was conducted using frequency of occurrence and volumetric

methods of analyses.

then fully dried in a laboratory oven at 175°C for three hours and

processed separately except for Pb and Cd. For Pb and Cd 60°C

were considered. A solution of aqua regia (3:1 hydrochloric to nitric

acid) was prepared as per Nwani et al. (2010). One gram of dried

muscle and 0.5 grams of liver and kidney were added to a 100 ml

ﬂask with 10 ml of aqua regia and reﬂuxed overnight to dissolve

organic materials and release trace metals, following Muinde et al.

(2013) method. After reﬂuxing, samples were digested at 60°C for

three hours to enhance reaction kinetics. Each sample was digested

in triplicates and diluted to a ﬁnal volume in a 50 ml volumetric

ﬂask and ﬁltered with the attached monochromater ﬁlter.

Frequency of occurrence (%FO): the number of stomach samples

containing one or more individuals of each food category was

expressed as a percentage of all stomachs containing food (Hyslop,

1980). In volumetric analysis (%V), the food items found in each

stomach were sorted into diﬀerent food categories, and the water

displaced by the group of volume of items in each category was

measured (Bowen, 1996). The relative importance of each category

was then expressed as a percentage of the total volume of food

categories.

2.7 Determination of Heavy Metals

2.4 Ontogenetic Dietary Shift and Dietary

Overlap

The digested ﬁsh organ and muscle were analyzed for Cd,

Co, Cu, Mn, Ni, Pb, and Zn using ﬂame atomic absorption

spectrometry (FAAS) with a dual background correction system

(BUCK SCIENTIFIC, Model 210VGP, USA). An air-acetylene ﬂame

was employed, utilizing aqueous calibration standards from

stock solutions of the metals. Three standard solutions and

a blank solution, made from the acid used in digestion, were

prepared to minimize errors and avoid overestimating heavy

To assess ontogenetic dietary shifts, specimens were categorized

into ﬁve size classes (Class I: 5- 9.9 cm; Class II: 10-14.9 cm; Class

III: 15-19.9 cm; Class IV: 20-24.9 cm; Class V: 25- 29.9 cm). The

total volume of food items in each size class was determined and

volumetric contribution of each category of food items was then

expressed as a percentage of total volume of food consumed in each

size class. The dietary overlap between diﬀerent size-classes was

calculated as percentage overlap using the Schoener Diet Overlap

Index (SDOI) (Schoener, 1970; Wallace Jr, 1981) based on Eq.(1) :

metal concentrations due to contamination.

The trace metal

concentrations in the organs were calculated by subtracting the

levels in the stock solution from those measured in the acid. Each

sample was aspirated into the FAAS for direct readings, and the

blank was created by combining all reagents in a 50 ml volumetric

ﬂask and diluting with deionized water. Finally, the FAAS was

adjusted with the following detection limit capacity of the element

as Cd (0.03 mg/kg), Co (0.02 mg/kg), Cu (0.005 mg/kg), Mn (0.03

mg/kg), Ni (0.02 mg/kg), Pb (0.03 mg/kg), and Zn (0.005 mg/kg),

respectively.

ꢀ

ꢁ

n

i=1

α = 100[l − 0.5]

|P_xi− P_yi|

(1)

where α is percentage overlap SDOI, between size group x and y,

Pxi and Pyi are proportions of food category (type) i used by size

group x and y, and n is the total number of food categories. Diet

overlap in the index is generally considered to be strong dietary

similarity and overlap when α value exceeds 0.60 (Mathur, 1977;

Zaret & Rand, 1971).

2.8 Statistical Analysis

The chi-square test was employed to compare the variations of the

frequency of occurrence of the diﬀerent food categories during the

dry and wet seasons (Worms and Touati, 2017). For volumetric

data, the Mann-Whitney U test was used to assess seasonal

diﬀerences (Worms & Touati, 2017). This non-parametric test was

used because the data violated the assumption of homogeneity

of variance required for parametric test. For the comparisons of

ontogenetic dietary overlap between diﬀerent size classes, their

schooner dietary overlap index was considered depending on the

benchmark (0.6). The concentration of considered trace metals

from the muscle, kidney, and liver of S. schall was compared using

one-way analysis of variance with the aid of SPSS v20 software at a

95% conﬁdence interval.

2.5 Fish Samples Collection and

Preservation for Determination of

Heavy Metals

A total of 20 S. schall specimens were collected from Lake Abaya

for heavy metal analysis using gill nets and a beach seine. For

each specimen, fork length (FL) and total weight (TW) of each

ﬁsh were recorded to the nearest 0.1 cm and 0.1 g, respectively.

Fish dissection for muscle, kidney, and liver samples followed the

EMERGE protocol, ensuring proper handling (Gupta & Mullins,

2010). The separated organs and muscles were quickly wrapped

in aluminum foil and then placed in plastic bags. These bags

were subsequently stored in an icebox and transported to the deep

freezer at Hawassa University Laboratory. Finally, the samples

were preserved at a temperature of -20 °C in the deep freeze.

3 Results

3.1 Diet Composition

2.6 Sample Preparation for Atomic

Absorption Spectroscopy (AAS)

Analysis

From a total of 849 S. schall specimens examined, 751 (88.5%)

contained food, while 98 (11.5%) had empty stomachs. The

sampled ﬁsh ranged in size from 5.4 to 33.0 cm in fork length

and weighed between 2.6 and 566 g in total weight. The diet of

S. schall in Lake Abaya was diverse, consisting of phytoplankton,

Muscle, liver, and kidney tissues were mechanically crushed with

a stainless steel knife and partially air-dried overnight. They were

Dadebo et.al (2026)

36

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

zooplankton, insects, macrophytes, detritus, ostracods, nematodes,

hydracarina and ﬁsh scales at diﬀerent proportion (Table 1).

Of these, the frequency of occurrences of detritus, insects,

macrophytes, zooplankton and phytoplankton were identiﬁed as

major food categories, while ostracods, nematodes, ﬁsh scales

and Hydracarina were found to be of minor importance (Table

1). Volumetrically, the contributions of Phytoplankton (31.20%),

detritus (20.10%), insects (16.80%), zooplankton (13.40%) and

macrophyte (12.80%) were dominant as a diet of S. schall. While, the

volumetric contributions of other identiﬁed food categories were

negligible (Table 1).

Table 1: Diet Compositions of S. schall (n = 751) in Lake Abaya, Ethiopia

Frequency of

occurrences

Volumetric

contribution

Food type

FO

%FO

VC(ML)

%VC

Phytoplankton

Blue green algae

Green algae

Diatoms

352

236

57

133

3

46.90

31.40

7.60

170

154.70

87.23

8.72

58.51

0.20

31.20

17.60

1.80

11.80

0.04

Euglinoids

0.40

Zooplankton

Cladocera

411

343

87

54.70

45.70

11.60

4.70

66.42

58.35

6.11

13.40

11.80

1.20

Calanoid copepods

Cyclopoid copepods

Rotifera

35

1.76

0.40

5

0.70

0.19

0.04

Insects

484

354

101

80

59

23

8

64.40

47.10

13.40

10.70

7.90

83.24

50.90

8.70

12.01

4.35

1.32

1.52

4.12

0.12

16.80

10.30

1.80

2.40

0.90

0.30

0.80

0.02

Diptera

Ephemeroptera

Plecoptera

Coleoptera

Hymenoptera

Tricoptera

Anisoptera

Hemiptera

3.10

1.10

7

0.90

8

1.10

Macrophytes

Detritus

Ostracods

Hydracarina

Nematodes

Fish scales

431

490

203

10

57.40

65.20

27.00

1.30

12.40

1.90

63.54

99.77

22.26

0.23

12.80

20.10

4.50

0.05

0.68

0.60

93

3.38

14

3.02

3.2 Seasonal Variation in the Diet

Composition

the occurrences of zooplanktons (70.50%), detritus (68.20%), and

macrophytes (58.80%) were the three most ingested food items of S.

schall. The remaining food categories in both dry and wet seasons

were negligible (Table 2). Volumetrically, during dry season the

contribution of phytoplankton (41.20%), detritus (19.60%), insects

(17.40%), and macrophytes (13.10%) were the major food categories

of S. schall. On the other hand, during wet season, zooplankton

(26.40%), detritus (21.00%), phytoplanktons (18.33%) were the

dominant food categories of S. schall. The contributions of other

food categories were relatively low (Table 2).

Signiﬁcant seasonal variations were observed in the diet of S.

schall in Lake Abaya (Table 2). The frequency of occurrence of

phytoplankton and zooplankton signiﬁcantly varied during the

dry (64.80%) and wet (29.30%) seasons (x²test, p < 0.05; Table

2). The occurrences of insects (68.8%), phytoplanktons (64.8%),

detritus (62.20%), and macrophytes (53.3%) were the major food

items of S. schall during dry season. While, during wet season,

Dadebo et.al (2026)

37

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

Table 2: Diet Compositions of S. schall during the Dry (n = 304) and Wet (n = 447) Seasons in Lake Abaya.

Frequency of occurrence (%) Volumetric contribution (%)

Food items

Dry

Wet

Dry

Wet

Phytoplankton

Blue green algae

Green algae

Diatoms

64.80

48.40

13.20

17.80

-

29.30

12.80

3.10

18.30

-

41.20

28.60

2.80

9.80

-

18.33

3.60

0.40

14.30

-

Euglinoids

Zooplankton

Cladocera

Copepoda

Rotifera

28.60

10.50

22.40

-

70.50

69.60

7.60

3.00

0.80

2.20

-

26.40

25.40

0.90

0.20

Insects

68.80

61.50

10.90

2.30

3.00

5.90

-

59.50

37.40

15.40

15.90

11.20

1.10

-

17.41

13.00

1.70

0.50

0.30

0.40

-

15.94

6.80

2.80

3.80

1.70

0.10

0.70

-

Diptera

Ephemeroptera

Plecoptera

Coleoptera

Hymenoptera

Tricoptera

Anisoptera

Hemiptera

2.30

0.70

1.50

0.010

0.90

0.040

Macrophytes

Detritus

Ostracods

Hydracarina

Nematodes

Fish scales

53.60

62.20

203

-

4.30

3.00

58.80

68.20

25.50

2.20

17.90

1.10

13.10

19.60

4.60

-

0.10

1.00

12.5

21.00

4.30

0.05

0.580

0.90

3.3 Ontogenetic Diet Shift and Dietary

Overlap

classes I and II (α = 52.0%), I and III (α = 58.8%), and II and IV (α

= 55.0%), suggesting a higher degree of dietary partitioning among

these groups.

The diet of S. schall exhibited distinct changes across the ﬁve size

classes (Figure 1). S. schall in size class 5-9.9 cm FL widely relied

on phytoplankton (60.3%) and zooplankton (17.4%) compared to

the contributions of other identiﬁed food categories (Figure 1).

When S. schall attained 10-14.9 cm FL size class, the importance

of detritus, insects, macrophytes and ostracods increased while

the contributions of phytoplankton and zooplankton decreased

(Figure 2). As the ﬁsh grew to the 15-19.9 cm FL size range,

it relied mainly on phytoplankton (30.6%), detritus (19.8%),

insects (17.4%), macrophytes (12.4%) and ostracods (4.1%). When

the ﬁsh further grew to 20-24.9 cm FL size range, it mainly

consumed phytoplankton (25.6%), detritus (21.5%), insects (18.2%),

macrophytes (11.6%) and ostracods (4.1%) (Figure 2). The major

food categories of the largest size class (25-29.9 cm FL) were

phytoplankton (45.5%), macrophytes (17.4%), detritus (16.5%) and

insects (16.1%) by volume (Figure 2). Other food categories, namely

ﬁsh scales and zooplankton had negligible role and unimportant in

the diet of the largest size class.

The Schoener Diet Overlap Index (SDOI) indicated signiﬁcant diet

similarity (> 60%) between several size classes, with the highest

overlap observed between classes III and IV (α = 93.3%) and II

and III α = 81.8%). Other signiﬁcant overlaps were recorded for

combinations III and V (α = 79.9%), IV and V (α = 77.8%), II and V

(α = 69.0%), I and IV (α α = 66.9%), and I and V (α = 65.1%) (Table

3). In contrast, diet overlap was not biologically signiﬁcant for size

Figure 2: Mean volume of food items consumed by diﬀerent size class of S. schall

sampled from Lake Abaya.

Dadebo et.al (2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

Table 3: Schoener Diet Overlap Index (SDOI) in ﬁve size classes of S. schall from Lake

and Mn - were detected in all tissue types at levels exceeding the

analytical detection limits (see the detection limit from material

and method at sub section of determination of heavy metals).

Conversely, Pb and Co were found at levels below the detection

threshold of the equipment, as detailed in Table 4. The distribution

of metals in the liver was ranked as Cu > Zn > Mn >Ni > Cd, while

in the muscle and kidney tissues, the order was Zn > Cu > Mn > Ni

> Cd.

Abaya, Ethiopia

Size class

SDOI (%)

I and II

52.0

I and III

I and IV

I and V

58.8

65.1*

66.9*

81.8*

55.0

69.0*

93.3*

79.6*

77.8*

II and III

II and IV

II and V

III and IV

III and V

IV and V

Mean concentrations exhibited signiﬁcant tissue speciﬁc variations

(Table 3). The mean concentration of Cu was notably higher

in the liver (3.847±0.341 mg/kg DW) compared to the kidney

(1.211±0.168 mg/kg DW) and muscle (0.944±0.028 mg/kg DW).

Similarly, Zn levels were elevated in the liver (1.85±0.153 mg/kg

DW) relative to the kidney (1.38±0.179 mg/kg DW) and muscle

(1.26±0.169 mg/kg DW). Conversely, more Cd concentration was

detected in the kidney (0.03±0.006 mg/kg DW), followed by the

liver (0.017±0.002 mg/kg DW) and muscle (0.011±0.00). The mean

concentrations of Cu in muscle (0.94), kidney (1.21), and liver (3.85)

exhibited signiﬁcant diﬀerences (p < 0.05). Likewise, Cd levels in

muscle (0.01), kidney (0.03), and liver (0.02) also showed signiﬁcant

variation (p < 0.05). The mean Zn concentrations in muscle (1.26),

kidney (1.38), and liver (1.85) indicated signiﬁcant diﬀerences (p <

0.05). In contrast, the concentrations of Ni and Mn did not show

signiﬁcant diﬀerences (p > 0.05) across the three tissues.

*indicated SDOI value showed strong dietary overlap between considered size classes

3.4 Concentration of Some Heavy Metals in

Muscle, Liver and Kidney

The concentrations of seven heavy metals in liver, muscle, and

kidney tissues of S. schall from Lake Abaya are summarized in Table

3. The ﬁndings indicated that ﬁve heavy metals - Cu, Cd, Ni, Zn,

Table 4: Mean concentrations of trace metals in muscle, kidney and liver (mean concentration in mg/kg dry weight ± standard error) of S. schall in Lake Abaya.

Element

Muscle

Kidney

Liver

Cu

Cd

Ni

0.94 ± 0.028^a

0.01 ± 0.000^a

0.05 ± 0.006^a

1.26 ± 0.169^a

0.07 ± 0.008^a

ND

1.21 ± 0.168^a

0.03 ± 0.006^b

0.05 ± 0.007^a

1.38 ± 0.179^a

0.09 ± 0.011^a

ND

3.85 ± 0.341^b

0.02 ± 0.002^a

0.04 ± 0.006^a

1.85 ± 0.153^b

0.09 ± 0.009^a

ND

Zn

Mn

Pb

Co

ND

Note: Superscript represented by diﬀerent letters indicate signiﬁcant diﬀerence (p < 0.05), ND–Not detected.

3.5 Discussion

gastropods, and ﬁsh scales. Moreover, various other workers

studying the food and feeding habits of S. schall in diﬀerent African

water bodies have indicated the polyphagus nature of the species

(Adeyemi, 2010; Akombo et al., 2014; Arame et al., 2021) .

The results of the present study indicated that S. schall feeds on

various food items including phytoplankton, zooplankton, insects,

detritus, macrophytes, ostracods, nematodes, ﬁsh scales and

Hydracarina in Lake Abaya (Table 1). From the various food items,

phytoplankton, detritus, insects, zooplankton and macrophytes

were the major food items while ostracods, nematodes, ﬁsh scales

and Hydracarina were of minor importance. Various authors

studied the feeding habits of S. schall and reported its polyphagous

nature. Hickley and Bailey (1987) studying S. schall in the Sudd

Swamps of River Nile (Sudan) have pointed out the importance of

detritus, benthic algae, macrophytes, benthic crustaceans, insects

and ﬁsh scales in its diet. Ofori-Danson (1992) working on the

ecology of some Synodontis species in Kpong Head-pond (Ghana)

have reported the dominant food items of S. schall as detritus,

insects, oligochaets, nematodes and Hirudinae. Dadebo et al.

(2012) reported a similarly diverse diet for S. schall in Lake Chamo,

including zooplankton, plant materials, insects, ﬁsh fry, ﬁsh eggs,

The high frequency and substantial volumetric contributions of

both plant materials and macro- invertebrates in the stomachs of

S. schall were a good evidence for its omnivorous feeding habits in

Lake Abaya. Various authors have also reported the omnivorous

feeding habits of S. schall in diﬀerent African inland water bodies.

Baras and Laleye (2003) reported the omnivorous feeding habit of

S. schall with a strong tendency to predation. Willoughby (1974)

described S. schall as an omnivorous species feeding on insect larvae

and nymph, ﬁsh eggs and detritus. Dadebo et al. (2012) also

reported the omnivorous feeding habits of S. schall in Lake Chamo.

Seasonal variations in the diet of S. schall were observed during

the present study (Table 2). During the dry season, the diet was

dominated by phytoplankton, detritus, insects and macrophytes

Dadebo et.al (2026)

39

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

(Table 2). The reason for the abundance of phytoplankton during

the dry season might be attributed to higher light penetration

and reduced water turbulence, which favor autotrophic production

(Drakare et al., 2002). Detritus was the second important food

of S. schall in the dry season (Table 2). The contribution of

insects was considerable during the dry season (17.4%). Among

insects, Diptera (Chironomidae larvae) was the most important

contributing more than 70% by volume of all insect groups. This

high contribution of Diptera was probably due to the ease of capture

and also their ability to ﬂourish in wide range of environmental

conditions (Drakare et al., 2002). Ofori-Danson (1992) also reported

the importance of Diptera and other insects in the diet of S, schall

in the Kpong Head-pond in Ghana.

depending on food availability and their size. According to Lalèyè

et al. (2006), large size S. schall browse on benthic deposit as can

be seen from the presence of detritus and mud in the stomachs

of large ﬁsh. The same authors also noted the importance of

ﬁsh scales in the diet of S. schall as its size increases. Similarly,

Dadebo et al. (2012) reported that ﬁsh scales become important

food items in large size S. schall in the neighboring Lake Chamo.

Bishai and GideirI (1965) found a signiﬁcant diﬀerence between

the diets of large and small S. schall in Khartoum. Several other

investigators also demonstrated that S. schall showed an ontogenetic

diet shift as a result of the change in habitat use in diﬀerent

water bodies (Araoye, 1999; Dadebo et al., 2012; Ofori-Danson,

1992). The other probable factor for such dietary variations across

size classes might be aligned with the habitat that they survive.

Juvenile S. schall hide themselves from the risk of predators (Baras

& Laleye, 2003). Similar to the present ﬁnding, Araoye (1999)

and Dadebo et al. (2012) reported that, fry and ﬁngerlings of S.

schall were usually restricted to the ﬂooded littoral zone of the lake

where they feed mainly on zooplankton, insect larvae and other

macro-invertebrates.

Macrophytes were ingested in considerable quantities during the

dry season. It is probable that the ﬁsh might ingest part of

macrophytes incidentally as they pursue their prey in the littoral

region where the prey normally seek refuge from predators

(Thomaz et al., 2025). More focused study is needed to determine

the importance of macrophytes to the nourishment of the species

by comparing the nutritive values of the plant fragments in the

fore and hind guts of the ﬁsh (Thomaz et al., 2025). During the

wet season, the contributions of zooplanktons were dominated and

widely represented by Daphnia (Table 2). The reason for this was

probably due to seasonal reproductive cycle of the cladocerans

population, which often peak during rainy season in tropical lakes

(Choedchim et al., 2017). Detritus was also an important food

item during the wet season. The source of this food category

could be the ﬂoods that introduce diﬀerent plant materials into the

lake and plant leaves falling into the lake and undergoing partial

decomposition. Araoye (1999) reported that the contribution of

plant materials and detritus in the diet of S. schall during the wet

season was high, and such food items were dispersed along the

surface water column at this period due to ﬂoods and overturn.

The concentrations of the ﬁve heavy metals detected were generally

higher in the kidney and liver compared to muscle tissue (Table 5).

For example, Cu levels were elevated in the liver relative to both

the kidney and muscle tissues, consistent with ﬁndings of Gerenfes

et al. (2019) for Enteromious species in Lake Chamo, Ethiopia and

Shahid et al. (2016) for Cyprinus carpio. This distribution can be

explained by the liver’s function in detoxiﬁcation and synthesis of

copper-binding metallothioneins, highlighting its role as a crucial

bio-indicator for evaluating Cu levels in aquatic ecosystems (Javed

& Usmani, 2013). In terms of Cd levels, S. schall from Lake Abaya

exhibited a muscle tissue concentration of 0.01 mg/kg, which is

higher than that of bream (0.009 mg/kg) and mandarin ﬁsh (0.0009

mg/kg) from Poyang Lake (Wei et al., 2014). Additionally, Cd

concentrations ranging from 0.001 to 0.009 mg/kg were identiﬁed

in eleven ﬁsh species from Rio de Janeiro State, Brazil (Medeiros

et al., 2012), but the ﬁnding in this study is lower than the 0.03 to 1.57

mg/kg detected in ﬁsh from the Pearl River Delta (Cheung et al.,

2008). All observed concentrations of the detected heavy metals fall

below the limits set by the EU (2001), TFC (2002) and FAO (1983)

guideline for human consumption.

From the results of the present study, it was evident that S.

schall showed a clear ontogenetic dietary shift during its life cycle

(Figure 1). Smaller individuals relied widely on phytoplankton

and zooplankton, whereas larger ﬁsh incorporated more insects,

detritus, and macrophytes. Bishai and GideirI (1965) reported that

some members of the genus Synodontis switch from benthic feeding

to surface feeding or vice versa by using ventrally positioned mouth

Table 5: Comparisons of Concentration of Trace Metals in Fish Muscle Relative to the Standards (mg/kg dry weight).

Parameter (guidelines)

Cu

Zn

Mn

Cd

0.01

–

Ni

0.05

–

References

Present study in ﬁsh muscle

0.94

1.26

0.07

FAO

EU

30

4

50

–

30

–

FAO (1983)

EU (2002)

TFC (2002)

Turkish Food Codex

20

50

20

Mn were found in all three tissues, while Pb and Co were absent.

In the liver, the concentration ranking was Cu > Zn > Mn > Ni >

Cd, whereas in muscle and kidney tissues, it was Zn > Cu > Mn

> Ni > Cd. From the present study, S. schall is an omnivorous

in its feeding strategy. Overall, heavy metal concentrations were

generally higher in the kidney and liver than in muscle. The level

in muscle showed below FAO and EU maximum acceptable limits

for human consumption. The present result suggested conducting

4 Conclusion

This study has clearly shown that S. schall in Lake Abaya ingests

a wide range of plant based and animal origin of food categories.

However, the diet composition of S. schall was diﬀerent based on

their size classes and season of sampling. With the exception of

some size classes, strong dietary overlap was seen across diﬀerent

size classes. Trace metals analysis revealed that Cu, Cd, Ni, Zn, and

Dadebo et.al (2026)

40

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 34-42

further heavy metal analysis is required based on simultaneous

study including water quality analysis, sediment analysis, and the

interaction between feeding ecology with trace metal concentration

for further comparisons.

River in Northern Benin. International Journal of Aquatic

Biology, 9, 41–54. https://doi.org/10.22034/ijab.v9i1.973

Araoye, P. (1999). Spatio-temporal distribution of the ﬁsh

Synodontis schall (Teleostei: Mochokidae) in Asa lake,

Ilorin, Nigeria. Revista De Biología Tropical, 47, 1061–1066.

https://doi.org/10.15517/rbt.v47i4.19311

Baras, E., & Laleye, P. (2003). Ecology and behavior of catﬁsh. In

G. Arratia, B. G. Kapoor, M. Chardon, & R. Diogo (Eds.),

Catﬁsh. Science Publishers.

Baxter, R. (2002). In: Ethiopian Rift Valley Lakes. In C. Tudorancea

& W. D. Taylor (Eds.). Backhuys Publishers.

Conflict of Interest

None declared.

Funding

Bishai, H., & GideirI, Y. (1965). Studies on the biology of the genus

Synodontis at Khartoum: II. Food and feeding habits.

Hydrobiologia, 26, 98–113. https : / / doi . org / 10 . 1007 /

BF00731551

This research was supported by NORAD project

Acknowledgements

Bowen, S. H. (1996). Quantitative description of the diet. In Fisheries

techniques (2nd, pp. 513–532). American Fisheries Society.

Cheung, K., Leung, H., & Wong, M. H. (2008). Metal concentrations

of common freshwater and marine ﬁsh from the Pearl

River Delta, South China. Archives of Environmental

Contamination and Toxicology, 54, 705–715. https : / / doi .

org/10.1007/s00244-007-9064-7

The authors are grateful to the Department of Aquatic Sciences,

Fisheries and Aquaculture for providing laboratory facilities and

the logistics for the ﬁeld trips. Dr. Andargachew Gedebo, NORAD

project coordinator and Hawassa University are acknowledged

for providing a vehicle for the ﬁeld trips. We thank ﬁsherman

Asaminew Matte for his assistance during sample collection.

Choedchim, W., Van-Damme, K., & Maiphae, S. (2017). Spatial and

temporal variation of Cladocera in a tropical shallow lake.

International Journal of Limnology, 53, 233–252. https://doi.

org/10.1051/limn/2017006

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ARTICLE

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.5S

Intuitionistic Fuzzy Multi-objective

Optimization Method for Determination of

ARTICLE HISTORY

Received: 03 March, 2026

Accepted: 04 June June, 2026

Published Online: 10 June, 2025

Optimal Cropping Pattern

Habtamu Tsegaye Teferi ^1,∗

�

CITATION

Teferi H.T (2026). Intuitionistic Fuzzy

Multi-objective Optimization Method for

Determination of Optimal Cropping

Pattern. East African Journal of

¹Department of Mathematics, College of Natural and Computational Sciences, Wolkite University, Ethiopia

^∗Corresponding author: hbtmtsgy@gmail.com

Biophysical and Computational

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.5S.43-63

Abstract

OPEN ACCESS

Agriculture has become a difficult occupation due to inadequate farming resources and cultivation risks.

Thus, proficient utilization of resources alongside risk-alleviation strategies is essential aspect to realize

sustainable farm benefits. Most of the earlier studies have reported the capability of Operations research

in solving agricultural problems and enhancing farm productivity. However, they have not addressed

effectively the distinctive nature of decision-makers, uncertainties, and associated risks of agriculture. This

study mainly aims to fill these lacunae by applying an intuitionistic fuzzy optimization method to determine

optimal cropping pattern that maximizes overall net benefits, minimizing cultivation costs and workforce

concurrently with regard to procurable agricultural resources. For this purpose, an effective multi-objective

optimization method is formulated, and its effectiveness is verified through proof and numerical example.

The comparison between existing and proposed cropping patterns showed that the proposed patterns

offer several advantages in enhancing overall agricultural benefits sustainably for farmers in the study

area.

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Multi-objective optimization; Intuitionistic fuzzy optimization; Agricultural production

planning; Cropping pattern.

models to support farm activities and contribute to feeding the steadily

growing population (Carravilla & Oliveira, 2013; Weintraub & Romero,

1 Introduction

Today, the burgeoning world’s population increases the demand for

agricultural products and this in turn increasing pressure on the resources

required for production (Wang, 2022). However, to meet the ever-escalating

demand for nourishment, crop production must be boosted either by

increasing land area for cultivation or by enhancing production per unit

area of land (FAO, 2017; Mirkarimi et al., 2013). Since agricultural

resources for farming are very limited all over the world (Guo et al., 2021),

increasing land area for crop production regardless of limiting factors

causes deterioration of available resources. Moreover, climate change,

drought, political disputes, and disease are holding back agricultural

advancement and remain key determinants of food security and poverty

alleviation (Luo et al., 2023; Zerssa et al., 2021). Therefore, it is crucial

to design means of eﬃcient utilization of resources and risk alleviation

strategies to improve the overall agricultural returns sustainably.

2006).

Agriculture is the primary sector of the Ethiopian economy, employing

approximately 85% of the country’s population as workers (Zerssa et al.,

2021). It contributes 50% of Ethiopia’s gross domestic product and earns

over 90% of the foreign exchange (Haile & Kasa, 2015; Zerssa et al., 2021).

The country’s goal for achieving overall economic growth mainly depends

on the accomplishment of agriculture sector (Haile & Kasa, 2015).

Despite the country’s vast irrigable land and water resources, farming is

weather dependent and the production of which depends heavily on the

availability of rainfall (Awulachew & Ayana, 2011). Consequently, most

farmers are exposed to inconsistent rainfall patterns and weather conditions

(Zerssa et al., 2021). Even though Ethiopia is vulnerable to the vagaries of

natural weather conditions, it has substantial agricultural potential because

of its vast areas of fertile land, enormous labor, abundant water resources,

and diverse climate conditions (Awulachew & Ayana, 2011; Kelbore, 2014).

In this regard, the decision of agricultural stakeholders has its own impact

on the achievement of the desired objectives. Even if decision-making for

the optimum utilization of resources is a challenging task for farmers and

agricultural managers, it can be scientiﬁcally addressed using optimization

Due to seasonal variation, Ethiopia has three distinct seasons: Kiremt

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

(June – September), Bega (October – January), and Belg (February – May)

(Gebremichael et al., 2014). Kiremt is the main rainy season, accounting for

about 90% of agricultural production, while Bega is dry and Belg is a short

rainy season contributing the remaining 10% of production (Kelbore, 2014).

literature is reviewed. The optimization method for addressing the practical

problem is outlined in Section 3. A comparative analysis of the proposed

method is presented in Section 4. Section 5 provides the model application

in three subsections: study area description, data collection and analysis,

and problem formulation. A detailed analysis of the results is given in

Section 6. Section 7 summarizes the ﬁndings, discusses limitations, and

provides recommendations for future research.

A cropping pattern is the proportion of land area under various crops that

changes over space and time (G. Singh, 2012). It is the annual succession of

diﬀerent crops and fallow in a particular region and can be reported at the

farm level to address the collective issues of a farming system (Andrews &

Kassam, 1976).

2 Literature Review

Agriculture is one of the ﬁelds where Operations research (OR) models were

ﬁrst employed and they have been extensively applied (Rădulescu et al.,

2014). The capability of addressing MOO problems for decision-making

helps out OR to play a fundamental role in agriculture (Rădulescu et al.,

2014; Weintraub & Romero, 2006).

The principal objective of optimal cropping pattern (OCP) is to identify

the combination of several crops to be cultivated which maximizes the net

return of farming by managing agricultural risks with respect to available

resources (Ouda et al., 2017).

Carravilla and Oliveira (2013) reviewed studies demonstrating the

applications of OR in agriculture at farm and sector levels. Depending

on problem complexity, some used linear programming (LP), while others

employed multi-objective and fuzzy optimization models to address APP

problems.

As the population grows, agricultural resources have been decreasing

and the situation worsens with the spread of drought (Carravilla &

Oliveira, 2013). These circumstances continue to generate many agricultural

inquiries in search for more productive alternatives on a given land area

for the optimal utilization of other agricultural resources (Paudel, 2016).

Therefore, OCP is one of the important and feasible mechanisms to increase

productivity with other integrated scientiﬁc agricultural practices (Luo

et al., 2023; Ouda et al., 2017).

Weintraub and Romero (2006) demonstrated the potential of OR in the

management of agricultural resources and forestry, and their advantages

in simplifying DM in farming activities. In their article, the applications of

OR in APP problems at the farm and regional levels were comprehensively

reviewed. They address uncertainties, risks, environmental conservation,

and discuss future research directions in these areas.

Determining the OCP remains a complex task for farmers and managers

(Pawar et al., 2026), as decisions on machinery selection, input use,

operation timing, and cultivation practices must be made in each cropping

season (Duan et al., 2021).

Environmental, social, and economic factors make agricultural data

inherently uncertain (Bairwa et al., 2013), leading to decision-making under

ill-deﬁned objectives and constraints (Li et al., 2017). Accordingly, risks in

agriculture are better represented with fuzzy numbers than crisp values,

prompting the development and application of fuzzy multi-objective

optimization (FMOO) methods to handle APP problems.

In Ethiopia, where most farmers practice traditional agriculture, these

decisions are largely based on experience, ﬂuctuating market prices,

and other operational factors (Kelbore, 2014). Conversely, policies and

guidelines delivered through extension services often fail to account for

multiple agricultural objectives and constraints, such as weather variability

and production risk. Therefore, more eﬀective strategies are required

through the use of optimization models.

Although many researchers employed FMOO approaches to deal with APP

problems (Amini, 2015; Biswas & Pal, 2005; Gupta et al., 2000; Mirajkar &

Patel, 2012; Mirkarimi et al., 2013; Rasikh et al., 2024; Wang, 2022; X. Zeng

et al., 2010) their studies were limited in scope and did not represent the

real nature of the problems very well (Mahapatra & Roy, 2009). This

is due to insuﬃcient information, incomplete attributes, ill-deﬁnedness,

uncertainties, and vagueness in every aspect of MOO problems (Sen et al.,

2018). Consequently, diﬀerent advanced generalizations are ascertained.

From that, IFS, which is originated by Atanassov (1986), is an eﬀective

generalization of fuzzy sets. It has been used in a wide range of operations

because of its ability to address uncertainties and vagueness in practical

problems. Thus, intuitionistic fuzzy optimization (IFO) (Angelov, 1995) was

introduced to handle diﬀerent pragmatic problems.

The problem of optimizing multiple goals simultaneously under given

constraints is called multi-objective optimization (MOO). It involves

concurrent optimization of incommensurate and conﬂicting goals subject

to diﬀerent constraints (V. Singh & Yadav, 2018).

Many real-life problems are inherently characterized by multiple and

conﬂicting goals with uncertainties (Gupta et al., 2000). So, it is diﬃcult

to deal with such problems using classical optimization techniques.

Hence, intuitionistic fuzzy sets (IFSs) can be used to represent insuﬃcient

information, imprecise concept, uncertainties and the diverse perspectives

of decision-makers (DMs) in a more generalized way compared to fuzzy

sets (Roszkowska et al., 2024). Thus, intuitionistic fuzzy modelling is more

relevant than other classical optimization methods (Pawar et al., 2026).

IFO technique is a relatively recent research ﬁeld in contrast to fuzzy

optimization approaches (Angelov, 1995). It enhances understanding of

the addressed problems and provides valuable insights into their nature

(Roszkowska et al., 2024). Moreover, the output of an investigation

employing IFO is a more valuable analytical means for researchers,

practitioners, and experts.

Several researchers have attempted to address agricultural production

planning (APP) problems employing various MOO methods to recommend

an alternative cropping system for improved outcomes (Luo et al., 2023;

Weintraub & Romero, 2006). However, the ambiguity of the parameters

in the problem, inconsistent natural conditions, the distinctive perspective

of DMs, and associated operational risks in agriculture have not been well

addressed in their studies.

In APP problem, interactions among various natural entities and correlated

factors complicate the management process. These factors enforce DMs to

use advanced optimization models for proﬁcient usage of resources and to

gain better overall beneﬁts.

To overcome the above diﬃculties, it is imperative to formulate an eﬀective

optimization model and design OCP at the farm level by considering the

available resources to assist sustainable crop production.

This study aims to investigate the application of an intuitionistic fuzzy

multi-objective optimization (IFMOO) model and to propose OCP for the

study area that improves farmers’ welfare concerning scarce resources and

reduces various challenges of farming.

Nishad and Singh (2015) employed an intuitionistic fuzzy goal

programming to resolve the land use planning problem. They considered

the agricultural system undertaken by Biswas and Pal, 2005 in which several

seasonal crops were cultivated in a year and diﬀerent productive resources

are taken into account in the model. Their study revealed that IFO method

gives better results in all aspects compared to the results obtained by the

fuzzy optimization method.

The rest of this paper is organized as follows. In Section 2, the related

Teferi H.T.,(2026)

44

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Li et al. (2017) formulated an IFMOO model that incorporates MOO,

nonlinear programming (NLP) and intuitionistic fuzzy number (IFN) to

deal with the uncertainties of conﬂicting targets in irrigated agriculture to

support sustainable farming. They employed an IFMOO model to allocate

limited accessible water to rice during growth stages to maximize crop yield,

minimize utilized water and cost of water supply in Heping irrigation area

of northeast China.

3 IFO Method

3.1 Preliminaries

˜

Deﬁnition 3.1.1 (Atanassov, 1986). An IFS ꢀ in the universe - is given by

˜

ꢀ = {hG, ꢀ (G), ꢁ (G)i:G ∈ -}, where ꢀ (G), ꢁ (G), ꢂ (G):

˜

ꢀ

˜

ꢀ

˜

ꢀ

- → [0, 1] such that

0 ≤ ꢀ (G) + ꢁ (G) ≤ 1 and ꢂ (G) = 1 − (ꢀ (G) + ꢁ (G)),

˜

ꢀ

˜

ꢀ

∈

ꢀ

∀G

-. The values ꢀ^ꢀ(G), ꢁ (G) and ꢂ (G) describe the degree

˜

ꢀ

˜

ꢀ

Li et al. (2019) used a multi-objective NLP model in an intuitionistic fuzzy

environment (IFE) for the management of the water-energy-food nexus in

irrigational agriculture, considering the cropping system of the Heihe River

basin in northwest China. In their study, both optimistic and pessimistic

views of DMs were considered under diﬀerent scenarios to maximize

system proﬁt and minimize carbon dioxide emissions, subject to water,

energy, land, and other input resources for cultivating wheat, corn, and

vegetables in the three regions of the basin.

˜

of membership, non-membership, and indeterminacy of G being in ꢀ,

respectively.

˜

Deﬁnition 3.1.2 (Mahapatra & Roy, 2009). A triangular IFN ꢀ is an IFS,

represented as ꢀ = h0 , 0 , 0 ; 0 , 0 , 0 i, where 0 ≤ 0 ≤ 0 ≤ 0 ≤ 0₃⁰,

0

˜

1

2

3

2

1

2

3

1

3

1

with

G−0

1

−G







,

if 0 < G ≤ 0

1

2

3

0

−0

2

1

0

3

ꢀ (G) =

if 0 ≤ G < 0

˜

2

ꢀ

0

−0

2

0³,

otherwise,





Pawar et al., 2022 applied an IFMOO method to determine the OCP of the

Ukai irrigation area in India. Their approach combined minimizing the

aggregated hesitation level of objectives with maximizing the minimum

membership degree and minimizing the maximum non-membership

and

0

−G

2







,

if 0₁⁰< G ≤ 0

0

2

0

−0

2

1

degree.

This produced an OCP that maximized net returns and

G−0

if 0 ≤ G < 0⁰

2

ꢁ (G) =

˜

ꢀ

0

2

3

employment, while minimizing farming expenditures under constraints of

arable land and irrigation water.

0

−0

2



1³,

otherwise.





Li et al., 2020 developed an optimization model for sustainable irrigated

agriculture combining IFMOO, nonlinear mixed-integer, and fuzzy

credibility-constrained programming. Their study aimed to allocate water

and farmland to crops across subareas and seasons to optimize net returns

while considering socioeconomic and ecological objectives. The algorithm

was successfully applied to crop planning in the Heping irrigation area of

China.

Deﬁnition 3.1.3 (S. K. Singh & Yadav, 2015a).

0

˜

Let ꢀ = h0 , 0 , 0 ; 0 , 0 , 0 i be a triangular IFN, the accuracy function of

1

2

3

2

1

3

0

+0 +40 +0 +0

3

1

2

3

1

˜

ꢀ is denoted by Γ(ꢀ) and deﬁned as Γ(ꢀ) =

Accuracy function Γ is used to defuzzify IFNs.

.

8

In handling of practical problems, vagueness and uncertainty can be

addressed in an IFE by considering parameters as IFNs and treating

inequality and equality as intuitionistic fuzzy inequalities and equality.

Based on this principle, an IFMOO problem is formulated (V. Singh & Yadav,

2018)

Kousar et al., 2022 formulated an IFO method considering all parameters

and variables as IFN to optimize the production of ﬁve types of fruits

in Baluchistan region, Pakistan.Their results showed that the fully IFO

technique has an imperative advantage to consider the ﬂuctuating nature

of prices and input resources more eﬃciently.

e

f

max { 5 (-), 5 (-), ..., 5 ₁(-)},

1

2

:

e

min { 5_{: +1}(-), 5_{: +2}(-), ..., 5_:(-)}

1

L. Zeng et al. (2020) studied sustainable resource management for the

Zhanghe Reservoir irrigation in central China using an interval stochastic

multi-objective mixed-integer model in an IFE. Their objectives were to

maximize crop production, hydroelectricity, and water allocation while

optimizing cropland under constraints of water availability, crop demand,

land policy, and electricity generation. The results demonstrated eﬃcient

farmland and water management to support food security and mitigate

global warming sustainably.

subject to

(1)

,e(-) - e2₈, 8 = 1, 2, . . . , < ,

,e(-) ¥ e2 , 8 = <₁+ 1, <₁¹+ 2, . . . , < ,

8

2

8

,e(-) ≈ e2₈, 8 = <₂+ 1, <₂+ 2, . . . , <,

8

- ≥ 0,

e

where 5₉(-) and ,e(-) are intuitionistic fuzzy functions, - is n-dimensional

8

A few researchers have used IFMOO models to handle uncertainties and

risks in APP problems. As noted above, investigators have applied various

IFMOO techniques to suggest alternative cropping patterns for better

outcomes. However, they have not eﬀectively addressed diﬀerences among

DMs, uncertainties, and risks that severely aﬀect crop production.

For instance, in most studies, parameters are not consistently considered

as IFNs, constraints are not incorporated into the solution framework,

DM preferences and stakeholder interactions are overlooked, and the

sustainability of objectives is inadequately addressed.

variable ∀9 = 1, 2, ..., : and ∀8 = 1, 2, ..., <.

To reformulate problem (1) as an equivalent crisp MOO problem, each

parameter has to be defuzziﬁed applying the accuracy function (Rădulescu

et al., 2014). Then, the degrees of acceptance and rejection of the objectives

have to be described to form a single objective optimization problem.

Accordingly, the solution of the IFMOO problem can be found by solving

such an equivalent single objective problem (S. K. Singh & Yadav, 2015b).

The IFMOO problem (1) can be changed into the following equivalent

deterministic MOO problem (V. Singh & Yadav, 2018):

To overcome these shortcomings, the present study formulates an IFMOO

model to design an OCP that maximizes sustainable net beneﬁts for a

large-scale farm (LSF) and farmers in Gefersa kebele. It also aids farmers,

development agents, and extension workers in complex decision-making

under multiple perspectives and scenarios for a LSF in Abeshge district.

max { 5 (-), 5 (-), ..., 5 (-)},

1

2

:

min { 5_{: +1}(-), 5_{: +2}(-)¹, ..., 5_:(-)}

1

subject to

,₈(-) ≤ 2₈, 8 = 1, 2, . . . , < ,

(2)

, (-) ≥ 2 , 8 = <₁+ 1, <₁¹+ 2, . . . , < ,

2

8

,₈(-) = 2₈, 8 = <₂+ 1, <₂+ 2, ..., <,

- ≥ 0,

Teferi H.T.,(2026)

45

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

where 5₉(-) and ,₈(-) are real-valued functions and 2₈∈ R,

∀9 = 1, 2, ..., : and ∀8 = 1, 2, ..., <.

of the maximization problem are respectively described as follows:



0,

if 5₉(-) ≤ !₉



Deﬁnition 3.1.4 (Cristofari et al., 2024). Let S be the set of all feasible

solutions of problem (2), and let -, -^∗∈ S. Then, -^∗is said to be a

Pareto optimal solution (POS) for problem (2) if and only if there does





5 (-)−!

9

ꢀ_*( 5₉(-)) =

(4)

,

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,

9

* −!

1,

9



not exist - ∈ S such that 5₉(-^∗) ≤ 5₉(-), ∀9 = 1, 2, . . . , : , and ∃9 ∈



1

{1, 2, . . . , : } such that 5 (-^∗) < 5 (-).

1

9

and



1,

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,

T heorem 3.1.1 (Xu & Cai, 2012; Xu & Yager, 2006).





* − 5 (-)

˜

9

Let ꢁ = (ꢀ , ꢁ ), 9 = 1, 2, ..., : be a collection of intuitionistic fuzzy values,

˜

9

ꢁ

9

ꢁ_*( 5₉(-)) =

,

(5)

9

* −(! −ꢄ )

0,

9



and let F , F , ..., F be the corresponding weights, where F₉∈ (0, 1) and

1

2

:

Í



:

₉₌₁F₉= 1. Then, the aggregated value using the intuitionistic fuzzy

where ꢄ₉is a tolerance value of the 9th objective and deﬁned as ꢄ₉

ꢃ(*₉− ! ) and ꢃ ∈ (0, 1), ∀9 = 1, 2, ..., : . If *₉= !₉, then we deﬁne

=

weighted geometric (IFWG) operator is given by

1

9

ꢀ_*( 5₉(-)) = 1.

9

The respective exponential membership function (ꢀ_*^ꢂ( 5₉(-))) and

:

9

Ö

IFWG (ꢁ , ꢁ , ..., ꢁ ) =

ꢀ^F_˜⁹, 1 −

(1 − ꢁ )

(3)

non-membership function (ꢁ_*^ꢂ( 5₉(-))) for the 9th objective respectively

©

-

ª

®

F

9

˜

F

1

2

˜

:

ꢁ

9

ꢁ

9

9=1

«

¬

deﬁned as



0₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,



3.2 The Solution Method





((* − 5 (-))/(* −! )) −3

9

ꢀ_*^ꢂ( 5₉(-)) =

(6)

4

−4

,

Most existing studies consider the optimistic variant of the problem (Kis

et al., 2021), paying little attention to alternative perspectives in the solution

process. However, this consideration has its own limitations in addressing

practical problems. Due to the inconsistent nature of human judgment,

DMs may deviate from their initial standpoint after evaluating the obtained

solution against the intended goals. Consequently, existing methods

overlook this important aspect of DM judgment and insights. Therefore,

identifying the inﬂuence of optimistic, pessimistic, and mixed perspectives

is highly valuable for obtaining consistent and robust solutions to MOO

problems (Chen et al., 2023; Mahajan & Gupta, 2021b). The main diﬀerence

among these three perspectives arises from the choice of violation and

tolerance values used to determine the non-membership degrees, while the

membership function remains identical across all cases.

−3

9



1−4



1,



and





1₋, ₃

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉.





(( 5 (-)+ꢄ −! )/(* +ꢄ −! )) −3

9

ꢁ_*^ꢂ( 5₉(-)) =

4

−4

,

−3

9



1−4



0,



(7)

where, 3₉is the shape parameter.

The linear membership and non-membership functions for minimization

objectives can be described as

In an IFE, parameter values, aspiration levels, coeﬃcients of the objectives

and constraints, as well as the equalities and inequalities in the model, are

ﬂexible. Consequently, the maximum and minimum allowable values of the

constraints and objectives can vary.



1,

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,





* − 5 (-)

9

ꢀ_!( 5₉(-)) =

(8)

,

9

* −!

0,

9





To incorporate ﬂexibility in the constraints, a violation parameter ℓ₈is

and

associated with the 8th constraint, 8 = 1, 2, . . . , < , as speciﬁed by the DMs.

1

For constraints of the form ≤, the upper bound 2₈is relaxed to 2₈+ ꢃ(ℓ₈) in



0,

if 5₉(-) ≤ !₉





the solution procedure (Tsegaye et al., 2021).

5 (-)−!

9

ꢁ_!( 5₉(-)) =

,

if !₉< 5₉(-) < *₉+ ꢄ₉

(9)

9

(* +ꢄ )−!

1,

9



if 5₉(-) ≥ *₉+ ꢄ₉,

The membership and non-membership functions for each objective are

described based on the diﬀerence between the maximum *₉and minimum

!₉achievable goals, which are identiﬁed using a table of extreme solutions.

Then, the tolerance variables ꢄ₉and ꢅ₉are obtained by using



where ꢄ₉is a tolerance value of the 9th objective and deﬁned as ꢄ₉

=

ꢃ(* − ! ) and ꢃ ∈ (0, 1), ∀9 = : + 1, : + 2, ..., :. If *₉= !₉, then we

1

deﬁne ꢀ_!⁹( 5₉(-)) = 1.

9

The corresponding exponential non-membership function (ꢀ^ꢂ_!( 5₉(-))) and

ꢄ₉= ꢃ(*₉− !₉) and ꢅ₉= ꢃ(*₉− (!₉− ꢄ₉)) = ꢄ₉(1 + ꢃ) where

9

!₉= min{ 5₉(-)}, *₉= max{ 5₉(-)}, and ꢃ ∈ (0, 1), 9 = 1, 2, ..., :.

non-membership function (ꢁ_!^ꢂ( 5₉(-))) for the 9th objective respectively

9

deﬁned as

The membership and non-membership functions are generally

characterized by nonlinear behavior due to instantaneous variation at each

solution point. Among nonlinear functions, the exponential function is

preferable for the IFMOO problem because of its eﬃciency and ﬂexibility in

evaluating the marginal values of objectives and constraints (Ahmadini

& Ahmad, 2021; Mahajan & Gupta, 2021a). Accordingly, exponential

membership and non-membership functions are formulated to describe

the optimistic, pessimistic, and mixed features and to obtain an eﬃcient

solution for the IFMOO problem.



0₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,





(( 5 (-)−! )/(* −! )) −3

9

ꢀ^ꢂ_!( 5₉(-)) =

(10)

4

−4

,

−3

9



1−4



1,



and





1₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉.





((* +ꢄ − 5 (-))/(* +ꢄ −! )) −3

9 9

9

ꢁ_!^ꢂ( 5₉(-)) =

4

−4

,

3.2.1 The optimistic approach

−3

9



1−4



In the optimistic approach, DM takes a liberal view of rejection (V.

0,



Singh et al., 2021). Therefore, the linear membership (ꢀ_*( 5₉(-))) and

(11)

9

non-membership (ꢁ_*( 5₉(-))) functions for the 9th objective function 5₉(-)

Their general shape is shown in Figure 1.

9

Teferi H.T.,(2026)

46

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

(a)

(b)

(a)

(b)

Figure 2: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under a pessimistic approach.

Figure 1: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under the optimistic approach.

3.2.2 The mixed approach

In a mixed approach, the DM is not ﬂexible in rejecting and is not capable

3.2.2 The pessimistic approach

of extra acceptance (V. Singh & Yadav, 2018). The linear non-membership

In the pessimistic approach, the DM is presumably extra cautious about

function ꢁ_*( 5₉(-))) of the 9th objective function 5₉(-) to the maximization

9

acceptance (V. Singh et al., 2021). The linear non-membership function

problem is deﬁned as

ꢁ_*( 5₉(-)) of the 9th objective 5₉(-) under a pessimistic approach to the

9

maximization problem is expressed as



1₍,_{! +ꢅ )− 5 (-)}

if 5₉(-) ≤ !₉− ꢄ₉





9



ꢁ_*( 5₉(-)) =

,

if !₉− ꢄ₉< 5₉(-) < !₉+ ꢅ₉

if 5₉(-) ≥ !₉+ ꢅ₉,

(16)

1,

if 5₉(-) ≤ !₉





9

(! +ꢅ )−(! −ꢄ )

9



(! +ꢄ )− 5 (-)

9

0,

ꢁ_*( 5₉(-)) =

,

if !₉< 5₉(-) < !₉+ ꢄ₉

if 5₉(-) ≥ !₉+ ꢄ₉,

(12)



9

(! +ꢄ )−!

0,

9





where ꢄ₉and ꢅ₉are the tolerance variables of the 9th objective and deﬁned

as ꢄ₉= ꢃ(*₉− !₉),

∀9 = 1, 2, ..., : .

1

ꢅ = ꢄ (1 + ꢃ), ꢃ ∈ (0, 1), 9 = 1, 2, ..., : .

1

9

The corresponding exponential non-membership function ꢁ_*^ꢂ( 5₉(-)) is

9

deﬁned as

constructed as



1₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < !₉+ ꢄ₉

if 5₉(-) ≥ !₉+ ꢄ₉.









1₋, ₃

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < !₉+ ꢅ₉

if 5₉(-) ≥ !₉+ ꢅ₉.

(( 5 (-)−! )/ꢄ

)

−3

9

ꢁ_*^ꢂ( 5₉(-)) =

(13)

(( 5 (-)−(! −ꢄ ))/(ꢅ +ꢄ )) −3

4

−4





,

9

−3

ꢁ_*^ꢂ( 5₉(-)) =

4

−4

9



1−4

,

−3

9



0,

1−4



0,



(17)

The linear non-membership function ꢁ_!( 5₉(-)) of the 9th objective 5₉(-)

9

The linear non-membership function ꢁ_!( 5₉(-))) of the 9th objective function

9

under a pessimistic approach to the minimization problem is expressed as

5₉(-) to the minimization problem is deﬁned as



0,

if 5₉(-) ≤ *₉− ꢄ₉

if *₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,







5 (-)−(* −ꢄ )

1,

if 5₉(-) ≤ *₉− ꢅ₉

if *₉− ꢅ₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉,

9





ꢁ_!( 5₉(-)) =

,

(14)

5 (-)−(* −ꢅ )

9

* −(* −ꢄ )

1,

9



ꢁ_!( 5₉(-)) =

,

(18)

9

(* +ꢄ )−(* −ꢅ )

0,

9









∀9 = :₁+ 1, :₁+ 2, ..., :.

The corresponding exponential non-membership function ꢁ_!^ꢂ( 5₉(-)) is

where ꢄ₉and ꢅ₉are the tolerance variables of the 9th objective and deﬁned

9

as ꢄ₉= ꢃ(*₉− !₉), ꢅ₉= ꢄ (1 + ꢃ), ꢃ ∈ (0, 1), 9 = : + 1, : + 2, ..., :.

1

9

deﬁned as

The corresponding exponential non-membership function ꢁ_!^ꢂ( 5₉(-)) is

9



0₋, ₃

if 5₉(-) ≤ *₉− ꢄ₉

if *₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉.



constructed as





(* − 5 (-))/ꢄ

)

−3

9

ꢁ_!^ꢂ( 5₉(-)) =

(15)

4

−4

,

−3

9





0₋, ₃

if 5₉(-) ≤ *₉− ꢅ₉

if *₉− ꢅ₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉.

1−4





1,

((* +ꢄ )− 5 (-))/(ꢅ +ꢄ )) −3



9

ꢁ_!^ꢂ( 5₉(-)) =

4

−4

,

−3

9



1−4



1,



Their possible shape is shown in Figure 2.

(19)

Their general shape is shown in Figure 3.

Teferi H.T.,(2026)

47

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Problem (21) can be expressed based on the DM’s perspectives, using

the membership and non-membership functions constructed from Sections

3.2.1 to 3.2.3, as presented below.

From an optimistic viewpoint, problem (21) can be expressed as

:

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

«

¬

«

¬

subject to

3

(1−((* − 5 (-))/(* −! )))

3

9

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

>

1

9

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(22)

(23)

(24)

(1−(( 5 (-)−(! −ꢄ ))/(* −(! −ꢄ ))))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

(1−(((* +ꢄ )− 5 (-))/((* +ꢄ )−! )))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(a)

(b)

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

Figure 3: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under the mixed approach.

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

8

- ≥ 0.

After converting an IFMOO problem into its equivalent crisp MOO

problem using the accuracy function (S. K. Singh & Yadav, 2015a), the

membership and non-membership functions are constructed based on the

DMs’ viewpoints. Accordingly, aggregation operators have been proposed

(Mahajan & Gupta, 2021b; S. K. Singh & Yadav, 2015b; V. Singh &

Yadav, 2018) to combine the membership and non-membership functions.

However, such aggregation methods have certain limitations, as they fail

to consider for both the satisfaction and dissatisfaction associated with

all objectives. To address this limitation, the IFWG operator, previously

applied to various multicriteria decision-making problems, is extended in

this study to the MOO problem.

The pessimistic variant of problem (21) described as

:

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

«

¬

«

¬

subject to

3

(1−((* − 5 (-))/(* −! )))

3

9

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

>

1

9

To formulate a single aggregation operator based on Theorem 3.1.1, a

multiplicative combination is employed to integrate the independently

deﬁned degrees of acceptance and rejection. This approach helps to

emphasize the interaction between the two degrees. Accordingly, the

aggregation operator can be expressed as

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(1−(( 5 (-)−! )/ꢄ ))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

(1−((* − 5 (-))/ꢄ ))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

:

1

9

Ö

©

-

ª

©

ª

®

9

F

9

/(ꢀ, ꢁ) =

ꢀ^F₉× 1 − (1 −

(1 − ꢁ₉)

)

®

-

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

9=1

«

¬

«

¬

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

8

(20)

:

- ≥ 0.

Ö

ꢀ^F₉

×

(1 − ꢁ₉)

©

-

ª

®

©

-

ª

9

F

9

=

®

9=1

«

¬

«

¬

For the mixed approach, problem (21) is expressed as

Therefore, the IFMOO problem (1) can be solved using an equivalent crisp

model by employing the aggregation operator (20) as follows:

:

Ö

©

-

ª

®

©

-

ª

®

9

F

9

:

max

ꢆ^F_>

×

(1 − ꢇ_>

)

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

9

«

¬

«

¬

9=1

«

¬

«

¬

subject to

ꢀ_*^ꢂ( 5 (-)) ≥ ꢆ_>, 9 = 1, 2, ..., :

ꢀ^ꢂ_!( 5₉(-)) ≥ ꢆ_>, 9 = :₁+ 1, :₁+ 2, ..., :

ꢁ_*^ꢂ( 5 (-)) ≤ ꢇ_>, 9 = 1, 2, ..., :

ꢁ_!^ꢂ( 5₉(-)) ≤ ꢇ_>, 9 = :₁+ 1, :₁+ 2, ..., :

0 ≤ ꢆ_>+ ꢇ_>≤ 1, 9 = 1, 2, ..., : , :₁+ 1, :₁+ 2, ..., :

0 ≤ ꢇ_>≤ ꢆ_>, 9 = 1, 2, ..., : , :₁+ 1, :₁+ 2, ..., :

3

(1−((* − 5 (-))/(* −! )))

3

9

1

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

9

>

9

1

9

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(1−(( 5 (-)−(! −ꢄ ))/(ꢄ +ꢅ )))

3

9

(21)

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

1

9

(1−(((* +ꢄ )− 5 (-))/(ꢄ +ꢅ )))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

1

9

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

1

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

9

8

- ≥ 0.

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

1

8

- ≥ 0,

T heorem 3.2.1. A unique optimal solution (-^∗, ꢆ^∗, ꢇ^∗) of problem (22)

where F₉is the weight assigned to the 9th objective such that

F₉∈ (0, 1) and ₉₌₁F₉= 1.

Í

:

corresponds to a POS -^∗of problem (2), where ꢆ^∗= (ꢆ^∗_>, ꢆ^∗_>, ..., ꢆ^∗_>

)

1

2

:

1

Teferi H.T.,(2026)

48

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

and ꢇ^∗= (ꢇ^∗_>, ꢇ^∗_>, ..., ꢇ^∗_>).

Then, using (ii) and (iii), we get

1

2

:

1

Proof. Since

ꢆˆ_>≥ ꢆ^∗_>, ∀9 ∈ {1, 2, ..., : } and ꢆˆ_>> ꢆ^∗_>for at least one 9 ∈ {1, 2, ..., : }.

:

1

Ö

1

(iv)

/(ꢆ, ꢇ) =

ꢆ^F_>

×

(1 − ꢇ_>

)

9

©

-

ª

®

©

-

ª

®

9

F

9

9=1

ꢇ_>≤ ꢇ^∗_>⇒ 1 − ꢇ_>≥ 1 − ꢇ^∗_>, ∀9 ∈ {1, 2, ..., : },

«

¬

«

¬

ˆ

1

9

Let (-^∗, ꢆ^∗, ꢇ^∗) be a unique solution of problem (22). Then

(v)

ꢇ_>< ꢇ^∗_>⇒ 1 − ꢇ_>> 1 − ꢇ^∗_>, for at least one 9 ∈ {1, 2, ..., : }.

ˆ

1

9

/(ꢆ^∗, ꢇ^∗) > /(ꢆ, ꢇ), ∀(-, ꢆ, ꢇ) in the feasible space of (22).

Suppose -^∗is not a POS of problem (2). Then, by Deﬁnition 3.1.4, ∃ - in

Now, using (iv), we get

ˆ

the feasible space of (2) such that

:

1

Ö

ꢆˆ^F_>

>

(ꢆ^∗_>

)

9

F

9

.

(vi)

9

∗

ˆ

9=1

5 (-) ≥ 5 (- ), ∀9 = 1, 2, . . . , : and 5 (-) > 5 (- ), for at least one9 ∈ {1, 2, ..., : } (8)

1

9

Thus,

Using (v), we obtain

5₉(-^∗) − !₉

ˆ

5₉(-) − !₉

≥

,

∀9 = 1, 2, ..., : ,

1

*₉− !₉

:

1

Ö

and

F

(1 − ꢇ_>

)

>

(1 − ꢇ^∗_>

)

.

(vii)

9

ˆ

9

5₉(-) − !₉

5₉(-^∗) − !₉

ˆ

9=1

>

for at least one 9 ∈ {1, 2, ..., : }.

1

*₉− !₉

Finally, from (vi) and (vii), we conclude:

Since 3₉> 0,

∗

ˆ

9

:

4

^{−3 ((* − 5 (-))/(* −! ))}− 4⁻³

4

^{−3 ((* − 5 (- ))/(* −! ))}− 4⁻³

1

Ö

ꢆˆ^F_>

(1 − ꢇ_>

)

>

(ꢆ^∗_>

)

(1 − ꢇ^∗_>

)

=⇒

≥

,

∀9 ∈ {1, 2, ..., : },

9

F

9

F

1

9

ˆ

1 − 4⁻³

9

9=1

and

∗

ˆ

−

3

(

*

−

5

(

-

)

/

(

*

−

!

)

−

3

4

−

4

^{−3 ((* − 5 (- ))/(* −! ))}− 4⁻³

9

>

,

∗

ˆ

1 − 4⁻³

⇒ /(ꢆˆ, ꢇ) > /(ꢆ , ꢇ ),

9

for at least one 9 ∈ {1, 2, ..., : }.

Similarly, for non-membersh¹ip functions:

(ii)

which contradicts the optimality of (-^∗, ꢆ^∗, ꢇ^∗) for problem (22). Hence, no

∗

ˆ

such - exists, and therefore - is a POS of problem (2).

*₉− 5₉(-)

*₉− 5₉(-^∗)

ˆ

The theorem can be proved similarly for the remaining two perspectives of

≤

, ∀9 ∈ {1, 2, ..., : },

1

the DMs and for minimization objectives.

*₉− (!₉− ꢄ₉) *₉− (!₉− ꢄ₉)

and

The overall solution procedure of the proposed method for solving an

IFMOO problem can be summarized as follows:

ˆ

*₉− 5₉(-)

*₉− 5₉(-^∗)

<

,

for at least one 9 ∈ {1, 2, ..., : },

1

*₉− (!₉− ꢄ₉)

Step 1.

Step 2.

Formulate the IFMOO problem (1).

∗

ˆ

−

3

(

5

(

-

)

+

ꢄ

−

!

)

/

(

*

+

ꢄ

−

!

)

−

3

4

−

4

^{−3 (( 5 (- )+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

9

=⇒

≤

,

Transform the IFMOO problem into the equivalent crisp

MOOP (2) by employing the accuracy function.

1 − 4⁻³

9

∀9 ∈ {1, 2, ..., : } , and

4^{−3 (( 5 (-)+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

1

Step 3.

Step 4.

Step 5.

Solve each objective function independently and construct

the pay-oﬀ table to determine the lower and upper bounds,

denoted by !₉and *₉, respectively, for each objective

function 5₉(-), where 9 = 1, 2, ..., :.

∗

ˆ

^{−3 (( 5 (- )+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

9

4

<

,

1 − 4⁻³

9

for at least one 9 ∈ {1, 2, ..., : }.

(iii)

1

Deﬁne

Construct the exponential membership and non-membership

functions corresponding to each objective function under

the optimistic, pessimistic, and mixed approaches using

the tolerance parameters ꢄ₉and ꢅ₉, where 9 = 1, 2, ..., :.

!

ˆ

9

*

− 5 (-)

9

−3

9

*

−!

9

− 4⁻³

9

4

ꢆˆ_>

=

,

9

1 − 4⁻³

9

Relax or reduce the ≤ type and ≥ type constraints by

employing the assigned violation parameter ;₈in the forms

2₈+ ꢃ(;₈) and 2₈− ꢃ(;₈), respectively, where ꢃ ∈ (0, 1) and

8 = 1, 2, ..., <.

ꢀ

ꢁ

∗

*

− 5 (-

)

9

*

−!

− 4⁻³

9

ꢆ^∗_>

,

1 − 4⁻³

9

!

Step 6.

Step 7.

Develop and solve the problem according to the DM’s

preference under the optimistic (22) or pessimistic (23) or

mixed (24) model.

ˆ

9

5 (-)+ꢄ −!

9

*

+ꢄ −!

9

− 4⁻³

9

4

ˆ

ꢇ_>

=

,

9

1 − 4⁻³

9

If the obtained solution satisﬁes the DM, then terminate the

solution procedure. Otherwise, reformulate the problem

and repeat the process until a satisfactory solution is

obtained.

ꢀ

ꢁ

∗

5 (- )+ꢄ −!

9

*

+ꢄ −!

− 4⁻³

9

ꢇ^∗_>

.

1 − 4⁻³

9

Teferi H.T.,(2026)

49

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving each objective independently, subject to the constraints, the

ideal solutions and the extreme values of the objectives are identiﬁed.

4 Comparative Analysis

To verify the eﬀectiveness of the proposed method, a comparative analysis

was conducted using the approach proposed by V. Singh and Yadav (2018).

The manufacturing problem presented in their study was employed for this

purpose. Accordingly, the transformed equivalent crisp problem is

Accordingly, - = (0, 84.09, 52.66) and - = (0, 120.85, 0) which give

1

!

= 1223.62, * = 1272.69, and ! = 468.3²0, * = 595.73.

1

2

For an optimisti¹c DM, problem (25) is transformed into a single-objective

problem of the form (26) using model (22), by assigning equal weights to

the objectives and considering violation parameters of 25 and 20 units for

the ﬁrst and second constraints, respectively.

max 5 (-) = 7.5G₁+ 10.125G₂+ 8G ,

1

min 5 (-) = 2.9375G₁+ 3.8750G₂+³5.1250G

2

3

subject to

2.9375G₁+ 2.0625G₂+ 2.9375G ≤ 328.125,

(25)

3

3.875G₁+ 2.9375G₂+ 2.0625G ≤ 355.625,

3

2.0625G₁+ 2.9375G₂+ 2.9375G ≥ 355,

3

- = (G , G , G ) ≥ 0.

1

2

3

max / = ꢆ⁰₁^.5ꢆ⁰₂^.5(1 − ꢇ )^0.5(1 − ꢇ )^0.5

1

2

subject to

+10.125G +8G

4^{0.2(1−((1272.69−(7.5G}

+3.8750G +5.1250G

3

^{))/(1272.69−1223.62)))}+ ꢆ (1 − 4^0.2) ≥ 1,

1

2

3

1

4^{0.2(1−((2.9375G}

^{−468.30)/(595.73−468.30)))}+ ꢆ (1 − 4^0.2) ≥ 1,

1

2

+10.125G +8G −(1223.62−(ꢃ(1272.69−1223.62))))/(1272.69−(1223.62−(ꢃ(1272.69−1223.62)))))

4^{0.2(1−((7.5G}

+

1

2

3

ꢇ (1 − 4^0.2) ≤ 1,

1

+3.8750G +5.1250G ))/((595.73+(ꢃ(595.73−468.30)))−468.30)))

4^{0.2(1−(((595.73+(ꢃ(595.73−468.30)))−(2.9375G}

+

1

2

3

ꢇ (1 − 4^0.2) ≤ 1,

2

2.0625G₁+ 3.8750G₂+ 2.9375G ≤ 333.125 + ꢃ(25),

3

3.8750G₁+ 2.0625G₂+ 2.0625G ≤ 365.625 + ꢃ(20),

(26)

3

2.9375G₁+ 2.0625G₂+ 2.9375G ≥ 360,

3

0 ≤ ꢆ₉+ ꢇ₉≤ 1, 9 = 1, 2

0 ≤ ꢆ₉≤ 1, 9 = 1, 2

0 ≤ ꢇ₉≤ 1, 9 = 1, 2

G , G , G ≥ 0.

1

2

3

Solving this problem, using LINGO (LINDO Systems Inc., 2017) version 21,

the solution is presented in Table 1.

Table 1: Solutions of problem (26) under diﬀerent ꢃ values.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 121.49, 3.30)

(0.00, 123.79, 0.00)

(0.00, 124.46, 0.00)

(0.00, 125.14, 0.00)

(0.00, 125.70, 0.00)

1248.50

1253.35

1260.24

1267.13

1272.69

482.56

479.68

482.31

484.95

487.08

257.33

255.31

256.72

258.12

259.25

361.63

363.62

365.62

367.62

369.24

363.63

363.62

365.62

367.62

369.24

As ꢃ increases, 5 (-) improves while 5 (-) reaches its lowest value around

Similarly, by reformulating problem (25) for pessimistic and mixed DM

perspectives using models (23) and (24), respectively, we obtain the

following solutions presented in Tables 2 and 3, respectively.

1

2

ꢃ = 0.4 and then slightly worsens for larger values of ꢃ. This shows a

trade-oﬀ, where increasing emphasis shifts from minimizing 5 (-) toward

2

maximizing 5 (-). The solution becomes stable for larger ꢃ.

1

Table 2: Solutions of Problem (25) under the pessimistic perspective.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 114.81, 11.82)

(0.00, 118.60, 7.39)

(0.00, 122.39, 2.96)

(2.02, 118.03, 6.35)

(2.91, 118.91, 4.40)

(3.47, 118.96, 4.23)

1256.98

1259.93

1262.88

1260.97

1260.91

1264.38

505.45

497.44

489.44

495.83

491.84

492.86

271.51

266.32

261.13

268.01

266.70

267.98

361.62

363.62

365.63

367.62

369.62

371.62

371.97

370.09

368.22

369.52

409.67

369.04

Teferi H.T.,(2026)

50

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

For the pessimistic variants of the problem, as ꢃ increases, 5 (-) improves

ﬂuctuations in both objectives.

while 5 (-) remains relatively stable in the mid-range of ꢃ, w¹ith only minor

2

Table 3: Solutions of Problem (25) under the mixed perspective.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 114.81, 11.82)

(0.00, 118.60, 7.39)

(0.00, 119.26, 7.42)

(0.00, 125.15, 0.00)

(0.00, 125.49, 0.00)

(0.00, 125.56, 0.00)

1256.98

1259.93

1262.88

1267.13

1270.58

1271.13

505.45

497.44

500.15

484.95

486.27

486.54

271.51

266.32

267.76

258.12

258.82

258.96

361.62

363.62

365.62

367.62

368.62

368.82

371.97

370.09

372.12

367.62

368.62

368.82

For the mixed variants of the problem, as ꢃ increases, the solution steadily

ꢃ = 0.6, the solution minimizes 5 (-).

2

improves 5 (-), while 5 (-) stabilizes after an initial ﬂuctuation. Around

1

2

Table 4: Comparison of the proposed method with existing approach.

Variant

Proposed method (ꢃ = 0.3 to 0.8)

Existing method (from C = 1 to 5, under two reference

conditions)

Optimistic

Pessimistic

Mixed

5 (-) maximized within the range 1248.50 to 1272.69,

5 (-) maximized within the range 1244.91 to 1248.62,

1

5 (-) minimized within the range 487.08 to 479.68

5 (-) minimized within the range 530.80 to 520.33

2

5 (-) maximized within the range 1256.98 to 1264.38,

5 (-) maximized within the range 1248.63 to 1251.92,

1

5 (-) minimized within the range 505.45 to 489.44

5 (-) minimized within the range 539.37 to 530.7

2

5 (-) maximized within the range 1256.98 to 1271.13,

5 (-) = 1248.77 (maximized),

1

5 (-) minimized within the range 505.45 to 484.95

5 (-) = 530.66 (minimized)

2

The proposed method dominates the existing approaches across all

decision-making variants by achieving higher values of 5 (-) and lower

values of 5 (-) under diﬀerent values of ꢃ ∈ (0, 1). It sho¹ws a signiﬁcant

2

reduction in 5 (-), indicating improved minimization performance.

2

Generally, the results demonstrate that the proposed model provides more

eﬃcient and balanced trade-oﬀ between the two conﬂicting objectives.

5 Model Application

5.1 Description of the Study Area

The farming site is in Abeshge district, Gurage Zone, central Ethiopia,

between 8^◦19⁰–8^◦45⁰N latitude and 37^◦45⁰–38^◦7⁰E longitude (Nasir &

Hundie, 2014). Mean annual temperatures range from 18^◦C to 28.3^◦C, with

rainfall of 801–1400 mm, mostly during the Kiremt season (Dessie et al.,

2017). The soil is sandy loam, with a pH of 6.40–6.92.

Farming is mainly rainfed due to limited irrigation, though a few seasonal

rivers support perennial crops like mangoes and bananas along their

banks. Teﬀ (Eragrostis tef), maize (Zea mays), pepper (Capsicum annuum),

chickpea (Cicer arietinum), bean (Phaseolus vulgaris), and sorghum

(Sorghum bicolor) are the most widely cultivated crops in the area. These

crops dominate the farming pattern, accounting for about 85% of the

cropped area in the district.

The study area was chosen due to its high crop production potential and the

availability of accessible agricultural data. The considered LSF has detailed

information on existing cropping patterns, which makes the farming site

suitable for empirical analysis. Furthermore, the study area represents the

dominant farming system of the area, allowing the ﬁndings to be relevant

to smallholder farmers in the area. This LSF practices rainfed farming on

1,033ha of land in Gefersa kebele. The location map of the study area is

shown in Figure 4.

Figure 4: Map of the study area.

5.2 Methods of Data Collection and

Analysis

A stratiﬁed purposive sampling method was employed to intentionally

select the large-scale farm (LSF) site based on its suitability for the study.

The selection criteria included the availability of reliable and accessible

data, representative farming practices, adequate capital investment, and

suﬃcient manpower. These characteristics enabled the farm to eﬀectively

represent the target study population and provide meaningful insights into

the research objectives.

The existing farming pattern mainly targeted on the achievement of

maximum production. The required input allocation to the crops are mainly

determined by experience, even though they rarely apply the advice and

paradigm of the developmental agents and extension workers in the district.

Teferi H.T.,(2026)

51

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The data were collected from farmers, managers, and workers of the

LSF, extension experts, development agents, the zonal rural development

oﬃce, meteorological stations, and marketing agencies through surveys,

questionnaires, and key informant interviews.

expenditures for cultivation and agricultural input resources are presented

in Tables 5 to 9.

Crop data are presented in Supplementary Materials S.7 and S.8. These,

along with the soil data in Supplementary Material S.9, are used to calculate

crop water requirements.

The dataset incorporates environmental, climatic, hydrological,

agricultural, and socio-economic factors. Hydrological data cover eﬀective

rainfall and crop water demand. Eﬀective rainfall was calculated using the

dependable rain (FAO/AGLW formula) method Allen et al., 1998, while

crop water demand was estimated with the Penman-Monteith equation

(Allen et al., 1998; Smith, 1992) using crop evapotranspiration.

The considered crops are denoted as 2₁for maize, 2₂for teﬀ,2₃for sorghum,

2

for pepper, 2 for chickpea, 2 for bean.

4

5

6

In Table 5, ꢃ

12

and ꢃ

represent the fertilizers required for crop 2₌, where

22

=

= = 1, 2, 3, 4, 5, 6, in the ﬁrst and second rounds, respectively. Accordingly,

ꢃ

denotes NPS, while ꢃ

represents Urea. Similarly, ꢄ and ꢄ

12 22

=

12

22

rep⁼resent the required herbicides for crop 2₌in the ﬁrst and second rounds,

respectively. The pesticide required for crop 2₌is denoted by %₂₌. Thus, the

following symbols are used to denote diﬀerent herbicides and pesticides:

=

Crop data such as rooting depth, crop coeﬃcient, critical depletion, yield

response, crop height, and crop calendar were obtained from FAO Manual

56 (Allen et al., 1998) and related literature.

Socio-economic data include crop market prices, labor and machinery costs,

fertilizer, herbicide, and pesticide dosages and prices, and land resource

information.

ꢄ

and ꢄ

for Atrazine, ꢄ

12

for 2,4-D, ꢄ

for Pallas 45 OD, ꢄ

22 12

22

12

1

3

2

4

for Glyphosate, ꢄ

%

and ꢄ

for Pendimethalin, ꢄ₂for S-metolachlor,

22

for Diazinon 60⁵EC, %₂for Dimethoate, %₂for Karate 5 EC, %₂for

6

2

Ethiozinon 60 EC, %₂for Highway 50 EC, and %³₂for Proﬁt.

1

2

4

5

6

The gathered data from diﬀerent sources for a speciﬁc parameter are

arranged into ﬁve groups, viz., the extreme minimum, minimum, median,

maximum, and extreme maximum values, based on the level of their

deviation from the median value. The values less than the median value

are arranged into the extreme minimum and minimum values. Speciﬁcally,

the average of the highly deviated values from the median value is taken

as the extreme minimum value, while the average of the relatively less

deviated values from the median is considered as the minimum value. In

the same manner, the values greater than the median value are arranged into

the maximum and extreme maximum values in the agricultural production

problem. The average of the median values of a parameter is taken as the

mean value.

Table 5: Usage of fertilizer, herbicide and pesticide.

ꢃ

ꢄ

%

2

(!/ℎ0)

12

22

12

22

=

Crop

(:,/ℎ0)

(!/ℎ0)

g

e

maize (2 )

100

125

-

1

g

e

f

e

teﬀ (2 )

100

1

0.5

1

2

g

e

sorghum (2 )

100

-

1

3

g

f

pepper (2 )

100

200

1.5

-

1.5

4

g

e

chickpea (2 )

100

-

1

5

g

e

g

e

bean (2 )

100

1

0.75

1

6

The IFN in Table 5 describe the following numbers.

Using this principle, the aggregated values are used to construct triangular

IFNs to represent the APP problem more realistically.

g

f

200 = h196, 198, 208; 194, 198, 210i,

1.5 = h1.1, 1.6, 1.7; 1, 1.6, 1.8i,

g

e

125 = h123, 125, 127; 122, 125, 128i,

1 = h0.75, 1, 1.25; 0.5, 1, 1.5i,

g

100 = h97, 99, 105; 95.5, 99, 106.5i,

0.75 = h0.6, 0.7, 1; 0.3, 0.7, 1.3i,

In this study, the last twelve years climatic data from 2013 to 2024 were

gathered from Emdbir meteorological station, with an altitude of 2082<,

latitude 8.13^◦N and longitude 37.93^◦E.

In order to handle the current extreme variation in the cost of input

resources for crop cultivation and crop prices in the country, we have

employed closely related data for the proposed study. Accordingly, the

average data from 2022 to 2024 of the existing cropping schemes, crop yield

(FAO, 2022, 2023), labor force, crop prices, cost of seeds, and other related

f

e

2.5 = h2, 2.6, 2.8; 1.9, 2.8, 2.9i,

2 = h1.9, 2, 2.1; 1.5, 2, 2.5i

f

0.5 = h0.2, 0.5, 0.7; 0.1, 0.5, 1i.

The costs of cultivation, including plowing, threshing, land rent, seed, labor,

herbicides and pesticides, and fertilizers, are provided in Supplementary

Materials S.2, S.3, S.4, S.11, and S.12, respectively.

The overall cost of the cultivation of each crop is presented in Table 6.

Table 6: Cost of cultivation (ꢁ₂) and crop yield (. ₌).

2

=

ꢁ₂

.

(@C;/ℎ0)

2

=

Crop

(ꢂ)ꢅ/ℎ0)

maize (2 )

h35623, 35627, 35628; 35620, 35627, 35629i

h35698, 35705, 35714; 35689, 35705, 35719i

h26838, 26845, 26854; 26829, 26845, 26859i

h41113, 41120, 41129; 41104, 41120, 41134i

h20655, 20659, 20666; 20649, 20659, 20674i

h25124, 25131, 25140; 25115, 25131, 25145i

h50, 54, 62; 47, 54, 65i

1

teﬀ (2 )

h12.5, 13.5, 16.5; 12, 13.5, 17i

h23, 24.5, 27.5; 22.5, 24.5, 29i

h11, 13.5, 14; 10, 13.5, 16i

h16.5, 17.5, 20; 15.5, 17.5, 22i

h21, 23.5, 24; 20, 23.5, 25i

sorgh²um (2 )

pepper (2 )³

4

chickpea (2 )

5

bean (2 )

6

The climatic data are used to calculate the crop water requirements, which are presented in Supplementary Material S.10.

Table 7: Seed (ꢂ₂), labor (ꢅ₂) and water requirement (,₂₌).

=

ꢂ₂

(:,/ℎ0)

ꢅ₂

(<3/ℎ0)

,

2

(< /ℎ0)

=

Crop

3

maize (2 )

h22, 26, 25; 17, 26, 27i

h29, 30, 30.5; 28.5, 30, 32i

h14, 14.5, 16.5; 13.5, 14.5, 18i

h15, 15.5, 17.5; 14.5, 15.5, 19i

h26, 28, 30; 25, 28, 31i

h19, 21, 23; 18, 21, 24i

h80, 86, 89; 75, 86, 92i

h70, 76, 80; 64, 76, 82i

h35, 41, 45; 30, 41, 46i

h97, 99, 105; 94, 99, 108i

h25, 31, 35; 20, 31, 36i

h47, 51, 52; 45, 51, 55i

h3570, 3610, 3625; 3565, 3610, 3648i

h2580, 2625, 2640; 2564, 2625, 2660i

h3240, 3249, 3252; 3225, 3249, 3255i

h4750, 4766, 4770; 4705, 4766, 4783i

h2515, 2540, 2560; 2505, 2540, 2572i

h2355, 2360, 2386; 2347, 2360, 2400i

1

teﬀ (2 )

sorgh²um (2 )

pepper (2 )³

4

chickpea (2 )

5

bean (2 )

6

Teferi H.T.,(2026)

52

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The farming machines used in the area are a tractor and a combine harvester. Machine hours for plowing are considered until the land is ready for sowing;

details are in Supplementary Material S.1. The required total machine hours for each crop are shown in Table 8.

Table 8: Machine hours requirement ("₂) and proﬁt (#₂₌).

=

"

(ℎA/ℎ0)

#

2

(ꢂ)ꢅ/ℎ0)

2

=

Crop

maize (2 )

h1.83, 2.167, 2.33; 1.42, 2.167, 2.42i

h1.75, 2, 2.42; 1.5, 2, 2.5i

h33117, 33123, 33136; 33109, 33123, 33138i

h13293, 13295, 13297; 13290, 13295, 13300i

h18148, 18154, 18165; 18141, 18154, 18170i

h192873, 192881, 192888; 192862, 192881, 192893i

h25830, 25842, 25849; 25820, 25842, 25853i

h16262, 16269, 16278; 16253, 16269, 16283i

1

teﬀ (2 )

sorgh²um (2 )

pepper (2 )³

h1.75, 2, 2.42; 1.5, 2, 2.5i

4

chickpea (2 )

h1.83, 2.167, 2.33; 1.42, 2.167, 2.42i

h1.75, 2, 2.42; 1.5, 2, 2.5i

5

bean (2 )

6

The cropland allocation in the existing situation during 2020 - 2024 is given in the Supplementary Material S.5. Accordingly, the average land allocation of

the existing system and the attainability and non-attainability degrees of the intended objectives are depicted in Table 9. Where G , G , G , G , G , and G

6

2

3

5

denote the land area allocated to maize, teﬀ, sorghum, pepper, chickpea, and beans, respectively. The functions 5 (-), 5 (-), 5 (-),¹and 5 (-),⁴respectively,

1

2

3

4

represent the yield, proﬁt, cost, and labor objectives.

Table 9: Existing farm pattern.

- = (G , G , G , G , G , G )

5 (-)

6⁴4160

0.20

4

1

2

3

(221.02, 141.41,¹401².32,³106.42,⁵62.⁶29, 121.57)

29470.15

41510180

31413470

ꢀ

ꢁ

0.81

0.19

0.20

0.28

0.17

0.30

0.28

Some mathematical software is employed to solve the considered APP

problem and assist in analyzing the results of the study. CROPWAT

8.0 software (Smith, 1992) is used to generate and analyze the water

requirement of crops, and LINGO (LINDO Systems Inc., 2017) is used to

solve complex mathematical problems.

are described as IFN. Thus, the objectives and constraints of the intended

problem are expressed as intuitionistic fuzzy functions. Accordingly,

IFMOO model is employed for comprehensive and eﬃcacious investigation.

The Kiremt season is the widely practiced cropping season in the study area.

Crops such as maize, pepper, and sorghum are sown during the Belg season

and harvested by the end of Kiremt. Teﬀ and beans are sown in the Kiremt

season and harvested during the Bega season, while chickpea is planted in

the last days of Kiremt and harvested in the Bega season. Thus, all six crops

considered in this study are generally cultivated within a single cropping

season.

5.3 Problem Formulation

As a result of the inconsistent nature and imprecision of the pertained

agricultural data, crop planning rests under the inﬂuence of risk and

uncertainty (Luo et al., 2023). So the parameters in the APP problem

Table 10: Decision variable and parameters.

Parameters

)!

)!_A

Description

Total farmland for crop cultivation (ℎ0)

Minimum land area required for cultivation (ℎ0)

Average yield per unit area of crop 2 (@C;/ℎ0)

.

2

#

Average net proﬁt of crop 2 per hectare (ꢂ)ꢅ/ℎ0)

Average investment per unit area of crop 2 (ꢂ)ꢅ/ℎ0)

Average machine-hours required per unit area of crop 2 (ℎA/ℎ0)

Average labor required per unit area of crop 2 (<3/ℎ0)

Average water requirement per unit area of crop 2 (<³/ℎ0)

2

ꢁ₂

"

2

ꢅ₂

,

2

ꢂ₂

ꢃ_;2

%

;2

ꢄ_;2

)"

),

)ꢂ₂

)ꢃ_;

Seed required per unit area of crop 2 (:,/ℎ0)

Cℎ

;

type of fertilizer required per unit area for crop 2 (:,/ℎ0)

Cℎ

type of pesticide required per unit area for crop 2 (!/ℎ0)

Cℎ

;

type of herbicide required per unit area for crop 2 (!/ℎ0)

Total available machine-hours (ℎA)

Total available water (<³)

Total available seed for crop 2 (:,)

Total available ;^Cℎfertilizer (:,)

)%_;

)ꢄ_;

Total available ;^Cℎpesticide (!)

Total available ;^Cℎherbicide (!)

Decision

Description

Land area allocated to crop 2 (ℎ0)

variable

-

2

Note: <3 denotes man-days.

Teferi H.T.,(2026)

53

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Objective functions

during growth stage from diﬀerent pests and insects. This constraint is

formulated as

Based on their accessibility and regional importance, the following four

objectives are considered in this study.

6

Õ

f

g

%

_;2-₂- )%_;, ; = 1, 2, ..., !.

(34)

(i) Crop yield achievement

2=1

The estimated yield of a crop is equal to the product of the cultivable area

of land and the average yield produced per unit area of land.

Thus, the maximization of the total yield of the considered six crops can be

expressed as

(vi) Machine hours

Diﬀerent types of machines are needed for various tasks of agriculture,

such as tilling, plowing, sowing, cultivating, harvesting, threshing, etc. The

sum of the machine hours allocated to each season should not exceed the

machine hours required in a year. That means,

6

Õ

e

max 5 (-) ≈

. -₂.

(27)

1

2

2=1

6

Õ

(ii) Net proﬁt goal

g

"₂-₂- )".

(35)

The net proﬁt of various crops, is the product of the net proﬁt of each crop

per unit area of land and its respective utilized land, which is described as

2=1

(vii) Cultivable land availability

6

Õ

e

f

The sum of cultivable land for all crops must not exceed the total available

land. Furthermore, the total cultivable land should be less than the entire

arable land available in the study area. This is formulated as

max 5 (-) ≈

#₂-₂.

(28)

2

2=1

(iii) Cost of cultivation goal

6

To get the optimum production, farmers should invest a certain amount

of money for land rent, fertilizers, seeds, herbicides, pesticides, rental

machines, labor force, etc. Minimizing this working capital is another

important objective of farmers and mathematically given by

Õ

f

-₂- )!.

(36)

2=1

On the other hand, a minimum cultivable land area should be allocated to

crop production to maintain a minimum level of agricultural output and

proﬁt while ensuring eﬃcient utilization of available land resources. This

constraint is mathematically expressed as

6

Õ

e

min 5 (-) ≈

ꢁ₂-₂.

(29)

3

2=1

(vi) Labor requirement

The labor objective is described as

6

Õ

g

-₂¥ )!_A.

(37)

2=1

=

Õ

e

f

min 5 (-) ≈

ꢅ₂-₂.

(30)

4

(i) Seed requirement

2=1

Seed availability constraints were incorporated into the model to describe

the limitations in accessing selected seed for each crop. The maximum seed

availability for each crop was determined based on the recommended seed

requirement per hectare (Table 7), the maximum feasible cultivable area

for each crop, and crop suitability conditions under local agro-ecological

settings (Semu et al., 2022). This constraint mathematically expressed as

Constraints

The above four objectives are subject to the following eight constraints.

(ii) Water requirement

Additional water through irrigation is required to meet the crop’s

evapotranspiration needs and optimize yield. The constraint for water

supply can be described as

f

g

ꢂ₂-₂- )ꢂ₂, 2 = 1, 2, ..., 6.

(38)

6

Õ

f

g

,₂-₂- ),.

(31)

(viii) Non-negativity

2=1

In the modelling APP problem, all decision variables should be

non-negative.

-

≥ 0

2 = 1, 2, ..., 6.

(39)

2

For sustainable optimal crop yield and maximum proﬁt, agricultural input

resources should not be used at the expense of the environment (Li et al.,

2020). Therefore, to reduce the adverse eﬀects of fertilizers, pesticides, and

herbicides on the environment, the optimum amounts of these inputs must

be considered alongside the utilization of other favorable resources.

(iii) Dosage of fertilizer

The stated objectives and constraints align with economic, environmental,

and social goals, with aims to boost net beneﬁts, use resources wisely to

limit environmental harm, and increase local jobs (Li et al., 2020; Zhang &

Georgescu, 2022).

To maintain and improve the productivity of the soil, diﬀerent types of

fertilizers have to be used optimally according to the characteristics of the

crops, soil type and climate of the region. This constraint is expressed as

In this study, the three variants of the problem are considered

independently to address the interference of uncontrolled conditions and

associated risks, and individual diﬀerences in the DM process. This assists

farmers and managers from diﬀerent perspectives by proposing various

possible alternative management schemes.

6

Õ

f

g

ꢃ_;2-₂- )ꢃ_;, ; = 1, 2.

(32)

2=1

In the optimistic assumption, a farmer considers using farming resources

with varying degrees of acceptance and ﬂexibility to accommodate other

possible alternative operations, presuming that it oﬀers certain beneﬁts.

Conversely, a pessimist DM tends to be skeptical about implementing all

possible alternative farming tasks and partially considers those with a lower

degree of acceptance. From a mixed perspective, the farmer’s assumption

lies between the optimistic and pessimistic viewpoints.

As a result, the solution of the APP problem varies according to each

perspective (Kis et al., 2021), which leads to diﬀerences in the determination

of the OCP. Each viewpoint has its advantages and disadvantages.

Therefore, this study considers the optimistic, pessimistic, and mixed

aspects of DM for eﬃcient management of agricultural resources.

(iv) Amount of Herbicide

DM requires a certain level of herbicides for several crops during growth

stage for better yield taking the availability and environmental aspects into

account. Mathematically described as

6

Õ

g

ꢄ_;-₂- )ꢄ_;, ; = 1, 2, ..., !.

(33)

2

2=1

(v) Dosage of pesticide

Pesticide is another essential input resource of agriculture to protect crops

Teferi H.T.,(2026)

54

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

In this study, from 1, 033ℎ0 land of the LSF, nearly 1, 017ℎ0 of farmland

is considered for crop cultivation, and 6ℎ0 to 7ℎ0 of land is supposed to

be left permanently for forestation to maintain the ecological balance of

the environment. In the existing system, 5ℎ0 of cropland is occupied by

perennial crops. About 1.50ℎ0 of land of the LSF is permanently left over

as a residential place for workers and roads. Moreover, 2.5ℎ0 of land is not

suitable for farming and is currently used for animal grazing.

The land management of the LSF is a traditional approach based on the

weather conditions. They interchangeably use plots of land for diﬀerent

crops, and there is no reasonable pattern of operation on cropland.

In this work, three scenarios of crop cultivation are designed based on

the conventional cultivation pattern and suitability of the devised farming

system in the study area.

third scenarios are relaxed based on resource availability.

In the problem formulation of the three scenarios, the upper and lower

limits of the constraints, the value of violations, and tolerances are mainly

based on the availability of agricultural resources. For the second and third

scenarios, the land constraint attributed to chickpeas is expressed in terms

of the remaining crops.

The weights assigned to the objectives and constraints are estimated

based on the preferences rated by managers of the LSF, farmers, and

developmental agents of the study area.

6 Results and Discussions

We denote the land areas allocated to crop 2 by G₂, where 2 = 1, 2, ..., 6,

representing the land areas of maize, teﬀ, sorghum, pepper, chickpea, and

bean, respectively.

The ﬁrst scenario considers the case when chickpeas are planted on

the land leftover in the ﬁrst season of crop cultivation. Thus, in this

scenario, chickpea faces the same land rent as other crops. This system is

mainly adopted to fertilize unplowed land and to make use of ﬁelds left

uncultivated in the ﬁrst season due to factors like irregular rainfall, labor

shortages, limited seed varieties, and lack of capital or fertilizers.

The functions 5₉(-), where 9 = 1, 2, 3, 4, denote the objectives of production,

proﬁt, expenditure, and manpower, respectively. The constraints are

denoted by ,₈(-), where 8

=

1, 2, ..., 8, corresponding respectively to

water, ﬁrst-round fertilizer, second-round fertilizer, ﬁrst-round herbicide,

In the second scenario, farmers plant chickpeas on land right after

harvesting maize. This common practice helps to use the fertile soil left by

maize and saves farmers from the additional cost of renting extra land for

smallholder farmers with limited ﬁelds to cultivate multiple crops. Based

on the existing farming system in the study area, at least 33% of the land

allocated to maize is subsequently used for chickpea cultivation after maize

harvest.

second-round herbicide, pesticide, machine hours, and the maximum

available land area.

In addition, minimum land area cultivation

requirements are imposed to secure the minimum proﬁt and yield, while

seed constraints are included to account for limitations in seed availability.

Based on the available agricultural resources, the violation parameters of

the constraints are assigned as ℓ = 5235, ℓ = 935, ℓ = 940, ℓ = 96,

1

4

ℓ

= 88, ℓ = 78, ℓ = 104, and ℓ = 4. The vio²lation para³meters for the seed

In the third scenario, chickpeas are planted after harvesting maize,

sorghum, and beans. Like the second scenario, the most important reason

for farmers to use this cropping plan is to increase the yield from fertile land

and minimize additional cultivation expenses. In order to eﬀectively utilize

the land for chickpea after harvesting maize, sorghum, and bean, the land

allocated to chickpea should not be less than the combined land allocated to

teﬀ and pepper. Furthermore, the land allocated to bean should be at least

equal to that allocated to sorghum. However, according to the farmers, this

scenario is rarely practiced as land preparation for the succeeding crop after

harvesting sorghum is a relatively challenging activity.

c⁵onstrain⁶ts of mai⁷ze, teﬀ, sorghum, pepper, chickpea, and bean are assigned

as 66, 40, 10, 10, 44, and 38, respectively, based on their availabilities.

8

The weights assigned to the production, proﬁt, cost, and labor force

objectives are 30%, 28%, 28%, and 14%, respectively. Utilizing all these

values, the formulated problem is solved under optimistic, pessimistic, and

mixed viewpoints by employing models (22), (23), and (24), respectively.

The problem is deﬁned using tolerance values expressed as multiples of ꢃ,

with violations evaluated under the three approaches. To explore diﬀerent

solutions, several values of ꢃ ∈ (0, 1), speciﬁcally 0.30, 0.40, 0.50, 0.60, 0.70,

and 0.80, are considered in the solution process. Accordingly, the APP

problem is solved for the three scenarios based on the objectives and

constraints outlined in Section 5.3 and the data presented in Tables 5 to

8.

As chickpea is cultivated as a second crop after harvesting other crops

within the same cropping season, the land rent for chickpea under the

second and third scenarios decreases signiﬁcantly by 65% relative to the

annual land lease cost. Thus, to eﬀectively utilize the available land in this

production cycle, the upper limits of some constraints in the second and

Scenario 1: When chickpea is planted on the fallow land.

Teferi H.T.,(2026)

55

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The mathematical expression of this problem has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 33125G₁⊕ 13295G₂⊕ 18154G₃⊕ 192881G₄⊕ 24342G₅⊕ 16269G ,

2

6

e

min 5 (-) = 35625G₁⊕ 3705G₂⊕ 26845G₃⊕ 41119G₄⊕ 20659G₅⊕ 25132G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3410610,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 101700,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 108055,

4

e

f

e

(40)

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1100,

6

g

e

g

e

2.67G₁⊕ 2.67G₂⊕ 2G₃⊕ 2G₄⊕ 2.67G₅⊕ 2G - 2205,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G₅⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G₅⊕ 1G ¥ 1015,

6

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where,

1145 = h1140, 1145, 1146; 1139, 1145, 1147i,

1017 = h1016, 1017, 1018; 1015, 1017, 1019i,

1015 = h1014, 1015, 1016; 1013, 1015, 1017i,

1100 = h1099, 1101, 1107; 1098, 1101, 1108i,

e

2205 = h2200, 2205, 2207; 2197, 2205, 2208i,

1 = h0.95, 0.99, 1; 0.94, 0.99, 1.15i,

3410610 = h3410609, 3410610, 3410615; 3410607, 3410610, 3410617i,

101700 = h101697, 101699, 101705; 101695, 101699, 101707i,

108055 = h108052, 108057, 108058; 108050, 108057, 108060i,

21611 = h21605, 21610, 21611; 21604, 21610, 21612i,

15255 = h15250, 15255, 15256; 15249, 15255, 15257i,

99156 = h99151, 99156, 99157; 99150, 99156, 99158i,

15662 = h15657, 15662, 15663; 15656, 15662, 15664i,

12204 = h12199, 12204, 12205; 12198, 12204, 12206i,

28138 = h28132, 28138, 28140; 28130, 28138, 28142i,

6509 = h6504, 6509, 6510; 6503, 6509, 6511i.

The remaining values of the coeﬃcients of the variables are presented in

Tables 5 to 8.

The solutions of problem (40) obtained using the proposed optimistic

approach for diﬀerent ꢃ ∈ (0, 1) are presented in Table 11.

Table 11: Solutions of problem (40) under an optimistic perspective.

ꢃ

-

5 (-)

5⁴4150

54630

55064

55464

55761

55751

/

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(153.44, 0, 314.97, 78.79, 152.82, 314.97)

(162.93, 0, 305.66, 81.88, 158.86, 305.66)

(171.96, 0, 296.48, 84.73, 165.36, 296.48)

(180.58, 0, 287.40, 87.37, 172.26, 287.40)

(187.95, 0, 278.65, 89.34, 180.41, 278.65)

(191.60, 0, 270.90, 89.31, 192.28, 270.90)

27333.18

27557.00

27766.47

27963.23

28121.20

28163.46

34842180

35579290

36269800

36919230

37441020

37578060

28234760

28340800

28436290

28522470

28579940

28551380

0.2651

0.2811

0.2949

0.3070

0.3176

0.3274

From the above solutions, Table 11, the better compromised solution is obtained when ꢃ = 0.8. So, the compromised solution for the ﬁrst scenario is

- = (191.60, 0, 270.90, 89.31, 192.28, 270.90) and its detail is depicted in Table 12.

Teferi H.T.,(2026)

56

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Table 12: Compromise solution of problem (40).

- = (G , G , G , G , G , G )

5 (-)

5⁴5751

0.31

2

5

6

1

2

3

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

28163.46

37578060

28551380

ꢀ

ꢁ

0.48

0.20

0.50

0.20

0.37

0.26

0.29

- = (G , G , G , G , G , G )

, (-)

2

5

6

2

4

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

31¹24448

101500

6³8902

597.15

- = (G , G , G , G , G , G )

, (-)

2

5

6

5

7

8

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

665.68

10⁶59.65

2287

1015

Based on the results given in Table 12, the required input resources to cultivate the crops under the allocated cropland are presented in Table 13.

Table 13: Input resource requirements for allocated crops under scenario one.

Crop

Seed (kg)

4789.95

4063.58

1428.93

5383.90

5418.11

ꢃ

12

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

153.44

314.97

(L)

%

2

=

(L)

22

12

22

=

maize (2 )

15344

31497

7879

23949.79

-

153.44

314.97

133.96

152.82

314.97

1

sorghum (2 )

27090.56

pepper (2 )³

17861.63

133.96

192.28

270.90

-

4

chickpea (2 )

15282

31497

-

5

bean (2 )

203.17

6

Scenario 2: Chickpea is planted after harvesting maize.

The mathematical expression of this problem has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 36100G₁⊕ 13295G₂⊕ 18154G₃⊕ 192881G₄⊕ 29865G₅⊕ 16269G ,

2

6

e

min 5 (-) = 32650G₁⊕ 3705G₂⊕ 26845G₃⊕ 41119G₄⊕ 15135G₅⊕ 25132G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3667300,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 105900,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 127125,

4

e

f

e

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

(41)

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1525,

6

e

2.167G₁⊕ 2.167G₂⊕ 2G₃⊕ 2G₄⊕ 2.167G₅⊕ 2G - 2715,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G ¥ 1015,

6

e

g

1G ¥ 1G , 1G ¥ 0.33G ,

1

5

1

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where

3667300 = h3667293, 3667304, 3667307; 3667290, 3667304, 3667310i,

g

2715 = h2712, 2715, 2718; 2710, 2715, 2720i, 0.33 = h0.31, 0.33, 0.35; 0.30, 0.33, 0.36i,

105900 = h105895, 105900, 105905; 105890, 105900, 105910i,

127125 = h127122, 127125, 127128; 127120, 127125, 127130i,

29490 = h29491, 29492, 29498; 29488, 29492, 29499i,

36100 = h36096, 36098, 36104; 36095, 36098, 36105i,

15135 = h15132, 15134, 15140; 15130, 15134, 15142i,

29865 = h29862, 29865, 29868; 29860, 29865, 29870i,

32650 = h32648, 32650, 32655; 32645, 32650, 32660i,

1525 = h1522, 1525, 1528; 1520, 1525, 1530i.

Teferi H.T.,(2026)

57

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving problem (41) using the proposed method for the optimistic

variants of the problem, a better compromise solution is obtained for ꢃ =

0.8. The resulting compromised optimal solution for the second scenario

is - = (156.00, 0.00, 661.58, 122.85, 51.48, 74.57), with details presented in

Table 14.

Table 14: Compromise solution of problem (41).

- = (G , G , G , G , G , G )

5 (-)

5⁴7384

0.54

1

4

6

1

2

3

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

29358.32

44088180

30558420

ꢀ

ꢁ

0.92

0.03

0.30

0.36

0.26

0.18

- = (G , G , G , G , G , G )

, (-)

1

4

6

2

3

4

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

36¹01808

106648

110228

310.32

- = (G , G , G , G , G , G )

, (-)

1

4

6

5

8

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

873.51

11⁶27.90

22⁷71.97

1015

The required input resources to cultivate the crops under the allocated cropland are presented in Table 15.

Table 15: Input resource requirements for allocated crops under scenario two.

Crop

Seed (kg)

3900.00

9923.75

1965.58

1441.44

1491.36

ꢃ

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

156.00

661.58

(L)

%

2

=

(L)

12

22

19500

12

22

=

maize (2 )

15600

66158

12285

5148

-

156.00

661.58

184.27

51.48

1

sorghum (2 )

66158

24570

pepper (2 )³

184.27

51.48

74.57

-

4

chickpea (2 )

-

5

bean (2 )

7457

55.93

74.57

6

Scenario 3: Chickpea is planted after harvesting maize, sorghum and beans.

The mathematical expression of this scenario has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 36100G₁⊕ 13295G₂⊕ 21130G₃⊕ 192881G₄⊕ 29865G₅⊕ 19245G ,

2

6

e

min 5 (-) = 32650G₁⊕ 3705G₂⊕ 23870G₃⊕ 41119G₄⊕ 15135G₅⊕ 22155G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3667300,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 105900,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 127125,

4

e

f

e

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

(42)

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1525,

6

e

2.167G₁⊕ 2.167G₂⊕ 2G₃⊕ 2G₄⊕ 2.167G₅⊕ 2G - 2715,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G ¥ 1015,

6

e

1G - 1G₁⊕ 1G₃⊕ 1G , 1G ¥ 1G₂⊕ 1G , 1G ¥ 1G .

5

6

5

4

6

3

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where 21130 = h21126, 21130, 21134; 21124, 21130, 21136i,

23870 = h23864, 23870, 23876; 23862, 23870, 23878i,

19245 = h19240, 19245, 19247; 19237, 19245, 19248i,

22155 = h22153, 22155, 22160; 22152, 22155, 22163i.

Teferi H.T.,(2026)

58

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving problem (42) using the proposed method for the optimistic

- = (409.19, 0.00, 277.16, 51.48, 51.48, 277.16), with details presented in

variants of the problem, a compromise solution is obtained for ꢃ

=

Table 16.

0.8. The resulting compromised optimal solution for the third scenario is

Table 16: Compromise solution of problem (42).

- = (G , G , G , G , G , G )

5 (-)

6⁴6522

0.38

1

4

6

1

2

3

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

37405.40

37428600

29013310

ꢀ

ꢁ

0.41

0.24

0.72

0.10

0.33

0.28

, (-)

8³9162

, (-)

24⁷41.61

0.26

- = (G , G , G , G , G , G )

, (-)

1

4

6

2

4

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

34¹06694

106648

405.86

- = (G , G , G , G , G , G )

, (-)

1

4

6

5

8

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

894.23

10⁶92.22

1015

The required input resources to cultivate the crops under the allocated cropland are presented in Table 17.

Table 17: Input resource requirements for allocated crops under scenario three.

Crop

Seed (kg)

10229.87

4157.44

823.68

1441.44

5543.25

ꢃ

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

409.19

277.16

(L)

%

2

=

(L)

12

22

12

22

=

maize (2 )

40919

27716

5148

51149.35

-

409.19

277.16

77.22

51.48

277.16

1

sorghum (2 )

27716.26

pepper (2 )³

10296.00

77.22

51.48

277.16

-

4

chickpea (2 )

5148

27716

-

5

bean (2 )

207.87

6

Similarly, the considered problem under the three scenarios can also be

solved for pessimistic and mixed DMs. The compromise solutions for each

scenario under pessimistic and mixed viewpoints are presented in the upper

and lower parts of Table 18, respectively.

Table 18: Solutions under pessimistic and mixed perspectives.

Scenario

ꢃ

-

Z

1

2

3

1

2

3

0.40

0.30

0.40

0.30

(168.27, 0.00, 304.30, 84.61, 153.51, 304.30)

(144.67, 0.00, 661.32, 141.05, 47.74, 67.97)

(412.14, 0.00, 277.56, 47.74, 47.74, 277.56)

(189.12, 0.00, 317.99, 72.76, 117.14, 317.99)

(144.67, 0.00, 661.32, 104.06, 47.74, 104.95)

(414.56, 0.00, 276.82, 46.80, 46.80, 276.81)

0.4132

0.4626

0.4495

0.4067

0.4352

0.4491

Based on the obtained results, the arable land allocated to the six crops

under the three approaches regarding the conventional pattern and the

three scenarios is presented in Figure 5.

(a)

(b)

(c)

Figure 5: Allocated land area to the six crops under the optimistic (a), pessimistic (b) and mixed (c) perspectives relative to the existing pattern.

Let - ̲, - ̲, and -

̲

denote the compromised solutions for the ﬁrst,

non-membership degrees of the objectives for the pessimistic and mixed

variants of the problem, are presented in Table 19.

1

3

second, an²d third scenarios, respectively, under pessimistic and mixed

views (Table 18). The objective values, along with the membership and

Teferi H.T.,(2026)

59

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Table 19: Goal achievement under pessimistic and mixed perspectives.

Pessimistic perspective

Mixed perspective

5 (- ̲)

1

4

1

4

1

5(X₁̲)

27724¹.77

36²105370

28³462070

55064

28719¹.63

34²095870

28³677430

55719

ꢀ

ꢁ

0.44

0.45

0.38

0.34

0.54

0.40

0.35

0.31

0.00

0.02

5 (- ̲)

1

2

4

2

1

2

4

2

5(X₂̲)

28745².80

46²965540

30³707130

57780

0.52

29115².66

40²433450

30³115760

55931

0.58

ꢀ

ꢁ

0.78

0.34

0.86

0.25

0.41

0.00

0.21

0.06

0.00

5 (- ̲)

1

3

4

3

1

3

4

3

5(X₃̲)

37470³.45

36²717860

28³917350

66314

37538³.86

36²566940

28³909570

66329

ꢀ

ꢁ

0.42

0.00

0.68

0.35

0.00

0.38

0.42

0.00

0.67

0.35

0.38

0.00

Based on the obtained solutions using the pessimistic and mixed approaches (Table 18), the agricultural resource consumptions for the three scenarios

under each perspective, in order, are presented in Tables 20 and 21.

Table 20: Resource consumption under the pessimistic viewpoint.

, (- ̲)

1

2

1

6

1

7

1

8

1

31069¹67

101500

68³386.55

5⁴84.73

7⁵00.80

1057.30

2245.60

1015.00

, (- ̲)

1

2

3

2

6

2

7

2

8

2

36215²77

106274

112424.80

3⁴27.28

8⁵56.96

1133.26

2254.39

1015

, (- ̲)

1

2

3

6

3

7

3

8

3

33922³48

106274

88³821.33

3⁴96.91

8⁵97.87

1086.61

2433.60

1015

Table 21: Resource consumption under the mixed viewpoint.

, (- ̲)

1

2

1

6

1

7

1

8

1

31102¹09

101500

69³990.82

5⁴44.27

7⁵45.60

1051.38

2235.20

1015

, (- ̲)

1

2

3

2

6

2

7

2

8

2

35330²70

106274

105027.60

3⁴08.79

8⁵84.70

1114.77

2254.39

1015

, (- ̲)

1

2

3

6

3

7

3

8

3

33899³83

106180.50

88³862.42

3⁴93.83

8⁵98.99

1085.21

2432.72

1015

The OCP obtained using the proposed optimization method was compared

with existing patterns. Since the second scenario represents a commonly

practiced farming system in the district, a comparison was made between

the existing cropping pattern and the optimized plan under the optimistic

approach for this scenario.

29, 470.15@C; of yield was obtained under the existing cropping pattern,

but this is slightly reduced to 29, 358.32@C; in the proposed approach. If

we consider the remaining objectives, the total gain was 41, 510, 180ꢂ)ꢅ in

the existing pattern, but it can be increased to 44, 088, 180ꢂ)ꢅ applying the

proposed farming pattern. On the other hand, the cost of cultivation and

labor force were 31, 413, 470ꢂ)ꢅ and 64, 160<3, respectively, in the existing

farm plan, and these can be minimized to 30, 558, 420ꢂ)ꢅ and 57, 384<3,

respectively, by employing the suggested farming pattern.

As shown in Table 9, the area of land allocated to maize, teﬀ, chickpea and

bean in the existing situation was 221.03ℎ0, 141.41ℎ0, 62.29ℎ0, and 121.57,

respectively, while in the proposed pattern, as presented in Table 14, the

land area allocated to these crops respectively decreased to 156ℎ0, 0.00ℎ0,

51.45ℎ0, and 74.57. The area of land allocated to sorghum and pepper in

the existing situation was 401.33ℎ0 and 106.42ℎ0, respectively, while in

the proposed pattern, the land area allocated to these crops respectively

increased to 661.58ℎ0 and 122.85.

Employing the proposed cropping pattern, the objectives are achieved

with higher degrees of membership and lower degrees of non-membership

compared to the existing cropping plan, except for the maximization of the

yield target. However, in the existing situation, the production objective is

accomplished at the cost of agrarian assets and the remaining goals.

There are notable diﬀerences between the conventional farming system

and the suggested farming plan in the usage of manpower, fertilizers,

and agricultural machines; whereas there are slight diﬀerences in the

usage of herbicides and pesticides. If we consider the fertilizer constraint,

208, 591.20:, of fertilizer was required to cultivate 1, 054ℎ0 of farmland

in the existing farming pattern, while the suggested farming plan requires

216, 876:, of fertilizer to cultivate 1, 066.48ℎ0 of farmland within the two

production cycles. To cultivate the allocated crops on the respective areas

of land, 2, 392ℎAB (machine hrs.) were required using the existing plan,

whereas the suggested plan require 2, 272ℎAB.

In the conventional farming pattern, 991.76ℎ0 (96.00%) of 1, 033ℎ0 of land

is allocated to ﬁve crops in the ﬁrst round of farming, and 62.29ℎ0 (28.18%)

of 221.03ℎ0 of land is allocated to chickpea in the second round farming.

In the proposed cropping pattern, 1, 015ℎ0 (98.26%) of 1, 033ℎ0 of land is

allocated to four crops in the ﬁrst season of farming and 51.48ℎ0 (33%) of

156ℎ0 of land is allocated to chickpea in the second round of farming.

The results of the study showed that, including in the remaining scenarios

and DM approaches, teﬀ should not be included in the farming patterns

in favour of increasing the land area for other crops to attain better results

regarding all objectives and constraints.

Comparing the obtained results of optimistic approach under the ﬁrst,

second, and third scenarios, Tables 12, 14, and 16, respectively, the following

assessments have been made.

A comparatively wide area of land is allocated to sorghum and maize, under

From the allocated arable land in the existing farm patterns, a total of

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

the second and third scenarios, respectively, while pepper and bean are

allocated in a wide area of farmland under the second and third scenarios,

respectively. In contrast, Chickpeas share the largest area of land under

the ﬁrst scenario. Considering the resource constraints, water and fertilizer

consumption can be minimized by applying the ﬁrst farming scenario,

while herbicide and pesticide consumption can be reduced by applying

the second scenario. Whereas seed utilization can be minimized in the

second scenario. In the achievement of the considered objectives, enhanced

overall production and minimum expenditure are attained under the third

scenario. While the total gain is signiﬁcantly improved under the second

scenario and the number of manpower is suﬃciently minimized by applying

the ﬁrst farming scenario.

soil degradation unless resources are managed properly. All stakeholders

need to take action, especially in reducing fertilizer use. For example, using

manure and compost can help protect soil fertility, and practices like crop

rotation can lower the need for herbicides.

The study has numerous beneﬁts in assisting the managers of LSF and

farmers of the district for optimal management of agricultural resources. It

also indicates the advantages of OCP to overcome the potential disaster of

crops due to climate change and soil infertility. Furthermore, the study can

be used to predict promising cropping plans from a long-term perspective

as well.

A proﬁcient IFMOO model is proposed to address uncertainties and

associated risks of agriculture, aiming to achieve sustainable crop

production goals. Incorporating the risk management analysis model

into the IFMOO model can increase the eﬃciency and applicability of the

proposed approach to APP problems. Moreover, higher-order extensions of

IFO techniques, such as hesitant IFO (Teferi et al., 2025), are also helpful in

capturing the hesitation among DMs.

Based on the results of the three variants of the problem, as presented in

Tables 12, 14, 16, 18, 19, 20, and 21 there is signiﬁcant variation in the land

allocation to the six crops under each scenario. This proves the solution of

APP problem is contingent upon the DM’s perspective. For example, in the

ﬁrst scenario, the land allocated to maize under the optimistic approach

is reduced by 23.33ℎ0 and 2.48ℎ0, respectively, in the pessimistic and

mixed approaches. Whereas the cropland allocated to bean in the mixed

approach, respectively, decreased by 13.69ℎ0 and 47.09ℎ0 in the pessimistic

and optimistic approaches.

Data availability

The detailed experimental data used to support the ﬁndings of this study

are included in the supplementary information ﬁle(s).

Acknowledgements

The author gratefully acknowledges the Editor-in-Chief and the anonymous

reviewers for their careful evaluation, insightful comments, and

constructive suggestions. Their contributions have substantially improved

the presentation, clarity, and scientiﬁc rigor of this manuscript.

Conflict of Interest

The author declares that he has no competing ﬁnancial interests or personal

relationships that could have inﬂuenced the work reported in this paper.

The total cost of cultivation and manpower goals are better minimized

under a pessimistic approach, while total crop production and proﬁt are

better achieved under the mixed and optimistic approaches, respectively.In

the second scenario, a relatively equal large land area is allocated to

sorghum under the three approaches, whereas for beans, a wide area is

allocated under the mixed approach, but this is reduced by 30.38ha and

36.98ha under the optimistic and pessimistic approaches, respectively. The

farmland allocated to maize and chickpea remains the same under the

pessimistic and mixed approaches and diﬀers by 11.33ha for maize and

3.74ha for chickpea under the optimistic approach. In this scenario, the

yield maximization target is improved under the optimistic approach, while

the proﬁt maximization target is better achieved under the pessimistic

approach. In contrast, the cost and manpower minimization targets are

enhanced under the mixed approach.

Funding

The author did not receive any funding for this research.

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• References must be formatted consistently using standard journal abbreviations

(Index Medicus/MEDLINE) and listed in alphabetical order

5.5.1 In-text citations

• Use the author–year (APA 7th) style for in-text citations. Cite the author’s surname

followed by the year (e.g., Blanco, 2024).

• For two authors, include both names (Nevo and Chen, 2024); for three or more, use

the first author followed by et al. (e.g., Dave et al., 2024). Use lowercase letters (e.g.,

2024a, 2024b) to distinguish multiple works by the same author in the same year.

• Multiple citations should be listed alphabetically and separated by semicolons. In

narrative citations, include the author's name in the sentence; in parenthetical

citations, include both author and year in parentheses.

• Ethiopian names should follow the same format (e.g., Dereje, 2024; Dereje and

Bantewalu, 2024; Dereje et al., 2024).

East Afr. J. Biophys. Comput. Sci.

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5.5.2 In reference list

Journal article

• Szilard, R. (2004). Theories and applications of plate analysis: Classical, numerical

and engineering methods. Appl.Mech. Rev.,57(6), B32–B33

• Kitterød, N.-O., & Leblois, É. (2021). Estimation of sediment thickness by solving

poisson’s equation with bedrock outcrops as boundary conditions. Hydrology

Research, 52(3), 597–619.

• Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., & Russo, A.

(2013). Basic principles of virtual element methods. Mathematical Models and

Methods in Applied Sciences, 23(01), 199–214.

• Bastian, P., Blatt, M., Dedner, A., Dreier, N.-A., Engwer, C., Fritze, R., Gräser, C.,

Grüninger, C., Kempf, D., Klöfkorn, R., et al. (2021). The dune framework: Basic

concepts and recent developments. Computers & Mathematics with Applications,

81, 75–112.

Books

• de Boor, C. (2001). A practical guide to splines(3rd). Springer-Verlag

• Hoffman, J. D., & Frankel, S. (2018). Numerical methods for engineers and scientists.

CRC Press.

Book chapter

• Smith, J. R., & Taylor, S. D. (2022). Digital literacy in the classroom. In M. L. Anderson

(Ed.), Modern educational strategies (pp. 45–67). Academic Press.

Conference /workshop/seminar proceedings

• Roberts, L. (2024, June 12–14). The ethics of artificial intelligence in healthcare

[Paper presentation]. Global Health Summit, Berlin, Germany.

• Nguyen, T. (2025). Neural networks for urban planning. In K. Schmidt (Ed.),

Proceedings of the 12th International Workshop on AI & City Design (pp. 112–125).

Springer

Publications of organizations

• WHO (World Health Organization). (2023). World health statistics 2023: Monitoring

health for the SDGs. WHO, Geneva, Switzerland.

• CSA (Central Statistical Authority). (1991). Agricultural Statistics. 1991. Addis

Ababa, CTA Publications. 250 pp.

Dissertation or Thesis

• Kaufman, J. L. (2024). The impact of remote work on middle management

communication [Doctoral dissertation, Stanford University]. ProQuest Dissertations

and Theses Global

Author Guidlines

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• Moreno, G. (2025). Urban heat islands and public health in coastal cities [Master's

thesis, University of Miami]. Scholarly Repository. https://scholarship.miami.edu/

theses/1234

Publications from websites (URLs)

• National Institute of Mental Health. (2022, May 10). Anxiety disorders. https://www.

nimh.nih.gov/health/topics/anxiety-disorders

• FAO (Food and Agriculture Organization) (2000). Crop and Food Supply

Assessment Mission to Ethiopia. FAOIWFP. Rome. (http://www.fao.org~/GIEWS ).

(Accessed on 21 July 2000).

6. Authorship and Contributor Roles

6.1 Authorship Criteria

EAJBCS adheres to the ICMJE recommendations for authorship:

• Substantial contributions to the conception or design of the work; or the acquisition,

analysis, or interpretation of data

• Drafting the work or revising it critically for important intellectual content

• Final approval of the version to be published

• Agreement to be accountable for all aspects of the work

To qualify as an author, an individual (including any co-author) must have made a

substantial contribution to at least one of the above criteria. Individuals who do not meet

this requirement should be acknowledged rather than listed as authors.

6.2 Contributor Roles (CRediT Taxonomy)

Authors must specify their individual contributions using the CRediT taxonomy (e.g.,

conceptualization, methodology, data curation, writing – original draft, writing – review &

editing, supervision, funding acquisition).

6.3 Corresponding Author Responsibilities

• Ensures all authors have reviewed and approved the manuscript

• Manages all communications with the journal

• Handles post-publication queries and corrections

6.4 Changes to Authorship

• Any changes to the author list after submission require written consent from all

authors, with a clear explanation for the change.

East Afr. J. Biophys. Comput. Sci.

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7. Ethical Standards and Research Integrity

7.1 Publication Ethics

EAJBCS upholds the highest standards of publication ethics, guided by COPE and ICMJE

principles.

• Plagiarism, data fabrication, image manipulation, and other forms of misconduct

are strictly prohibited.

• All submissions are screened using Turnitin software.

• Manuscripts with a similarity index >20% (excluding references) may be rejected

7.2 Human and Animal Research

• Studies involving human participants must include a statement of ethical approval

from an appropriate institutional review board (IRB) and confirmation of informed

consent.

• Animal studies must comply with international, national, and institutional guidelines

for humane treatment and include a statement of ethical approval.

7.3 Clinical Trials and Reporting Guidelines

• Authors must adhere to relevant reporting guidelines and submit completed

checklists as supplementary files.

7.4 Data Availability and Transparency

• Authors must provide a Data Availability Statement describing where and how the

data supporting the findings can be accessed, or explain any restrictions.

• Where possible, datasets should be deposited in recognized repositories with

persistent identifiers (e.g., DOI).

7.5 Conflict of Interest and Funding Disclosure

• All authors must disclose any financial or non-financial conflicts of interest, or state

explicitly if none exist.

• Funding sources must be clearly identified, including grant numbers and the role of

funders in the research.

7.6 Declaration on Use of Generative AI and AI-Assisted

Technologies

• Authors must disclose any use of generative AI or AI-assisted technologies during

manuscript preparation at the time of submission. While such tools can support

efficiency by helping synthesize literature, identify research gaps, generate ideas, or

improve language and readability, they must never replace human expertise, critical

thinking, or scholarly judgment. All AI-generated content must be carefully

Author Guidlines

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reviewed, verified, and edited to ensure accuracy, originality, and alignment with the

author’s own analysis and insights.

• Authors remain fully responsible and accountable for the content of their work,

including safeguarding data privacy, intellectual property, and compliance with

ethical standards. The use of AI tools must be transparent to readers, and a

disclosure statement should be added in a dedicated section before the references

list.

• AI tools must not be listed or cited as authors or co-authors, as authorship implies

responsibilities that only humans can fulfill. Basic tools for grammar, spelling, or

reference checking do not require disclosure.

Declaration statement on Generative AI and AI-Assisted Technologies

• During the preparation of this work, the author(s) used [NAME OF TOOL/SERVICE]

to [REASON]. After using this tool/service, the author(s) reviewed and edited the

content as needed and take(s) full responsibility for the content of the published

article.

7.7 Corrections, Retractions, and Expressions of Concern

• EAJBCS follows COPE guidelines for handling corrections, retractions, and

expressions of concern.

• Authors are responsible for notifying the journal promptly if errors are identified

post-publication.

8. Peer Review Process

8.1 Peer Review Model

• EAJBCS operates a double-blind peer review process: both authors and reviewers

remain anonymous to each other, minimizing bias and ensuring objective

evaluation.

• In exceptional cases, open peer review or post-publication review may be

considered, with appropriate disclosure.

8.2 Reviewer Selection and Recognition

• Reviewers are selected based on subject expertise, publication record, and absence

of conflicts of interest.

• The journal maintains a diverse reviewer pool, reflecting geographic, institutional,

and demographic diversity.

• Reviewers receive formal recognition, including certificates and eligibility for annual

reviewer awards.

East Afr. J. Biophys. Comput. Sci.

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8.3 Peer Review Workflow

1. Initial Editorial Screening: Manuscripts are assessed for scope, originality, and

compliance with submission guidelines.

2. Assignment to Section Editor: Suitable manuscripts are assigned to a section

editor for oversight.

3. Reviewer Invitation: At least two independent reviewers are invited.

4. Review and Decision: Reviewers evaluate the manuscript for scientific rigor,

originality, clarity, and ethical compliance, providing detailed reports and

recommendations.

5. Editorial Decision: Based on reviewer feedback, the editor decides to accept,

request revisions, or reject the manuscript.

6. Revision and Re-Review: Authors respond to reviewer comments and submit

revised manuscripts, which may undergo further review.

7. Final Decision and Acceptance: Upon satisfactory revision, the manuscript is

accepted for publication.

9. Data, Code, and Materials Sharing

• EAJBCS encourages the sharing of data, code, and materials to promote

transparency and reproducibility.

• Authors should deposit datasets and code in recognized repositories and provide

persistent identifiers.

• If data cannot be shared due to ethical, legal, or privacy constraints, authors must

provide a clear explanation in the Data Availability Statement.

10. Equity, Diversity, and Inclusion (EDI)

• EAJBCS is committed to promoting equity, diversity, and inclusion in its editorial

practices, reviewer selection, and published content.

• Authors are encouraged to:

▪

Use inclusive language and avoid bias in reporting

Provide detailed demographic descriptions of study populations

Justify sample selection and discuss generalizability constraints

Include positionality or reflexivity statements where relevant

Cite a diverse range of sources, including underrepresented scholars

• The journal periodically reviews its policies and practices to ensure alignment with

evolving EDI standards.

Author Guidlines

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11. Discoverability, Metadata, and Indexing

• All articles are assigned Digital Object Identifiers (DOIs) and registered with

CrossRef for persistent citation and discoverability.

• Metadata, including abstracts, keywords, author affiliations, and funding

information, are provided in machine-readable formats to facilitate indexing in

major databases such as AJOL, DOAJ, CABI Abstracts, and FAO AGRIS.

• The journal maintains up-to-date ISSN registration for both print and electronic

versions.

12. Submission Process

12.1 Online Submission System

• Manuscripts must be submitted via the journal’s online submission platform by

clicking https://journals.hu.edu.et/hu-journals/index.php/eajbcs/submission

• Authors are required to complete all metadata fields, upload the main manuscript

and supplementary files, and provide signed author agreements and ethical

declarations.

12.2 Submission Checklist

Authors must ensure that:

• The manuscript adheres to the journal’s formatting and structure requirements

• All authors meet the authorship criteria and have approved the submission

• Ethical approvals and informed consent statements are included, as applicable

• Data availability, conflict of interest, and funding statements are provided

• Relevant reporting checklists are completed and uploaded

• Figures and tables are submitted in the correct format and resolution

12.3 Copyright Transfer and Author Agreement

• Upon acceptance, authors are required to sign a copyright transfer or license

agreement, specifying the terms of publication and reuse.

• The agreement template is available on the journal website.

13. Post-Acceptance and Publication

13.1 Proofs and Corrections

• Authors will receive proofs for review before publication.

• Only minor corrections (e.g., typographical errors) are permitted at this stage.

• Substantive changes require editorial approval.

East Afr. J. Biophys. Comput. Sci.

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13.2 Publication Frequency and Timeliness

• EAJBCS publishes articles on a biannual basis, with continuous online publication

to ensure timely dissemination

• The journal adheres to its stated publication schedule and promptly communicates

any delays to authors.

13.3 Post-Publication Updates

• Corrections, retractions, and expressions of concern are published in accordance

with COPE and ICMJE guidelines, with clear linkage to the original article.

14. Contact Information

For inquiries regarding submissions, editorial policies, or technical support, please contact:

Editorial Office

Principal Contact

Dr. Admasu Tadesse

Editor in chief, EAJBCS

Support Contact

East African Journal of

Biophysical and

Computational Sciences

Dr. Abnet Woldesenbet

Editorial Manager, EAJBCS

Phone +251- 911811819

Email: abnetm@hu.edu.et

College of Natural and

Computational Sciences,

Hawassa University

Phone: +251-913-267054

Email: eajbcs@hu.edu.et

Phone +251-913267054

Email: admasut@hu.edu.et