EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6423
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Spatiotemporal Framework for Malaria Control Integrating
Exposure Heterogeneity and Optimal Strategy Deployment

Nadeem Abbas1, Wasfi Shatanawi1,2, Syeda Alishwa Zanib3,∗

1 Department of Mathematics and Sciences, College of Humanities and Sciences,
Prince Sultan University, Riyadh, 11586, Saudi Arabia
2 Department of Mathematics, Faculty of Science, The Hashemite University, P.O Box 330127, Zarqa
13133, Jordan
3Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 44000,
Pakistan

Abstract. Malaria remains one of the most serious and widespread vector-borne infectious diseases world-
wide, caused by Plasmodium protozoa and transmitted through the bites of infected female Anopheles
mosquitoes. In this study, we present a novel, integrative, bioinformatics-driven deterministic mathematical
model to investigate the complex transmission dynamics of malaria. The model uniquely distinguishes be-
tween homogeneous and heterogeneous exposed human compartments (Ehm, Eht) and explicitly incorporates
mosquito population dynamics. The coupled system includes human compartments SM , Ehm, Eht, IM , HM ,
RM , and mosquito compartments SF , EF , IF . We perform a comprehensive stability analysis of both the
malaria-free and endemic equilibrium points, assessing global stability in relation to the basic reproduction
number R0. Sensitivity analysis reveals that the mosquito biting rate (αM ) and the transmission probability
(βM ) are key parameters influencing malaria spread. To evaluate the effectiveness of intervention strategies,
we incorporate time-dependent control variables and formulate an optimal control problem using Pontrya-
gin’s Maximum Principle. The control interventions include bed net usage (m1), antimalarial medication
treatment (m2), and insecticide spraying (m3). Numerical simulations, implemented using a fourth-order
Runge-Kutta method in Python and MATLAB, demonstrate the significant impact of these strategies in reduc-
ing both exposed and infected populations. Our findings emphasize the importance of timely and targeted
control measures and highlight the effectiveness of integrating bioinformatics with mathematical modeling
to support informed decision-making in malaria control policies.

2020 Mathematics Subject Classifications: 26A33, 34A08, 03C65

Key Words and Phrases: Malaria Disease, Mathematical Model, Optimal Control, Homogeneous and
Heterogeneous Individuals, Sensitivity Analysis

1. Introduction

One of the most serious infectious diseases in the world is malaria, a potentially fatal vector-borne
disease spread by the protozoan Plasmodium that infects female Anopheles mosquitoes and bites

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6423

Email addresses: nabbas@psu.edu.sa (N. Abbas),
wshatanawi@psu.edu.sa (W. Shatanawi),19907@riphahfsd.edu.pk (S. A. Zanib)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 2 of 35

people [1–5]. The causative agents, including Plasmodium falciparum, Plasmodium vivax, Plasmod-
ium ovale, and Plasmodium malariae, exhibit differences in microscopic appearance, geographical
distribution, and clinical characteristics, particularly in terms of infection potential, severity, and
propensity for relapse shown in Figure 1. Among them, P. falciparum is recognized as the most
perilous to humans [6]. Malaria is spread from person to person by mosquito bites. Plasmodium
vivax and P. falciparum are the two most common malaria species that infect people among the
five species that affect malaria overall. More than 400,000 deaths a year are caused by these two
species, with 90% of the deaths occurring in Africa.

Malaria 
Transmission 

dynamic

Figure 1: Malaria Transmission

According to the latest World Malaria Report of December 2019, there were 228 million reported
cases of malaria in 2018, a slight decline from 231 million cases in 2017. In 2018, an estimated
405,000 people succumbed to the disease, compared to 416,000 in 2017. The burden of malaria
remains disproportionately high in the WHO African Region, accounting for 93 % of cases and
94% of deaths in 2018. Malaria prevalence spans over 100 countries, resulting in approximately 216
million cases and 655,000 deaths in 2010 [7, 8]. Despite longstanding efforts to combat malaria, it
persists as a major public health challenge in affected areas, primarily in tropical and subtropical
regions of Africa, the Eastern Mediterranean, Asia, and South America. Vulnerable populations in-
clude pregnant women and non-immune travelers. Beyond its health-related implications, malaria
poses a significant socioeconomic threat to endemic nations. In Africa alone, the annual economic
burden of malaria was estimated at 8 billion [2, 5, 9]. Consequently, there has been a critical
need to devise various intervention strategies to mitigate the impact of malaria. While global ef-
forts are underway to develop a perfect vaccine against malaria, none currently exists [5, 10–16].
To control the spread of the disease, preventive measures include mosquito-reduction strategies



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 3 of 35

and personal protection against mosquito bites, facilitated by the use of insecticide-treated bed
nets (ITNs), intermittent preventive treatment (IPT), and the elimination of vector breeding sites
[2, 17–19]. Additional intervention strategies encompass indoor residual spraying (IRS) to eliminate
infected indoor mosquitoes and the sterile insect technique. The use of anti-malarial drugs also
plays a role in regulating malaria [6, 20–23]. Numerous mathematical models have been developed
to investigate fluid dynamics, neurological processes, and the transmission dynamics of epidemic
diseases over time [24–32]. Oke et al. (2020)[20] studied the malaria dynamics transmission us-
ing a non-linear mathematical model. He studied stability theory, optimum control techniques
and numerical simulations to show how different control strategies affect the spread of disease.
Tchoumi et al., (2021) [33] investigated the co-dynamics of malaria and COVID-19 in order to
determine the best control techniques and stability conditions. Results indicated that treating
both infections at the same time prevented their spread more successfully, which shed light on
the difficulties encountered during the COVID-19 pandemic. In sense of fractional derivatives ,
Sinan et al., (2022) [34] used and non-linear mathematical model to study the malaria dynamics
spread by mosquito vectors. The effectiveness of control measures like bed nets and pesticides in
the spread of infections was demonstrated using numerical simulations. Alhaj (2023) [35] studied
a deterministic model for Malaria transmission, analyzing the basic reproduction number (R0) and
stability conditions. Simulation and sensitivity analysis highlight control interventions’ impact, pro-
viding recommendations for eradicating Malaria transmission. Adegbite et al. (2023)[36] explored
malaria importation’s impact globally, using a novel ODEs system to evaluate vigilance-driven con-
ventional and traditional control strategies, emphasizing the need for 98% vigilance for effective
malaria management. Alqahtani et al, (2024) [37] modeled using a deterministic fractional-order
system incorporating a novel hospitalized compartment to simulate transmission dynamics between
humans and mosquitoes. The basic reproduction number was derived using the next-generation
matrix method, and numerical simulations were performed using the fourth-order Runge–Kutta
method in MAPLE. Olutimo et al., (2024) [38] explored the influence of environmental immunity
on malaria transmission through a developed SIR-SI mathematical model. The analysis revealed
that the malaria-free equilibrium was locally asymptotically stable when the reproduction number
was below unity. Numerical simulations demonstrated that acquired environmental immunity, in-
fluenced by nutrition and medicinal herbs, significantly diminished malaria spread by bolstering the
recovered class and reducing the infected class. This paper presents a dynamic mathematical model
to investigate malaria transmission dynamics, building upon earlier studies. Our primary goal is to
identify and assess effective control strategies for disease management. To this end, we develop an
optimal control model that incorporates interventions such as treated bed nets, pesticide applica-
tion, hospitalization, and preventive treatment. A novel aspect of our research is the division of the
exposed human population into two distinct classes based on genetic heterogeneity: homogeneous
and heterogeneous individuals, reflecting differences in resistance to malaria. The genetic basis of
hemoglobin variants, particularly the sickle cell allele, is highly prevalent in African populations,
where malaria transmission is also endemic. heterogeneous individuals carrying the sickle cell trait
exhibit increased resistance to Plasmodium falciparum infection and often remain asymptomatic
despite exposure, whereas homogeneous individuals are more susceptible to infection and disease
symptoms [39]. Additionally, our model introduces a hospitalized class, which is a novel variable
rarely considered in malaria modeling literature. This inclusion allows us to better capture the
dynamics of severe cases and healthcare interventions. To determine the optimal control strate-
gies, we apply Pontryagin’s Maximum Principle (PMP), a powerful tool in optimal control theory.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 4 of 35

Furthermore, we calculate the basic reproduction number R0 and perform sensitivity analysis to
evaluate the influence of key parameters on disease transmission and control efficacy. This model
serves as a valuable framework for evaluating the effectiveness of various malaria control measures
and enhances understanding of malaria dynamics post-intervention within populations.

2. Model Analysis

We develop an SMEhmEhtIMHMRM–SFEF IF compartmental model to simulate malaria trans-
mission dynamics between humans and mosquitoes. The human population is structured into six
distinct compartments: susceptible individuals (SM ), heterogeneously exposed individuals (Eht),
homogeneously exposed individuals (Ehm), infected individuals (IM ), hospitalized individuals (HM ),
and recovered individuals (RM ). Recovered individuals acquire temporary immunity but eventu-
ally re-enter the susceptible compartment (SM ) due to waning immunity. The mosquito population
(female Anopheles) is categorized into three compartments: susceptible mosquitoes (SF ), exposed
mosquitoes (EF ), and infectious mosquitoes (IF ). Crucially, both infected and recovered humans
transmit the malaria parasite to mosquitoes during blood-feeding events, as illustrated in the trans-
mission flowchart (Figure 2). The total human population at time t is given by:

TM = SM + Ehm + Eht + IM +HM +RM , (2.1)

while the total mosquito population is:

TF = SF + EF + IF . (2.2)

The parameters governing the model are systematically defined in Table 1. The mosquito lifecycle
is explicitly modeled through three developmental stages: immature/susceptible (SF ), exposed
(EF ), and infected (IF ). The resulting malaria transmission dynamics are captured by a coupled
system of nonlinear ordinary differential equations:

dSM
dt

= λM − αM βM SM IF
TM

− ϵM SM + ωM RM − γMSM , (2.3)

dEhm

dt
=
αM βM SM IM

TM
− ϵM Ehm − φM Ehm, (2.4)

dEht

dt
= γMSM − (ϵM + ηM )Eht, (2.5)

dIM
dt

= φM Ehm − (µM + ϵM + ρM ) IM , (2.6)

dHM

dt
= ρM IM − (µM + ϵM + δM )HM , (2.7)

dRM

dt
= ηMEht + δM HM − ϵM RM − ωMRM , (2.8)

dSF
dt

= λF − αM βF SF IM
TF

+
αM βM RM SF

TF
− ϵF SF , (2.9)

dEF

dt
=
αM βF SF IM

TF
+
αM βM RM SF

TF
− ϕM EF − ϵF EF , (2.10)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 5 of 35

𝑺𝑴

𝑬𝒉𝒎

𝑰𝑴 𝑯𝑴 𝑹𝑴

𝑺𝑭 𝑬𝑭 𝑰𝑭

𝝐𝑴

𝝐𝑴 + 𝝁𝑴 𝝐𝑴 + 𝝁𝑴

𝝐𝑴

𝝐𝑭 𝝐𝑭𝝐𝑭

𝝀𝑭

𝝀𝑴
𝜸𝒉𝒎

𝜶𝑭𝜷𝑭𝑰𝑴
𝑻𝑭

+
𝜶𝑭𝜷𝑭𝑹𝑴

𝑻𝑭

𝝐𝑴
𝝍𝑴

𝝆𝑴 𝜹𝑴

𝝓𝑭

𝝎𝑴

𝑬𝒉𝒕
𝝐𝑴

𝜼𝑴

𝜸𝒉𝒎

Figure 2: Schematic representation of malaria transmission dynamics. Solid arrows indicate natural pro-
gression; dashed arrows denote infection pathways.

dIF
dt

= −ϵF IF + ϕM EF , (2.11)

having initial conditions,

SM ≥ 0, Ehm ≥ 0, Eht ≥ 0, IM ≥ 0, HM ≥ 0, RM ≥ 0, SF ≥ 0, EF ≥ 0, IF ≥ 0. (2.12)

2.1. Feasible Region and Positivity of Solutions

Theorem 1. There exists a feasible region ΩT in which the solutions of the model (2.3)–(2.11)
remain bounded and positively invariant.

Proof. Summing the equations for the human compartments in system (2.3)–(2.11), we define
the total human population as

TM = SM + Ehm + Eht + IM +HM +RM .

Differentiating with respect to time yields

dTM
dt

=
dSM
dt

+
dEhm

dt
+
dEht

dt
+
dIM
dt

+
dHM

dt
+
dRM

dt
.

From the model equations, this becomes

dTM
dt

= λM − ϵM (SM + Ehm + Eht + IM +HM +RM )− µM (IM +HM ).



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 6 of 35

In the absence of disease-induced mortality (i.e., µM = 0), the equation simplifies to

dTM
dt

= λM − ϵMTM .

Solving this linear differential inequality gives

TM (t) ≤
(
TM (0)− λM

ϵM

)
e−ϵM t +

λM
ϵM

.

Taking the limit as t→ ∞, we have

0 ≤ TM (t) ≤ λM
ϵM

.

Thus, the feasible region for the human population is

ΩM =

{
(SM , Ehm, Eht, IM , HM , RM ) ∈ R6

+ : SM + Ehm + Eht + IM +HM +RM ≤ λM
ϵM

}
.

Similarly, for the mosquito population, defining

TF = SF + EF + IF ,

we obtain the feasible region

ΩF =

{
(SF , EF , IF ) ∈ R3

+ : SF + EF + IF ≤ λF
ϵF

}
.

Therefore, the overall feasible region for the system is

ΩT = ΩM × ΩF ,

which is positively invariant and bounded.

Theorem 2. The solutions of the system of equations (2.3)–(2.11) with initial conditions (2)
remain non-negative for all time t > 0.

Proof. Suppose, for the sake of contradiction, that there exists a time t̂ > 0 such that

SM (t̂) = 0,

while for all t ∈ (0, t̂),

SM (t) > 0, Ehm(t) > 0, Eht(t) > 0, IM (t) > 0, HM (t) > 0,

RM (t) > 0, SF (t) > 0, EF (t) > 0, IF (t) > 0.

From the system (2.3)–(2.11), the equation for the susceptible human population SM at time t̂ is

dSM
dt

∣∣∣∣
t=t̂

= λM − αMβMSM (t̂)IF (t̂)

TM
− ϵMSM (t̂) + ωMRM (t̂)− γMSM (t̂).



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 7 of 35

Since SM (t̂) = 0, this reduces to

dSM
dt

∣∣∣∣
t=t̂

= λM + ωMRM (t̂) ≥ 0,

because λM > 0 and RM (t̂) ≥ 0. Moreover, for t ∈ (0, t̂), we have

dSM
dt

+ ϵMSM (t) ≤ λM + ωMRM (t).

Multiplying both sides by the integrating factor eϵM t and integrating over [0, t̂], we obtain

SM (t̂) ≥ SM (0)e−ϵM t̂ > 0,

which contradicts the assumption that SM (t̂) = 0. By similar arguments, positivity of the other
state variables Ehm, Eht, IM , HM , RM , SF , EF , IF can be established for all t > 0. Therefore, the
solutions remain non-negative for all t > 0.

2.2. Malaria-Free Equilibrium Point

The steady state, also known as the malaria-free equilibrium (MFE), is attained when there is
no infection present in the population; that is, all exposed and infected compartments are zero. At
this equilibrium, the system is at a steady state where the disease does not persist. Setting the
derivatives of all compartments to zero,

dSM
dt

=
dEhm

dt
=
dEht

dt
=
dIM
dt

=
dHM

dt
=
dRM

dt
=
dSF
dt

=
dEF

dt
=
dIF
dt

= 0,

and solving the resulting algebraic system yields the malaria-free equilibrium point:

(SM , Ehm, Eht, IM , HM , RM , SF , EF , IF ) =

(
λM
ϵM

, 0, 0, 0, 0, 0,
λF
ϵF
, 0, 0

)
.

At this equilibrium, the human and mosquito populations consist entirely of susceptible individuals,
with no exposed or infected individuals present. This equilibrium plays a crucial role in stability
analysis: if it is stable, malaria will eventually be eradicated from the population. The stability
typically depends on the basic reproduction number R0, which determines whether the disease can
invade and persist.

3. Malaria-Present (Endemic) Equilibrium Point

The endemic equilibrium represents the steady state where malaria persists in both human
and mosquito populations. Setting the derivatives in the system (2.3-2.11) to zero and solving
algebraically yields the following expressions for each compartment. Equilibrium values are denoted



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 8 of 35

with an asterisk (e.g., E∗).

S∗
M =

λM (ϵM + φM )(µM + ϵM + ρM )

αMβMϕMφME∗
F /T

∗
M + (ϵM + φM )(µM + ϵM + ρM )(ϵM + γM )

E∗
hm =

αMβMS
∗
MI

∗
F

T ∗
M (ϵM + φM )

,

E∗
ht =

γMS
∗
M

ϵM + ηM

I∗M =
φME

∗
hm

µM + ϵM + ρM
,

H∗
M =

ρMI
∗
M

µM + ϵM + δM

R∗
M =

ηME
∗
ht + δMH

∗
M

ϵM + ωM

S∗
F =

λF

ϵF +
αMβF I∗M

TF
− αMβMR∗

M
TF

E∗
F =

S∗
F

(
αMβF I∗M

TF
+

αMβMR∗
M

TF

)
ϕM + ϵF

I∗F =
ϕME

∗
F

ϵF

(3.13)

3.1. Basic Reproduction Number

The basic reproduction number, R0, quantifies the average number of secondary infections
generated by a single infected mosquito or human in a completely susceptible population. To
calculate R0, we employ the next-generation matrix method as presented by Van den Driessche
and Watmough [40]. This approach focuses on the infectious compartments and incorporates
both transmission and transition processes within and between these classes. The next-generation
matrix is defined as FV −1, where the matrix F includes terms associated with new infections, and
V consists of transition terms excluding new infections. The basic reproduction number R0 is given
by the spectral radius (dominant eigenvalue) of FV −1, denoted by ρ(FV −1). The matrices F and
V are expressed as:

F =



0 0 0 0 0 αMβM
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 αMβM 0 0 0 0
0 0 0 0 0 0

 , (3.14)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 9 of 35

V =



ϵM + φM 0 0 0 0 0
−φM µM + ϵM + ρM 0 0 0 0
0 −ρM µM + ϵM + δM 0 0 0
0 0 −δM ϵM + ωM 0 0
0 0 0 0 ϕM + ϵF 0
0 0 0 0 −ϕM ϵF

 . (3.15)

The product FV −1 is

FV −1 =



0 0 0 0 0
αMβM
ϵM + φM

0 0 0 0 0
φMαMβM

(ϵM + φM )(µM + ϵM + ρM )

0 0 0 0 0
ρMφMαMβM

(ϵM + φM )(µM + ϵM + ρM )(µM + ϵM + δM )

0 0 0 0 0
δMρMφMαMβM

(ϵM + φM )(µM + ϵM + ρM )(µM + ϵM + δM )(ϵM + ωM )

0
αMβM
ϕM + ϵF

0 0 0 0

0
ϕMαMβM

(ϕM + ϵF )ϵF
0 0 0 0



.

(3.16)
The dominant eigenvalue, which defines the basic reproduction number R0, is

R0 =

√
ϵF (ϕF + ϵF )(ϵM + φM )(µM + ϵM + ρM )ϕFαFβFφMαMβM

(ϕF + ϵF )ϵF (ϵM + φM )(µM + ϵM + ρM )
. (3.17)

The behavior of R0 with respect to different parameter values is illustrated in Figure 3. This
analysis provides insight into how each parameter influences malaria transmission dynamics, which
is essential for designing effective control strategies.

3.2. Global Stability of the Malaria-Free Equilibrium Point

Following the approach in [41], the system can be represented in a triangular form to analyze
the global stability of an equilibrium point:{

dA1
dt = X(A1, A2),
dA2
dt = Y (A1, A2),

(3.18)

where
H(A1, A2) = 0.

At the malaria-free equilibrium point (MFEP), the uninfected population is denoted by A1 ∈ R2,
while the infected population is represented by A2 ∈ R7. The criterion for global stability at the
IFEP is given by

dA1

dt
= X(A1, 0) = 0, (3.19)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 10 of 35

(a) Behavior of R0 between αM and βF

10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0
M

0.2

0.4

0.6

0.8

1.0

1.2

1.4

F

9.00 12.00
15.00

18.00
21.00

24.00
27.00

30.00

33.00

36.00

39.00

42.00

45.00

48.00

(b) Effect of varying αM and βM on the reproduction
number R0

(c) Behavior of R0 between ϕF and ϵM

0.06 0.08 0.10 0.12 0.14
F

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0.00008

0.00009

0.00010

M

27
.0

0
27

.3
0

27
.6

0
27

.9
0

28
.2

0
28

.5
0

28
.8

0
29

.1
0

29
.4

0

29
.7

0

30
.0

0

30
.3

0

30
.6

0

30
.9

0

31
.2

0

(d) Effect of varying ϕF and ϵM on the reproduction
number R0

(e) Behavior of R0 between ϵF and βM

0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050
F

0.010

0.015

0.020

0.025

0.030

0.035

0.040

M

20.00

25.00

30
.00

35
.00

40
.00

45
.00

50
.0

0

55
.0

0
60

.0
0

65
.0

0
70

.0
0

75
.0

0

(f) Effect of varying βM and ϵF on the reproduction
number R0

Figure 3: Behavior of R0 at sensitive parameters



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 11 of 35

and
H(A1, A2) = P1NG∗ − Ĥ(A1, A2). (3.20)

Theorem 3. The system of equations (2.3)–(2.11) is globally asymptotically stable at the malaria-
free equilibrium point if R0 < 1.

Proof. To verify condition (3.19), consider the malaria-free equilibrium point

E0 =
(
K0, 0

)
= (SM , Ehm, Eht, IM , HM , RM , SF , EF , IF ) =

(
λM
ϵM

, 0, 0, 0, 0, 0,
λF
ϵF
, 0, 0

)
.

The subsystem for the uninfected classes reduces to

dA1

dt
= X(A1, 0), (3.21)

with {
dS∗

M
dt = λM − ϵMS

∗
M ,

dS∗
F

dt = λF − ϵFS
∗
F .

(3.22)

Solving (3.22) yields the unique equilibrium

(S∗
M , Ehm, Eht, IM , HM , RM , S

∗
F , EF , IF ) =

(
λM
ϵM

, 0, 0, 0, 0, 0,
λF
ϵF
, 0, 0

)
,

which satisfies the global asymptotic stability condition for X0. Next, to verify condition (3.20),
define

F (A1, A2) = P1NA∗ − Ŷ (A1, A2),

where
Ŷ (A1, A2) ≥ 0,

and

H(A1, A2) =



αMβMSM IF
TM

− ϵMEhm − φMEhm

γMSM − (ϵM + ηM )Eht

φMEhm − (µM + ϵM + ρM )IM
ρMIM − (µM + ϵM + δM )HM

δMHM − ϵMRM − ωMRM + ηMEht
αF βFSF IM

TF
+ αF βF IMRM

TF
− ϕFEF − ϵFEF

ϕFEF − ϵF IF


,

N∗
A =



(−ϵM − φM )Ehm +
αMβMS∗

M IF
TM

(−ϵM − ηM )Eht

φMEhm + (−µM − ϵM − ρM )IM
ρMIM + (−µM − ϵM − δM )HM

ηMω + δMHM + (−ϵM − ωM )RM(
αF βFS∗

F
TF

+ αF βFRM
TF

)
IM + αF βF IMRM

TF
+ (−ϕF − ϵF )EF

ϕFEF − ϵF IF


,



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 12 of 35

and

Ŷ (A1, A2) =



βM cαM (S∗
M−SM )

TM

0
0
0
0

βFαF IM (S∗
F−SF+RM )
TF

0


≥ 0.

Since Ŷ (A1, A2) ≥ 0, conditions (3.19) and (3.20) hold. Therefore, the system is globally asymp-
totically stable at the malaria-free equilibrium point when R0 < 1, which completes the proof of
Theorem 3.

Table 1: Descriptions of Model Parameters

Parameter Meaning

λM Growth rate of the human population.
βF Transmission rate of malaria parasites from infected humans to suscep-

tible mosquitoes.
βM Transmission rate of malaria parasites from infected mosquitoes to sus-

ceptible humans.
αM Rate at which mosquitoes bite humans.
ϵM Natural mortality rate of humans.
ψM Transition rate of humans from exposed to infected class.
ρM Rate at which infected humans move to hospitalization.
µM Malaria-induced mortality rate in humans.
δM Rate of recovery from hospitalization to recovered class.
λF Recruitment rate of the mosquito population.
ϵF Natural death rate of mosquitoes.
ϕF Rate at which mosquitoes progress from exposed to infectious stage.
ηM Recovery rate of heterogeneous exposed humans.

4. Global Stability of the Malaria-Present Equilibrium Point

Theorem 4. Consider the malaria transmission model defined by Equations (2.3-2.11). If the
basic reproduction number R0 > 1, then the endemic equilibrium E∗ = (S∗

M , E
∗
hm, . . . , I

∗
F ) is globally

asymptotically stable in the interior of the feasible region ΩT = {(SM , . . . , IF ) ∈ R9
+}.

Proof. We prove global stability using a Lyapunov function and LaSalle’s Invariance Principle.
Define the Lyapunov function [42]:

L =
∑
i

(
xi − x∗i − x∗i ln

xi
x∗i

)
, (4.23)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 13 of 35

where xi represents the model compartments and x∗i their endemic equilibrium values. The time
derivative of L is:

dL

dt
=
∑
i

(
1− x∗i

xi

)
dxi
dt
. (4.24)

Substituting the model equations and simplifying, we obtain:(
1−

S∗
M

SM

)
dSM
dt

=

(
1−

S∗
M

SM

)[
λM − αMβMSMIF

TM
− ϵMSM + ωMRM − γMSM

]
,(

1−
E∗

hm

Ehm

)
dEhm

dt
=

(
1−

E∗
hm

Ehm

)[
αMβMSMIF

TM
− (ϵM + φM )Ehm

]
,(

1−
I∗M
IM

)
dIM
dt

=

(
1−

I∗M
IM

)
[φMEhm − (µM + ϵM + ρM )IM ] ,(

1−
S∗
F

SF

)
dSF
dt

=

(
1−

S∗
F

SF

)[
λF − αMβFSF IM

TF
+
αMβMRMSF

TF
− ϵFSF

]
,(

1−
E∗

F

EF

)
dEF

dt
=

(
1−

E∗
F

EF

)[
αMβFSF IM

TF
+
αMβMRMSF

TF
− (ϕM + ϵF )EF

]
.

(4.25)

Using equilibrium conditions and algebraic identities:

dL

dt
=− ϵM

(SM − S∗
M )2

SM
− ϵF

(SF − S∗
F )

2

SF

+
αMβMS

∗
MI

∗
F

T ∗
M

[
4−

S∗
M

SM
−
EhmSMI

∗
F

E∗
hmS

∗
MIF

−
E∗

hmIF
EhmI

∗
F

− SMIF
S∗
MI

∗
F

]
+
αMβFS

∗
F I

∗
M

T ∗
F

[
3−

S∗
F

SF
−
EFSF I

∗
M

E∗
FS

∗
F IM

−
E∗

F IM
EF I∗M

]
− ΩT ,

where ΩT > 0 contains mortality/progression terms. By the Arithmetic Mean-Geometric Mean
(AM-GM) inequality:

4−
S∗
M

SM
−
EhmSMI

∗
F

E∗
hmS

∗
MIF

−
E∗

hmIF
EhmI

∗
F

− SMIF
S∗
MI

∗
F

≤ 0

3−
S∗
F

SF
−
EFSF I

∗
M

E∗
FS

∗
F IM

−
E∗

F IM
EF I∗M

≤ 0

Equality holds iff:
SM
S∗
M

=
Ehm

E∗
hm

=
IF
I∗F

and
SF
S∗
F

=
EF

E∗
F

=
IM
I∗M

,

Thus, dL
dt ≤ 0 for all (SM , . . . , IF ) ∈ Γ◦, with equality only when:

SM = S∗
M , Ehm = E∗

hm, IF = I∗F , SF = S∗
F , EF = E∗

F , IM = I∗M .

The largest invariant set where dL
dt = 0 is E∗. By LaSalle’s principle, all trajectories in ΩT converge

to E∗ as t→ ∞. Thus, for R0 > 1, E∗ is globally asymptotically stable in ΩT .



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 14 of 35

4.1. Sensitivity Analysis

Understanding the parameters that influence the basic reproduction number R0 is crucial for
guiding effective malaria control strategies. Sensitivity analysis [43–45] plays a pivotal role in
identifying key parameters that should be prioritized, especially when resources are limited. The
normalized forward sensitivity index of a variable W , which depends on a parameter p, is defined
as [46]:

ωW
p =

∂W

∂p
× p

W
. (4.26)

According to this definition, a larger magnitude of the sensitivity index indicates a greater influence
of the parameter p on the variable W . Applying this to the basic reproduction number R0, the
sensitivity index is given by:

ωR0
p =

∂R0

∂p
× p

R0
. (4.27)

The sensitivity indices of key parameters affecting R0 are computed as follows:

ωR0
ϕM

=
ϵF

2ϕM + 2ϵF
> 0, ωR0

ϵF
= − ϕM + 2ϵF

2ϕM + 2ϵF
< 0, ωR0

φM
=

ϵM
2ϵM + 2φM

> 0,

ωR0
ρM

= − ρM
2µM + 2ϵM + 2ρM

< 0, ωR0
µM

= − µM
2µM + 2ϵM + 2ρM

< 0, ωR0
βM

= 1,

ωR0
ϵM

= − (µM + ρM + 2ϵM + φM )ϵM
2(ϵM + φM )(µM + ϵM + ρM )

, ωR0
αM

= 1.

(4.28)

M F M M M M M M

Model Parameters

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Se
ns

iti
vi

ty
 In

de
x 0.50

-0.30

0.20

-0.10 -0.10

1.00

-0.50

1.00

Sensitivity Analysis

Figure 4: Sensitivity Analysis of the Basic Reproduction Number R0.

Figure 4 illustrates the sensitivity indices of R0 with respect to various parameters. This analysis
highlights which parameters have the greatest impact on malaria transmission dynamics, thereby
informing targeted intervention strategies.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 15 of 35

5. Optimal Control

To control the transmission of malaria, we examine the effects of medicinal treatments and
preventive measures through a set of time-dependent control variables m1,m2, and m3, defined as
follows:

(i) Provision of treated pesticide bed nets to the susceptible population, represented by m1,

(ii) Treatment of the infected population, represented by m2,

(iii) Insecticide spray application, represented by m3.

The mathematical model incorporating these optimal controls m1,m2, and m3 is described by the
following non-autonomous system of nonlinear ordinary differential equations:

dSM
dt

= λM − m1αMβMSMIF
TM

− ϵMSM + ωMRM − γMSM , (5.29)

dEhm

dt
=
m1αMβMSMIF

TM
− ϵMEhm − φMEhm, (5.30)

dEht

dt
= γMSM − (ϵM + ηM )Eht, (5.31)

dIM
dt

= φMEhm − (µM + ϵM +m2ρM )IM , (5.32)

dHM

dt
= m2ρMIM − (µM + ϵM + δM )HM , (5.33)

dRM

dt
= ηMEht + δMHM − ϵMRM − ωMRM , (5.34)

dSF
dt

= λF −m3

(
αMβMSF IM

TF
+
αMβMRMSF

TF

)
− (m1 +m3)ϵFSF , (5.35)

dEF

dt
= m3

(
αMβMSF IM

TF
+
αMβMRMSF

TF

)
− ϕMEF − (m1 +m3)ϵFEF , (5.36)

dIF
dt

= ϕMEF − (m1 +m3)ϵF IF . (5.37)

Our goal is to find the optimal control functions (m∗
1,m

∗
2,m

∗
3) that minimize the objective

functional

J(m1,m2,m3) = min
m1,m2,m3

∫ T

0

[
P1Ehm + P2Eht + P3IM + P4EF + P5IF

+ L1m1(t)
2 + L2m2(t)

2 + L3m3(t)
2
]
dt,

(5.38)

where T is the final time, P1, P2, P3, P4, P5 are weight parameters representing the relative costs
associated with exposed and infected human and mosquito populations, and L1, L2, L3 are the
weight costs associated with implementing the control measures.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 16 of 35

Following the approach in [47, 48], the control costs are modeled by quadratic functions to ensure
convexity and satisfy optimality conditions.

The optimal controls satisfy

J(m∗
1,m

∗
2,m

∗
3) = min

(m1,m2,m3)∈U
J(m1,m2,m3),

where the admissible control set is defined as

U = {(m1,m2,m3) | mi(t) is Lebesgue measurable on [0, T ], 0 ≤ mi(t) ≤ 1, i = 1, 2, 3} .

5.1. Hamiltonian and Its Optimality Equation

Using Pontryangin’s Maximum Principle [PMP] [49], one may ascertain what conditions an
optimal control must meet. His principle converts equations (5.29 - 5.37) and (5.38) into a problem
of minimization of the point-wise hamiltonian (H) with regard to m1(t),m2(t), and m3(t).

H = P1Ehm + P2Eht + P3IM + P4EF + P5IF + L1 (m1 (t))
2 + L2 (m2 (t))

2 + L3 (m3 (t))
2

+ Ξ1[λM − m1αM βM SM IF
TM

− ϵM SM + ωMRM − γMSM ]

+ Ξ2[
m1αM βM SM IF

TM
− ϵM Ehm − φM Ehm]

+ Ξ3[γMSM − (ϵM + ηM )Eht]

+ Ξ4[φM Ehm − (µM + ϵM +m2ρM ) IM ]

+ Ξ5[m2ρM IM − (µM + ϵM + δM )HM ]

+ Ξ6[ηMEht + δM HM − ϵM RM − ωMRM ]

+ Ξ7[λF −m3

(
αM βM SF IM

TF
+
αM βM RM SF

TF

)
− (m1 +m3)ϵF SF ]

+ Ξ8[m3

(
αM βM SF IM

TF
+
αM βM RM SF

TF

)
− ϕM EF − (m1 +m3)ϵF EF ]

+ Ξ9[−(m1 +m3)ϵF IF + ϕM EF ].

(5.39)

Where (Ξi), i = 1, 2, . . . , 9. are adjoint variables associate with SM , Eht, Ehm, IM , HM , RM , SF , EF

and, IF .

Theorem 5. For the optimal control (a∗2, a
∗
2, a

∗
3) and, corresponding state solution,

SM , Eht, Ehm, IM , HM , RM , SF , EF , IF

that minimize J over U of the corresponding system of equations (2.3 - 2.11) having the adjoint
variable Ξ1, . . . ,Ξ8, such that,



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 17 of 35



dΞ1
dt = (Ξ1 − Ξ2)

m1αM βM IF
TM

+ ϵMΞ1 − (Ξ3 − Ξ1) γM ,
dΞ2
dt = (Ξ4 − Ξ2)φM + ϵMΞ2 − P1,
dΞ3
dt = (Ξ6 − Ξ3)ηM + ϵMΞ3 − P2,
dΞ4
dt = (Ξ8 − Ξ7)m3

(
αM βM SF

TF

)
+ (Ξ3 − Ξ4)m2ρM + (µM + ϵM )Ξ3 − P3,

dΞ5
dt = Ξ5 − Ξ4)δM + (µM + ϵM )Ξ4,
dΞ6
dt = (Ξ6 − Ξ5)

m1αM βM SF
TM

+ (Ξ1 − Ξ6) ηM + Ξ6ϵM ,
dΞ7
dt = (Ξ7 − Ξ8)m3

(
αM βM IM

TF
+ αM βM RM

TF

)
+ (m1 +m3)ϵFΞ7

dΞ8
dt = (Ξ8 − Ξ9)ϕM + (m1 +m3)ϵFΞ8 − P4,
dΞ9
dt = (Ξ7 − Ξ8)

m1αM βM SF
TM

+ (m1 +m3)ϵFΞ9 − P5.

(5.40)

With conditions, Ξi(T ) = 0, for i = 1, 2, 3, . . . , 9.
furthermore, the control set obtain (m∗

1,m
∗
2,m

∗
3) component of,

m∗
1 = max{0,min(1,Ω1)},

m∗
2 = max{0,min(1,Ω2)},

m∗
3 = max{0,min(1,Ω3)}.

(5.41)

Where,

Ω1 =
(Ξ2 − Ξ1)

αM βM SM IF
TM

− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L1
,

Ω2 =
(Ξ5 − Ξ4)ρM

2L2
,

Ω3 =
(Ξ8 − Ξ7)

(
αM βM SF IM

TF
+ αM βM RM SF

TF

)
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2Y3
.

(5.42)

Proof. Pontryagin’s Maximal Principle yields the standard conclusions of the adjoint equation
and transversality criteria [50].
The adjoint system is written as a result of differentiation of the Hamiltonian concerning the various
states SM , Ehm, Eht, IM , HM , RM , SF , EF and IF .

dΞ1

dt
= − dH

dSM
= (Ξ1 − Ξ2)

m1αMβMIF
TM

+ ϵMΞ1, (5.43)

dΞ2

dt
= − dH

dEhm
= (Ξ3 − Ξ2)φMS + ϵMΞ2 − P1, (5.44)

dΞ3

dt
= − dH

dEht
= (Ξ6 − Ξ3)ηM + ϵMΞ3 − P2, (5.45)

dΞ4

dt
= −(Ξ8 − Ξ7)m3

(
αMβMSF

TF

)
+ (Ξ3 − Ξ4)m2ρM + (µM + ϵM )Ξ3 − P3, (5.46)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 18 of 35

dΞ5

dt
= −(Ξ5 − Ξ4)δM + (µM + ϵM )Ξ4, (5.47)

dΞ6

dt
= −(Ξ6 − Ξ5)

m1αMβMSF
TM

+ (Ξ1 − Ξ6)ηM + Ξ6ϵM , (5.48)

dΞ7

dt
= −(Ξ7 − Ξ8)m3

(
αMβMIM

TF
+
αMβMRM

TF

)
+ (m1 +m3)ϵFΞ7, (5.49)

dΞ8

dt
= −(Ξ8 − Ξ9)ϕM + (m1 +m3)ϵFΞ8 − P4, (5.50)

dΞ9

dt
= −(Ξ7 − Ξ8)

m1αMβMSF
TM

+ (m1 +m3)ϵFΞ9 − P5, (5.51)

With conditions, Ξi(T ) = 0, for i = 1, 2, 3, . . . , 8.
The optimal controls (m∗

1,m
∗
2,m

∗
3) are characterised by adopting the strategy used by Pontryagin

[49], and the optimality equations are constructed based on the following conditions,
∂H
∂mi

, for i = 1, 2, 3, . . . , 8. which gives,

∂H
∂m1

=
(Ξ2 − Ξ1)

αM βM SM IF
TM

− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L1
,

∂H
∂m2

=
(Ξ4 − Ξ3)ρM

2L2
,

∂H
∂m3

=
(Ξ7 − Ξ6)

(
αM βM SF IM

TF
+ αM βM RM SF

TF

)
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L3
,

(5.52)

setting ∂H
∂mi

= 0,
for m∗

i , the results are,

m∗
1 =

(Ξ2 − Ξ1)
αM βM SM IF

TM
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L1
,

m∗
2 =

(Ξ4 − Ξ3)ρM
2L2

,

m∗
3 =

(Ξ7 − Ξ6)
(
αM βM SF IM

TF
+ αM βM RM SF

TF

)
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L3
,

(5.53)

The boundaries on the controls will be written using common control parameters. As a result,

m∗
1 =


Ω1, if 0 < Ω1 < 1;
0, if Ω1 ≤ 0;
1, if Ω1 ≥ 1.

, (5.54)

m∗
2 =


Ω2, if 0 < Ω2 < 1;
0, if Ω2 ≤ 0;
1, if Ω2 ≥ 1.

, (5.55)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 19 of 35

and,

m∗
3 =


Ω3, if 0 < Ω3 < 1;
0, if Ω3 ≤ 0;
1, if Ω3 ≥ 1.

. (5.56)

In concise notation,

m∗
1 = max{0,min(1,Ω1)},

m∗
2 = max{0,min(1,Ω2)},

m∗
3 = max{0,min(1,Ω3)}.

(5.57)

By using the values of Ω1, Ω2, and Ω3

m∗
1 =


(Ξ2−Ξ1)

αM βM SM IF
TM

−(Ξ7+Ξ6+Ξ8)ϵFSF

2L1
, if 0 <

(Ξ2−Ξ1)
αM βM SM IF

TM
−(Ξ7+Ξ6+Ξ8)ϵFSF

2L1
< 1;

0, if
(Ξ2−Ξ1)

αM βM SM IF
TM

−(Ξ7+Ξ6+Ξ8)ϵFSF

2L1
≤ 0;

1, if
(Ξ2−Ξ1)

αM βM SM IF
TM

−(Ξ7+Ξ6+Ξ8)ϵFSF

2L1
≥ 1.

,

(5.58)

m∗
2 =


(Ξ4−Ξ3)ρM

2L2
, if 0 < (Ξ4−Ξ3)ρM

2L2
< 1;

0, if (Ξ4−Ξ3)ρM
2L2

≤ 0;

1, if (Ξ4−Ξ3)ρM
2L2

≥ 1.

, (5.59)

and,

m∗
3 =


(Ξ7−Ξ6)

(
αM βM SF IM

TF
+

αM βM RM SF
TF

)
−(Ξ7+Ξ6+Ξ8)ϵFSF

2L3
, if 0 < Ω3 < 1;

0, if Ω3 ≤ 0;
1, if Ω3 ≥ 1.

. (5.60)

Briefly expressed,

m∗
1 = max{0,min

(
1,

(Ξ2 − Ξ1)
αM βM SM IF

TM
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L1

)
},

m∗
2 = max{0,min

(
1,

(Ξ4 − Ξ3)ρM
2L2

)
},

m∗
3 = max{0,min

1,
(Ξ7 − Ξ6)

(
αM βM SF IM

TF
+ αM βM RM SF

TF

)
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L3

}.

(5.61)

By including the described control set, initial, and transversal conditions, the optimality system is



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 20 of 35

generated from the adjoint variable system and the optimal control system.

dΞ1

dt
= − dH

dSM
= (Ξ1 − Ξ2)

m1αM βM IF
TM

+ ϵMΞ1,

dΞ2

dt
= − dH

dEM
= (Ξ3 − Ξ2)φMS + ϵMΞ2 − P1,

dΞ3

dt
= − dH

dIM
= (Ξ7 − Ξ6)m3

(
αM βM SF

TF

)
+ (Ξ3 − Ξ4)m2ρM + (µM + ϵM )Ξ3 − P2,

dΞ4

dt
= − dH

dHM
= (Ξ6 − Ξ5)δM + (µM + ϵM )Ξ4,

dΞ5

dt
= − dH

dRM
= (Ξ6 − Ξ7)

m1αM βM SF
TM

+ Ξ5ϵM ,

dΞ6

dt
= − dH

dSF
= (Ξ6 − Ξ7)m3

(
αM βM IM

TF
+
αM βM RM

TF

)
+ (m1 +m3)ϵFΞ6,

dΞ7

dt
= − dH

dEF
= (Ξ7 − Ξ8)ϕM + (m1 +m3)ϵFΞ7 − P3,

dΞ8

dt
= − dH

dIF
= (Ξ6 − Ξ7)

m1αM βM SF
TM

+ (m1 +m3)ϵFΞ8 − P4,

m∗
1 =

(Ξ2 − Ξ1)
αM βM SM IF

TM
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L1
,

m∗
2 =

(Ξ4 − Ξ3)ρM
2L2

,

m∗
3 =

(Ξ7 − Ξ6)
(
αM βM SF IM

TF
+ αM βM RM SF

TF

)
− (Ξ7 + Ξ6 + Ξ8)ϵFSF

2L3
,

Ξi(T ) = 0. i = 1, 2, 3, . . . , 8.

(5.62)

With the initial conditions,
SM (0) = SM0, Emh(0) = Emh0, Eht(0) = Eht0, IM (0) = IM0, HM (0) = HM0, R(0) = RM0, SF (0) =
SF0, EF (0) = EF0, IF (0) = IF0.

6. Numerical Simulation

In this section, we will explore how different control strategies can influence the control of
malaria and aim to identify the cost approach by employing numerical optimization techniques.
The optimization process involves solving a set of eight nonlinear Differential Equations. To carry
out this optimization we adopt a method. Initially, we make estimations for the control parameters
shown in Table 2. Then simulate the system over time using a fourth-order Runge Kutta numerical
integration method results shown in Figure 5.

6.1. Numerical Method

Here, we solve the optimality system from the previous sections using the Runge-Kutta fourth-
order approach, which was developed in [51]. The following is a summary of this approach:



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 21 of 35

(i) Select a random number for m∗ over [0, T ]; in general, (m∗ = 0).

(ii) Utilizing the beginning conditions x(t0) = x0 and the previously mentioned value of m∗, solve
the state system explicitly,

(iii) The adjoint system Ξ(t) can be solved implicitly by taking into account the transversality
criteria Ξ(T ) and the expressions of m∗ and x∗ that were previously estimated,

(iv) Replace x(t) and Ξ(t) with their respective expressions to update the expression of m∗,

(v) Check for convergence. If the values of the variables in the current iteration and previous
iterations are comparable enough, return the real values as solutions. Otherwise, return to
Step 2.

For the state and adjoint equations, we employ the following notations:

dSM
dt

= k1 (x(t),m(t))

dEhm

dt
= k2 (x(t),m(t))

dEht

dt
= k3 (x(t),m(t))

dIM
dt

= k4 (x(t),m(t))

dHM

dt
= k5 (x(t),m(t))

dRM

dt
= k6 (x(t),m(t))

dSF
dt

= k7 (x(t),m(t))

dEF

dt
= k8 (x(t),m(t))

dIF
dt

= k9 (x(t),m(t))

(6.63)

and

Ξ′
1(t) = h1 (x(t),Ξ(t),m(t))

Ξ′
2(t) = h2 (x(t),Ξ(t),m(t))

Ξ′
3(t) = h3 (x(t),Ξ(t),m(t))

Ξ′
4(t) = h4 (x(t),Ξ(t),m(t))

Ξ′
5(t) = h5 (x(t),Ξ(t),m(t))

Ξ′
6(t) = h6 (x(t),Ξ(t),m(t))

Ξ′
7(t) = h7 (x(t),Ξ(t),m(t))

Ξ′
8(t) = h8 (x(t),Ξ(t),m(t))

Ξ′
9(t) = h9 (x(t),Ξ(t),m(t))

(6.64)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 22 of 35

The approximation of each state variable xi, i = 1, 2, ..., 9 given a step size h is given by:

xi+1
n = xin +

h

6
(Ki1 + 2Ki2 + 2Ki3 +Ki4) , (6.65)

where

Ki1 = Ki(x),

Ki2 = fi

(
x+

h

2
Ki1

)
,

Ki3 = fi

(
x+

h

2
Ki2

)
,

Ki4 = fi (x+ hKi3) .

(6.66)

The approximation for the adjoint vector is of the following form and is provided backward in time:

Ξn−1
j = Ξn

j − h

(
1

6

(
Kj

1 + 2Kj
2 + 2Kj

3 +Kj
4

))
,

where

Kj
1 = gj(x),

Kj
2 = gj

(
x+

h

2
Kj

1

)
,

Kj
3 = gj

(
x+

h

2
Kj

2

)
,

Kj
4 = gj

(
x+ hKj

3

)
.

Following these procedures, the controls’ values m∗
i , i = 1, 2, 3 are updated by their corresponding

equations, Eqs (5.54-5.56).



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 23 of 35

0 10 20 30 40 50
Time

0

5000

10000

15000

20000

Po
pu

la
tio

n

SM

SM

0 10 20 30 40 50
Time

300

400

500

600

700

800

900

1000

1100

Po
pu

la
tio

n

Ehm

Ehm

0 10 20 30 40 50
Time

0

20000

40000

60000

80000

Po
pu

la
tio

n

Eht

Eht

0 10 20 30 40 50
Time

100

120

140

160

180

200

220

Po
pu

la
tio

n

IM

IM

0 10 20 30 40 50
Time

10

15

20

25

30

35

40

Po
pu

la
tio

n

HM

HM

0 10 20 30 40 50
Time

0

2000

4000

6000

8000

10000

12000

Po
pu

la
tio

n

RM

RM

0 10 20 30 40 50
Time

0

2000

4000

6000

8000

10000

12000

14000

16000

Po
pu

la
tio

n

SF

SF

0 10 20 30 40 50
Time

0

2000

4000

6000

8000

Po
pu

la
tio

n

EF

EF

0 10 20 30 40 50
Time

0

100

200

300

400

500

600

700

Po
pu

la
tio

n

IF

IF

Malaria Model Dynamics Over Time

Figure 5: Malaria Model Dynamics

6.2. Phase Plot

Figure 6 illustrates the relationship between the number of infected mosquitoes and the like-
lihood of malaria transmission to humans. As the infected mosquito population increases, the
risk of malaria transmission rises because infected mosquitoes are the primary vectors that bite
humans and transmit the parasite. Consequently, a higher number of infected mosquitoes leads
to more human infections. Similarly, an increase in susceptible mosquitoes results in a larger
pool of mosquitoes available to bite susceptible humans, which can maintain or elevate the sus-
ceptible human population and potentially facilitate future infections if those mosquitoes become
infected. Thus, the dynamics of mosquito infection directly influence human infection levels, while
the abundance of susceptible mosquitoes affects the susceptibility of the human population to
malaria transmission.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 24 of 35

0 2000 4000 6000 8000 10000120001400016000
SF

0

5000

10000

15000

20000

25000

S M

Start
End

0 500 1000 1500 2000 2500
IF

0

200

400

600

800

I M

Start
End

5 10 15 20 25 30 35 40
HM

0

200

400

600

800

I M

Start
End

500 1000 1500 2000 2500 3000 3500 4000
Ehm

0

25000

50000

75000

100000

125000

150000

175000

E h
t

Start
End

Figure 6: Phase plane plots showing the interaction between susceptible and infected mosquito populations
and their impact on malaria transmission.

6.3. Control Strategies

6.3.1. Optimal Medication Treatment, Bed-nets & Insecticide Spray

To maximize the objective function J(m1,m2,m3), we implement controls on treated bed-nets
(m1), medication (m2), and insecticide spray (m3). Figure 7 demonstrates a significant reduction
in the number of infected humans when these combined strategies are applied, especially after ten
days. Additionally, the figure shows a decrease in the susceptible, exposed, and infected mosquito
populations due to these interventions.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 25 of 35

0 20 40 60 80 100
Time

0

5000

10000

15000

20000

25000

Po
pu

la
tio

n

Susceptible Humans SM

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

500
1000
1500
2000
2500
3000
3500
4000

Po
pu

la
tio

n

Exposed Humans (hm) Ehm

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0

25000

50000

75000

100000

125000

150000

175000

Po
pu

la
tio

n

Exposed Humans (ht) Eht

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0

200

400

600

800

Po
pu

la
tio

n

Infected Humans IM
No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

5

10

15

20

25

30

35

40

Po
pu

la
tio

n

Hospitalized Humans HM

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0

10000

20000

30000

40000

Po
pu

la
tio

n

Recovered Humans RM

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Susceptible Mosquitoes SF

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Exposed Mosquitoes EF

No Control
Control (m1, m2, m3 0)

0 20 40 60 80 100
Time

0

500

1000

1500

2000

2500

Po
pu

la
tio

n

Infected Mosquitoes IF
No Control
Control (m1, m2, m3 0)

Figure 7: Effect of combined optimal medication treatment, bed-nets, and insecticide spray on malaria dy-
namics.

6.3.2. Medication & Insecticide Spray

This strategy employs medication (m2) and insecticide spray (m3) controls, with no bed-net inter-
vention (m1 = 0). Figure 8 shows a substantial decrease in infected humans after applying these
controls, with effectiveness increasing after ten days. It also illustrates reductions in susceptible,
exposed, and infected mosquito populations.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 26 of 35

0 20 40 60 80 100
Time

0

5000

10000

15000

20000

25000

Po
pu

la
tio

n

Susceptible Humans SM

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

500

1000

1500

2000

2500

3000

3500

4000

Po
pu

la
tio

n

Exposed Humans (hm) Ehm

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0

25000

50000

75000

100000

125000

150000

175000

Po
pu

la
tio

n

Exposed Humans (ht) Eht

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0

200

400

600

800

Po
pu

la
tio

n

Infected Humans IM
No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

5

10

15

20

25

30

35

40

Po
pu

la
tio

n

Hospitalized Humans HM

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0

10000

20000

30000

40000

Po
pu

la
tio

n

Recovered Humans RM

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Susceptible Mosquitoes SF

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Exposed Mosquitoes EF

No Control
Control (m2, m3 0, m1 = 0)

0 20 40 60 80 100
Time

0

500

1000

1500

2000

2500

Po
pu

la
tio

n

Infected Mosquitoes IF
No Control
Control (m2, m3 0, m1 = 0)

Figure 8: Impact of medication and insecticide spray without bed-nets on malaria transmission.

6.3.3. Optimal Bed-nets & Insecticide Spray

Here, controls on bed-nets (m1) and insecticide spray (m3) are applied, with no medication (m2 =
0). Figure 9 shows that this combination significantly reduces the number of infected humans,
especially after ten days. The figure also indicates decreases in susceptible, exposed, and infected
mosquito populations.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 27 of 35

0 20 40 60 80 100
Time

0

5000

10000

15000

20000

25000

Po
pu

la
tio

n

Susceptible Humans SM

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

500
1000
1500
2000
2500
3000
3500
4000

Po
pu

la
tio

n

Exposed Humans (hm) Ehm

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0

25000

50000

75000

100000

125000

150000

175000

Po
pu

la
tio

n

Exposed Humans (ht) Eht

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0

200

400

600

800

Po
pu

la
tio

n

Infected Humans IM
No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

5

10

15

20

25

30

35

40

Po
pu

la
tio

n

Hospitalized Humans HM

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0

10000

20000

30000

40000

Po
pu

la
tio

n

Recovered Humans RM

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Susceptible Mosquitoes SF

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Exposed Mosquitoes EF

No Control
Control (m1, m3 0, m2 = 0)

0 20 40 60 80 100
Time

0

500

1000

1500

2000

2500

Po
pu

la
tio

n

Infected Mosquitoes IF
No Control
Control (m1, m3 0, m2 = 0)

Figure 9: Optimal bed-nets and insecticide spray control strategy.

6.3.4. Optimal Bed-nets & Medication

This strategy combines treated bed-nets (m1) and medication (m2) controls, without insecticide
spray (m3 = 0). Figure 10 illustrates a marked reduction in infected humans, with increased
effectiveness after ten days. It also shows declines in susceptible, exposed, and infected mosquito
populations.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 28 of 35

0 20 40 60 80 100
Time

0

5000

10000

15000

20000

25000

Po
pu

la
tio

n

Susceptible Humans SM

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

500

1000

1500

2000

2500

3000

3500

4000

Po
pu

la
tio

n

Exposed Humans (hm) Ehm

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0

25000

50000

75000

100000

125000

150000

175000

Po
pu

la
tio

n

Exposed Humans (ht) Eht

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0

200

400

600

800

Po
pu

la
tio

n

Infected Humans IM
No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

5

10

15

20

25

30

35

40

Po
pu

la
tio

n

Hospitalized Humans HM

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0

10000

20000

30000

40000

Po
pu

la
tio

n

Recovered Humans RM

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0

2500

5000

7500

10000

12500

15000

17500

Po
pu

la
tio

n

Susceptible Mosquitoes SF

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0
2000
4000
6000
8000

10000
12000
14000
16000

Po
pu

la
tio

n

Exposed Mosquitoes EF

No Control
Control (m1, m2 0, m3 = 0)

0 20 40 60 80 100
Time

0

500

1000

1500

2000

2500

Po
pu

la
tio

n

Infected Mosquitoes IF
No Control
Control (m1, m2 0, m3 = 0)

Figure 10: Optimal bed-nets and medication control strategy.

7. Spatial Model Implementation

We extended the classical malaria transmission model into a spatial framework by dividing the
study area into a 10 × 10 grid of patches, each with its own human and mosquito compartments.
Within each patch, human classes (SM , Ehm, Eht, IM , HM , RM ) and mosquito classes (SF , EF , IF )
were modeled using coupled ordinary differential equations. Mosquito dispersal between adjacent
patches was represented as a diffusion process, allowing movement to four nearest neighbors at
a fixed rate. Human movement was not included. The system was solved numerically using the
odeint solver over 50 time units, with infection seeded in the central patch. The figure 11 displays
heatmaps of infected humans (IM ) across the grid at six time points. Initially, infection is localized
centrally, then spreads outward symmetrically due to mosquito movement and local transmission.
Darker red indicates higher infection levels. A consistent color scale allows comparison across times.
This illustrates how spatial processes shape malaria dynamics and highlights the importance of
spatial heterogeneity for understanding and controlling transmission.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 29 of 35

Time = 0.0 Time = 10.0 Time = 20.0

Time = 30.1 Time = 40.1 Time = 50.0

0 20 40 60 80 100 120
Infected Humans (IM)

Spatial Spread of Infected Humans (IM) Over Time

Figure 11: Spatial spread of infected humans (IM ) across a 10× 10 grid at six time points

Table 2: Parameters Description

Parameters Values Sources

λM 2500 [52]
βF 0.8333 [52]
βM 0.022 [52]
ηM 0.0083 assumed
ωM 0.021 assumed
αM 18 [52]
γM 0.123 assumed
ϵM 0.00004212 [52]
ψM 0.045 [52]
ρM 0.0083 [52]
ϕM 0.0071 assumed
µM 0.0000345 [52]
δM 0.1429 assumed
λF 1000 [52]
ϵF 0.033 [52]
ϕF 0.083 [52]



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 30 of 35

8. Discussion

Malaria remains one of the most serious infectious diseases worldwide, caused by the proto-
zoan Plasmodium and transmitted via bites from female Anopheles mosquitoes. Understanding
the transmission dynamics is vital for developing effective control strategies. In this study, we
constructed a deterministic compartmental model, denoted as SMEhmEhtIMHMRM −SFEF IF , to
represent the complex interactions between human and mosquito populations.
A primary focus of our analysis is the stability of the malaria-free equilibrium. Sensitivity analysis
of the basic reproduction number R0 revealed key parameters influencing disease transmission, such
as the mosquito biting rate αM and the transmission probability from humans to mosquitoes βF
(see Figures 12 and 13). These findings corroborate previous research emphasizing the importance
of vector-host contact rates and transmission probabilities in malaria dynamics [1–3]. Moreover, we
formulated an optimal control problem based on Pontryagin’s Maximum Principle, incorporating
three intervention strategies aimed at reducing the disease burden.

(i) Use of insecticide-treated bed nets (m1),

(ii) Medication treatment of infected individuals (m2),

(iii) Insecticide spraying to reduce mosquito populations (m3).

Given results demonstrate that the combined application of all three controls is more effective in
reducing malaria transmission than any single strategy alone. Among individual measures, the
use of bed nets (m1) is particularly impactful, as it provides a physical barrier that significantly
reduces human exposure to infectious mosquito bites. Consistent and widespread use of bed nets,
especially when integrated with other interventions, offers a promising approach to malaria control.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

M

10

15

20

25

30

35

40

45

R 0

Effect of Varying M on R0

M = 10
M = 18
M = 25

Figure 12: Effect of varying mosquito biting rate αM on the basic reproduction number R0.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 31 of 35

10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

F

20

25

30

35

40

45

R 0

Effect of Varying F on R0

F = 0.5
F = 0.8333
F = 1.2

Figure 13: Effect of varying transmission probability from humans to mosquitoes βF on the basic reproduction
number R0.

These insights are consistent with the broader literature, which emphasizes that maintaining R0 < 1
through targeted interventions is essential for malaria elimination [3, 6, 7]. Future work could extend
this model by incorporating spatial heterogeneity and stochastic effects to better capture real-world
complexities.

9. Conclusion

In this research, we developed a comprehensive mathematical model to simulate malaria trans-
mission dynamics by incorporating both homogeneous and heterogeneous exposed human com-
partments through a system of nonlinear differential equations. We derived the basic reproduction
number R0 and performed a detailed sensitivity analysis, identifying key parameters such as the
mosquito biting rate αM and the transmission probability βM as the most influential factors driv-
ing disease spread. To mitigate malaria transmission, we formulated an optimal control problem
using Pontryagin’s Maximum Principle, introducing three time-dependent control strategies: use
of treated bed nets (m1), medical treatment (m2), and insecticide spraying (m3). Our innovative
approach explored the combined effects of two strategies while gradually reducing the third, offering
valuable insights into the trade-offs involved in malaria control planning. The resulting optimal
control profiles exhibited increasing trends across all three interventions. Numerical simulations
confirmed that integrating multiple interventions yields the most substantial reduction in malaria
transmission. Specifically, the combined application of treated bed nets, medication, and insecticide
spraying proved to be the most effective strategy. These findings are consistent with existing litera-
ture emphasizing the importance of integrated vector management and coordinated preventive and
treatment measures to reduce the malaria burden. This study underscores the value of integrative
mathematical modeling in guiding evidence-based public health interventions for malaria control
and highlights the critical role of targeting vector-host interactions and key transmission param-
eters to achieve disease reduction and eventual elimination. Future research can further enhance



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6423 32 of 35

this framework by incorporating spatial heterogeneity to capture regional variations in transmission
intensity, stochastic effects to reflect environmental and demographic uncertainties, and additional
interventions such as vaccination or genetic vector control technologies. Moreover, coupling the
model with real-time epidemiological and climatic data through data assimilation techniques can
improve predictive accuracy and inform adaptive policy decisions in malaria-endemic regions.

Data Availability
No data availability

Conflict of Interest
The authors declare that there is no conflict of interest.

Funding
No funds.

Acknowledgements

Authors (Dr. Nadeem Abbas and Prof. Dr. Wasfi Shatanawi) would like to thank Prince Sultan
University for their support through the TAS research lab.

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