# Optimal Control and Cost-Effectiveness Strategies of Malaria Transmission with Impact of Climate Variability

Authors: Temesgen Duressa Keno; Lemesa Bedjisa Dano; Gamachu Adugna Ganati
Journal: Journal of Mathematics
 (2022)

---

## Abstract

We proposed in this study a deterministic mathematical model of malaria transmission with climate variation factor. In the first place, fundamental properties of the model, such as positivity of solution and boundedness of the biological feasibility of the model, were proved whenever all initial data of the states were nonnegative. The next-generation matrix method is used to compute a basic reproduction number with respect to the disease-free equilibrium point. The Jacobian matrix and the Lyapunov function are used to check the local and global stability of disease-free equilibriums. If the basic reproduction number is less than one, the model’s disease-free equilibrium points are both locally and globally asymptotically stable; otherwise, an endemic equilibrium occurs. The results of the sensitivity analysis of the basic reproduction numbers were obtained, and its biological interpretation was provided. The existence of bifurcation was discussed, and the model exhibits forward and backward bifurcations with respect to the first and second basic reproduction numbers, respectively. Secondly, using the maximum principle of Pontryagin, the optimal malaria reduction strategies are described with three control measures, namely, treated bed nets, infected human treatment, and indoor residual spraying. Finally, based on numerical simulations of the optimality system, the combination of treatment and indoor spraying is the most efficient and least expensive strategy for malaria eradication.


---

## Body

## 1. Introduction

Malaria is a serious vector-borne disease caused by a parasite calledPlasmodium and transmitted between humans (hosts) and mosquitoes (vectors) through bites from infected female Anopheles mosquitoes [1]. It can spread through blood transfusions, needle sharing, or congenital causes. The parasites then reproduce in the human liver and bloodstream. Malaria is often associated with poverty, which has a negative impact on the country’s economic development. According to the most recent global malaria report, published in December 2021, an estimated 241 million cases and 627000 deaths were reported worldwide, with the African region accounting for 94% of all cases in 2020 [2]. The use of insecticide-treated bed nets, antimalaria drug treatment, and indoor residual spraying are the most effective methods of preventing malaria transmission dynamics [3]. Most mosquito distributions worldwide have been influenced by climatic factors such as temperature, rainfall, and malaria transmission increases with temperatures ranging from 16℃ to 28℃, which create conducive conditions for mosquito breeding rates [4].Ross [5] initiated a mathematical model on the dynamics of malaria spread. He developed SISSI models for human and malaria populations. According to Ross, reducing the number of mosquitoes to below a certain threshold is sufficient for malaria control. Following the Ross model, several models were proposed by scholars who extended the Ross model by considering various factors such as the exposure time in humans and mosquitoes [6–8] and climate variability on mosquito death rate, contact rate, and birth rate [9–11].A group of researchers studied an optimal malaria transmission control model to determine the role of control measures in disease transmission. For example, Okosun et al. [12] presented the malaria transmission dynamics with optimal control and cost-effectiveness analysis. The authors used a combination of three control measures: treated bed nets, antimalarial drug treatment of infected humans, and indoor residual spraying. They conclude that the combination of drug treatment of infected humans and indoor insecticide spraying is the most effective and least expensive way to prevent malaria spread. Makinde et al. [13] proposed a malaria model for disease spread with optimal control and cost-effectiveness analysis based on three control measures. Using the incremental cost-effectiveness ratio methods, a cost-effectiveness analysis was performed to support the results of the optimal control problem. The authors used numerical simulations to supplement the theoretical results. Finally, they proposed that a combination of treated bed nets, drug treatment, and indoor residual spraying is the most effective and less expensive intervention strategy. Okosun et al. [14] presented the SEIRS-SEI malaria transmission model with optimal control and cost-effectiveness analysis using combinations of three control measures, including treated bed nets, infected human treatment, and indoor residual spraying. The authors presented the use of infected human treatment with drugs and indoor residual spraying as the most cost-effective controls to prevent malaria spread. Otieno et al. [15] presented a malaria transmission model with an optimal control problem, employing four time-dependent control measures. Pontryagin’s minimum principle was described in order to derive the necessary conditions for optimal malaria control. Their findings concluded that a combination of treated bed nets, treatment, and spraying is the most effective strategy for mitigating disease. Temesgen et al. [16] presented SIRS for the human population and SI malaria model for malaria transmission by incorporating three continuous controls. The authors suggested that combinations of treatment of infected humans and indoor residual spraying are the best strategies for the prevention of the disease.However, none of these models take into account the impact of climate variability on malaria epidemics with optimal control and cost-effectiveness analysis with a logistic growth of climate variation with respect to mosquito breeding and malaria infection. In this paper, the malaria transmission model [17] is extended to the optimal control problem in the presence of climate variability in mosquito breeding rate and malaria infection, and further analysis on the least cost strategy is considered using the concept of cost-effectiveness analysis.This paper is arranged as follows: In Section2, we propose a malaria transmission model that shows the role of climate variability in malaria epidemics. In Section 3, the model analysis of the model is illustrated. In Section 4, we describe the analysis of sensitivity. In Section 5, the optimal control of the malaria transmission model is analytically analyzed using the maximum principle of Pontryagin. In Section 6, we show the simulation of theoretical results. In Section 7, the analysis of cost-effectiveness is depicted. The conclusion of the work is discussed in Section 8.


## 2. Model Description and Formulation

In this section, we developed a malaria transmission dynamics model in which the human population is represented by the SIRS model and the population of mosquitoes is described by the SI model due to their short lifespan. The total population of human at timet denoted by Nht is divided into three subpopulations based on their disease status; susceptible humans Sh(t) are individuals who are vulnerable to risk of developing an infection from the mosquito. Infected humans Ih(t) are individuals who show the symptom of the disease and can transmit the disease to mosquito. Recovered humans Rh(t) are individuals who got temporary immunity, so that the total human populations is given by Nht=Sht+Iht+Rht. Individuals who are born or migrate are assumed to be susceptible human populations and considered as the recruitment rate denoted by Ψ. Susceptible human populations become infected if they come into contact with infected mosquitoes at the rate βhT,R which depends on climate change, β0h represents the human contact rate with vector when there is no climate variation, and β1h denotes the human incremental contact rate due to climate variation. Humans leave the total population with a death rate μh, and due to malaria disease, the induced death rate δ reduces the population of human. Infected human recovered through treatment rate γh using antimalarial drugs. The recovered human population whose immunity is not permanent can lose it and become susceptible to reinfection at the rate ωh. The total vector population given by Nmt at time t is subgrouped into susceptible mosquito Smt and infected mosquito Imt. Hence, the total vector population is given by Nmt=Smt+Imt. The vector population is recruited at the rate ΦT,R which depends on climate change, whereas Φ0 denotes the vector birth rate when there is no climate variation and Φ1m is the vector incremental birth rate due to variation of climate. A mosquito population gets an infection when it contacts with infected human at the rate βmT,R which depends on climate variation, β0m represents the vector contact rate with a human when there is no climate change, and β1m denotes incremental contact rate due to variations in climate. The vector natural death rate is μm. We assume that mosquitoes do not recover from malaria disease. Thus, mosquitoes do not die due to disease infection. The growth rate of the temperature r follows a logistic function, m is the temperature-dependent rate of precipitation, and Tmax represents the maximum temperature for the mosquito to be most active, whereas the minimum temperature for the mosquito to be less active is denoted by T0. Also, we have considered that the parameters stated in Table 1 are positive. Figure 1 illustrates the transmission dynamics of the malaria diagram.Table 1
Parameters’ description used in system (1).
ParametersParameters’ descriptionΨThe rate of new humans entering to populationΦ0Mosquito population recruitment rateγhRecover rate of the infected humanμhNatural death rate of the human populationδHuman population induced death rateμmMosquito natural death rateωhImmunity loss rate of human populationβ0hHuman contact rate with population of mosquitoΦ1mIncreasing amount of vector breeding rateβ0mContact rate of the mosquito with humanz∗Proportion of an antibody produced by humanβ1mIncreasing amount of vector contact ratem∗Proportion of an antibody produced by mosquitoβ1hIncreasing amount of human contact raterGrowth rate of the temperaturemTemperature-dependent rate of precipitationT0The smallest temperature that the vector is less activeTmaxMaximum temperature that the vector is the most activeFigure 1
Transmission of malaria disease diagram.Depending on the flowchart diagram in Figure1, we write a model that governs the transmission dynamics of malaria disease using a system of ordinary differential equations.(1)dShdt=Ψ−βhT,R1+z∗ImShIm−μhSh+ωhRh,dIhdt=Ψ−βhT,R1+z∗ImShIm−μhS+ϒhIh,dRdt=ϒhIh−μh+ωhRh,dSmdt=ΦΤ−βhT,R1+m∗InSmIh−μhSm,dImdt=r1−ΤΤmaxΤ−Τ0,dRdt=mdΤdt⇒Rt=mTt−T0+ε,where(2)βhT,R=β0h+β1hk,βmT,R=β0m+β1mk,ΦT,R=Φ0+Φ1mk,k=1+mT−T0Tmax,with initial conditions(3)Sh0=Sh0,Ih0=Ih0,Rh0=Rh0,Sm0=Sm0,Im0=Im0,T0=Tm0,R0=Rm0.Based on the fact that the average temperature at the Earth’s surface rises, more evaporation occurs, which in turn increases overall precipitation (rainfall). Therefore, a warming climate is expected to increase precipitation (rainfall) in many areas. This implies that rainfall pattern is an increasing function of temperature. Note thatm is the temperature-dependent rate of precipitation and ε is the amount of precipitation at T=T0. In essence,(4)R0=ε,whenT=T0,Rmax=mTmax−T0+R0,whenT=Tm.


## 3. Mathematical Analysis of the Model

### 3.1. Invariant Region

The invariant region is used to determine where the model’s solution is constrained. The proposed model (1) has two kinds of populations. Firstly, the total population of humans is given by Nht=Sht+Iht+Rht and differentiating Nht with respect to time. Then, by adding the first three equations from system (1), we obtain(5)ddtSh+Ih+Rh=Ψ−μhNh−δIh.This implies that equation (5) becomes(6)ddtSh+Ih+Rh≤Ψ−μhNh.By solving equation (6), we obtain that(7)0<Nh≤Ψμh.Thus, the invariant region of system (1) for the human population is a given by(8)Ωh=Sh,Ih,Rh∈ℝ+3:0<Sh+Ih+Rh≤Ψμh.Secondly, the total number of the mosquito populations from system (1) is given as(9)Nmt=Smt+Imt.Hence, the differentiatingNmt with respect to time t is given by equation(10)ddtSm+Im=ΦT,R−μmNm.By solving equation (10), we obtain 0<Nm≤ΦT,R/μm. Hence, the invariant region of system (1) for the mosquito population is given by(11)Ωm=Sm,Im∈ℝ+2:0<Sm+Im≤ΦT,Rμm.Consequently, the dynamics of system (1) is studied in the biological feasible region of the form(12)Ω=Ωh×Ωm=Sh,Ih,Rh,Sm,Im∈ℝ+5:Nh≤Ψμh,Nm≤ΦT,Rμm,which is a positive invariant set under the flow induced by the solution set of system (1).


### 3.2. Positivity of the Solution

For model (1), we will show that all solutions of system with positive initial data will remain positive for all times t≥0.Theorem 1.
IfSh0,Ih0,Rh0,Sm0, and Im0 are nonnegative, then the solutions Sht,Iht,Rht,Smt, and Imt of system (1) are nonnegative for t≥0.Proof.
The first equation from system (1) is given by(13)dShdt=Ψ−βhT,RShIm1+z∗Im−μhNh+ωhRh,dShdt≥−βhT,RIm1+z∗Im+μhSh.
Integrating equation (13) with respect to time and using the method of separation variables with applying the initial conditions, we obtain(14)Sht≥Sh0e−βhT,RIm/1+z∗Im+μht≥0.
Clearly, equation (13) is nonnegative at all time t≥0. The other state variables Iht,Rht,Smt, and Imt are nonnegative for all time t≥0 by the same procedure. As a result, the malaria transmission model stated in system (1) is both epidemiologically significant and mathematically well-posed in equation (12).


### 3.3. Basic Reproduction Numbers

In the first place, we calculate the disease-free equilibrium (DFE) points of system (1) before calculating an expression for basic reproduction numbers. To do so, we equate all equations of model (1) to zero with Ih=0 and Im=0. In this case, we obtain two malaria-free equilibrium points of model (1), which are given by E1=Sh0,Ih0,Rh0,Sm0,Im0,T0 or E2=Sh0,Ih0,Rh0,Sm0,Im0,Tmax where(15)E1=Ψμh,0,0,ΦT0μm,0,T0,orE2=Ψμh,0,0,ΦTmaxμm,0,Tmax.One advantage of determining the DFE of system (1) is to help us to calculate an expression for the basic reproduction number, which is defined as the average amount of secondary infections caused by a primary infectious in the given period [18]. It can be obtained using the approach of the next-generation matrix and that is the dominant eigenvalue of the next-generation matrix. For model (1), to obtain R01 and R02, we rewrite model (1), beginning with newly infective classes of humans and mosquitoes, which are given by(16)dIhdt=βhT,RShIm1+z∗Im−μh+δ+γhIh,dImdt=βmT,RSmIh1+m∗Ih−μmIm.The right-hand side (RHS) of equation (16) can be written in the form of f−v, where(17)f=βhT,RShIm1+z∗ImβmT,RSmIh1+m∗Ih,v=μh+δ+γhIhμmIm.The partial derivatives off and v in equation (17) evaluated at DFE of system (1) produce two matrices F and V, respectively,(18)F=0βhT,RΨμhβmT,RΦΤ,Rμm0,V=μh+δ+γh00μm.As a result, the basic reproduction numberR0=ρFV−1, where ρ is the dominant eigenvalue of FV−1. Thus, the first basic reproduction number R01 calculated at the first disease-free equilibrium points E1 is given by(19)R01=β0hΨβ0mΦ0μhμm2μh+δ+γh.Similarly, the second basic reproduction numberR02 calculated at the second disease-free equilibrium points E2 is given by(20)R02=β0h+β2hΨβ0m+β2mΦ0+Φ2mμhμm2μh+δ+γh,where β2h=β1hk,β2m=β1mk,Φ2m=Φ1mk, and k=1+mTmax−T0/Tmax.Generally, the basic reproduction number at maximum temperatureTmax and rainfall variation in terms of R01 is given by(21)R02=R012+Φ0β0hβ2m+Ψβ0m+β2mβ2hΦ0+Φ2m+Φ2mβ0hμhμm2μh+δ+γh,where R01 is the first basic reproduction number at T0,R0 for the mosquito to be less active to breed.


### 3.4. Local Stability of Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium point of system (1) is locally asymptotically stable in Ω if R01<R02<1.Proof.
We begin with finding that the Jacobian matrix of system (1) calculated at DFE is given by(22)JΕ∗=−μh0ωh0−βhT∗,R∗ψμh0−μh+δ+γh00βhT∗,R∗ψμh0γh−ωh+μh000−βmT∗,R∗ΦT∗,R∗μm0−μm00βmT∗,R∗ΦT∗,R∗μm00−μm.
From equation (22), we can obtain the characteristic equation of the following form:(23)−λ−μh−λ−μm−λ−μh+ωhλ2+a1λ+a2=0,where(24)a1=μm+μh+δ+γh,a2=μmγh+μmδ+μmμh−βhT∗,R∗ΨβmT∗,R∗ΦT∗,R∗μmμh.
Hence, from equation (23), we have(25)λ1=−μh<0,λ2=−μm<0,λ3=−μh+ωh<0.
Another equation from equation (23) will be a second-degree polynomial(26)λ2+a1λ+a2=0.
By using the Routh–Hurwitz criteria [19], equation (26) has a negative real root if a1>0 and a2>0. Hence, we can observe that a1>0 since it is the sum of nonnegative parameters and the value of a2 at T∗,R∗=Tmax,Rmax is given by(27)a2=μmγh+μmδ+μmμh−β0h+β2hΨβ0m+β2mΦ0+Φ2mμmμh=1−R022>0.
It is worth noting from equation (27) that a2<0 if R02<1. Since R01<R02, so the DFE is locally asymptotically stable if R01<R02<1.


### 3.5. Global Stability of Disease-Free Equilibrium

Theorem 3.
IfR01<R02<1, R01<R02<1, then the disease-free equilibrium point(s) of system (1) is globally asymptotically stable in Ω.Proof.
To discuss the global behavior of DFE of system (1), we use a technique implemented by the Lyapunov theorem [20] by first constructing a suitable Lyapunov function(28)V=μmβhT∗,R∗Ih+Im.
By differentiating equation (28) with respect to time (t), the result obtained becomes(29)dVdt=μmβhT∗,R∗dIhdt+dImdt=μmβhT∗,R∗βhT,R∗ShIm−μh+δ+γhIh+βmT∗,R∗SmIh−μmIm=μmShIm−μmβhT∗,R∗μh+δ+γhIh+βmT∗,R∗SmIh−μmIm=βmT∗,R∗Sm−μmβhT∗,R∗μh+δ+γhIh−μm1−ShIm≤βmT∗,R∗Sm−μmβhT∗,R∗μh+δ+γhIh=βmT∗,R∗ΦT∗,R∗μm−μmβhT∗,R∗μh+δ+γhIh=μmμh+δ+γhβ0h+β2hμhΨR022−1Ih.
Consequently, we observe thatdV/dt<0 if R02<1 and dV/dt=0 if and only if Ih=0,Im=0. So, the dominant bounded invariant set Γ=Sh,Ih,Rh,Sm,Im∈Ω:dV/dt=0 is the singleton which is DFE in Ω. Therefore, by the well-known LaSalle’s invariant principle [21], every solution that begins in the domain bounded invariant set approaches DFE as time tends to infinity when R01<R02. Therefore, the DFE is globally asymptotically stable in Ω if R01<R02<1.


### 3.6. Malaria Present Equilibrium

Malaria endemic equilibrium point is a situation where malaria exists in the human population regardless of time and other factors. The endemic equilibrium pointE∗=Sh∗,Ih∗,Rh∗,Sm∗,Im∗,T∗,R∗ can be obtained by setting RHS of all model equations of system (1) equal to zero. From model (1), the malaria present equilibrium point at (T∗,R∗) = (T0,R0) is given by(30)Sh∗=1+z∗Imψ+ωhRh∗β0hIm∗+μh1+z∗Im,Rh∗=γhIh∗ωh+μh,Sm∗=Φ01+m∗Ihβ0mIh∗+μm1+m∗Ih,Im∗=β0mSh∗Ih∗μm1+m∗Ih.From equation (30), the endemic equilibrium easily satisfies the following polynomial and Ih∗ is computed from the equation: (31)C1Ih∗2+C2Ih∗=0,where(32)C1=β0mβ0hΦ0ωhδ+μhγh+ωh+δ+μh+μhωh+μhγh+δ+μhμm,C2=ωh+μhμhμm2μh+δ+γh1−R012.Hence,C1>0 and C2≥0 whenever R01≥1. Solving for Ih∗ in equation (31), we obtain Ih∗=−C2/C1<0. Thus, the model has no nonnegative malaria present equilibrium whenever R01<1. This guarantees that backward bifurcation does not exist in the model if R01<1.Similarly, the endemic equilibrium point at (T∗,R∗) = (Tmax,Rmax), and solving for Ih∗s as expression of parameters, we obtain(33)Sh∗=1+z∗Imψ+ωhRh∗β0h+β2hIm∗+μh1+z∗Im,Rh∗=γhIh∗ωh+μh,Sm∗=Φ01+m∗Ihβ0m+β2mIh∗+μm1+m∗Ih,Im∗=β0m+β2mSh∗Ih∗μm1+m∗Ih,where β2h=β1k,β2m=β1k,Φ2m=Φ1k, and k=1+mT−T0/Tmax.From equation (33), the endemic equilibrium satisfies the following polynomial and Ih∗ is computed from the equation(34)B1Ih∗2+B2Ih∗=0,where(35)B1=β3mβ3hΦ2mωhδ+μhγh+ωh+δ+μh+μhωh+μhγh+δ+μhμm,B2=ωh+μhμhμm2μh+δ+γh1−R022,where β3h=β0h+β2h,β3m=β0m+β2m, and Φ3m=Φ0+Φ2m.From equation (35), we see that if R02<1, then it immediately implies that R01<1 and DFE exists for both R01 and R02. However, R01<1 does not immediately imply R02<1 as the value of R02 will be larger than unity that shows that whereas R01 present DFE. Thus, R02 may become an endemic situation or exhibit backward bifurcation while R01 undergoes forward bifurcation only.


## 3.1. Invariant Region

The invariant region is used to determine where the model’s solution is constrained. The proposed model (1) has two kinds of populations. Firstly, the total population of humans is given by Nht=Sht+Iht+Rht and differentiating Nht with respect to time. Then, by adding the first three equations from system (1), we obtain(5)ddtSh+Ih+Rh=Ψ−μhNh−δIh.This implies that equation (5) becomes(6)ddtSh+Ih+Rh≤Ψ−μhNh.By solving equation (6), we obtain that(7)0<Nh≤Ψμh.Thus, the invariant region of system (1) for the human population is a given by(8)Ωh=Sh,Ih,Rh∈ℝ+3:0<Sh+Ih+Rh≤Ψμh.Secondly, the total number of the mosquito populations from system (1) is given as(9)Nmt=Smt+Imt.Hence, the differentiatingNmt with respect to time t is given by equation(10)ddtSm+Im=ΦT,R−μmNm.By solving equation (10), we obtain 0<Nm≤ΦT,R/μm. Hence, the invariant region of system (1) for the mosquito population is given by(11)Ωm=Sm,Im∈ℝ+2:0<Sm+Im≤ΦT,Rμm.Consequently, the dynamics of system (1) is studied in the biological feasible region of the form(12)Ω=Ωh×Ωm=Sh,Ih,Rh,Sm,Im∈ℝ+5:Nh≤Ψμh,Nm≤ΦT,Rμm,which is a positive invariant set under the flow induced by the solution set of system (1).


## 3.2. Positivity of the Solution

For model (1), we will show that all solutions of system with positive initial data will remain positive for all times t≥0.Theorem 1.
IfSh0,Ih0,Rh0,Sm0, and Im0 are nonnegative, then the solutions Sht,Iht,Rht,Smt, and Imt of system (1) are nonnegative for t≥0.Proof.
The first equation from system (1) is given by(13)dShdt=Ψ−βhT,RShIm1+z∗Im−μhNh+ωhRh,dShdt≥−βhT,RIm1+z∗Im+μhSh.
Integrating equation (13) with respect to time and using the method of separation variables with applying the initial conditions, we obtain(14)Sht≥Sh0e−βhT,RIm/1+z∗Im+μht≥0.
Clearly, equation (13) is nonnegative at all time t≥0. The other state variables Iht,Rht,Smt, and Imt are nonnegative for all time t≥0 by the same procedure. As a result, the malaria transmission model stated in system (1) is both epidemiologically significant and mathematically well-posed in equation (12).


## 3.3. Basic Reproduction Numbers

In the first place, we calculate the disease-free equilibrium (DFE) points of system (1) before calculating an expression for basic reproduction numbers. To do so, we equate all equations of model (1) to zero with Ih=0 and Im=0. In this case, we obtain two malaria-free equilibrium points of model (1), which are given by E1=Sh0,Ih0,Rh0,Sm0,Im0,T0 or E2=Sh0,Ih0,Rh0,Sm0,Im0,Tmax where(15)E1=Ψμh,0,0,ΦT0μm,0,T0,orE2=Ψμh,0,0,ΦTmaxμm,0,Tmax.One advantage of determining the DFE of system (1) is to help us to calculate an expression for the basic reproduction number, which is defined as the average amount of secondary infections caused by a primary infectious in the given period [18]. It can be obtained using the approach of the next-generation matrix and that is the dominant eigenvalue of the next-generation matrix. For model (1), to obtain R01 and R02, we rewrite model (1), beginning with newly infective classes of humans and mosquitoes, which are given by(16)dIhdt=βhT,RShIm1+z∗Im−μh+δ+γhIh,dImdt=βmT,RSmIh1+m∗Ih−μmIm.The right-hand side (RHS) of equation (16) can be written in the form of f−v, where(17)f=βhT,RShIm1+z∗ImβmT,RSmIh1+m∗Ih,v=μh+δ+γhIhμmIm.The partial derivatives off and v in equation (17) evaluated at DFE of system (1) produce two matrices F and V, respectively,(18)F=0βhT,RΨμhβmT,RΦΤ,Rμm0,V=μh+δ+γh00μm.As a result, the basic reproduction numberR0=ρFV−1, where ρ is the dominant eigenvalue of FV−1. Thus, the first basic reproduction number R01 calculated at the first disease-free equilibrium points E1 is given by(19)R01=β0hΨβ0mΦ0μhμm2μh+δ+γh.Similarly, the second basic reproduction numberR02 calculated at the second disease-free equilibrium points E2 is given by(20)R02=β0h+β2hΨβ0m+β2mΦ0+Φ2mμhμm2μh+δ+γh,where β2h=β1hk,β2m=β1mk,Φ2m=Φ1mk, and k=1+mTmax−T0/Tmax.Generally, the basic reproduction number at maximum temperatureTmax and rainfall variation in terms of R01 is given by(21)R02=R012+Φ0β0hβ2m+Ψβ0m+β2mβ2hΦ0+Φ2m+Φ2mβ0hμhμm2μh+δ+γh,where R01 is the first basic reproduction number at T0,R0 for the mosquito to be less active to breed.


## 3.4. Local Stability of Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium point of system (1) is locally asymptotically stable in Ω if R01<R02<1.Proof.
We begin with finding that the Jacobian matrix of system (1) calculated at DFE is given by(22)JΕ∗=−μh0ωh0−βhT∗,R∗ψμh0−μh+δ+γh00βhT∗,R∗ψμh0γh−ωh+μh000−βmT∗,R∗ΦT∗,R∗μm0−μm00βmT∗,R∗ΦT∗,R∗μm00−μm.
From equation (22), we can obtain the characteristic equation of the following form:(23)−λ−μh−λ−μm−λ−μh+ωhλ2+a1λ+a2=0,where(24)a1=μm+μh+δ+γh,a2=μmγh+μmδ+μmμh−βhT∗,R∗ΨβmT∗,R∗ΦT∗,R∗μmμh.
Hence, from equation (23), we have(25)λ1=−μh<0,λ2=−μm<0,λ3=−μh+ωh<0.
Another equation from equation (23) will be a second-degree polynomial(26)λ2+a1λ+a2=0.
By using the Routh–Hurwitz criteria [19], equation (26) has a negative real root if a1>0 and a2>0. Hence, we can observe that a1>0 since it is the sum of nonnegative parameters and the value of a2 at T∗,R∗=Tmax,Rmax is given by(27)a2=μmγh+μmδ+μmμh−β0h+β2hΨβ0m+β2mΦ0+Φ2mμmμh=1−R022>0.
It is worth noting from equation (27) that a2<0 if R02<1. Since R01<R02, so the DFE is locally asymptotically stable if R01<R02<1.


## 3.5. Global Stability of Disease-Free Equilibrium

Theorem 3.
IfR01<R02<1, R01<R02<1, then the disease-free equilibrium point(s) of system (1) is globally asymptotically stable in Ω.Proof.
To discuss the global behavior of DFE of system (1), we use a technique implemented by the Lyapunov theorem [20] by first constructing a suitable Lyapunov function(28)V=μmβhT∗,R∗Ih+Im.
By differentiating equation (28) with respect to time (t), the result obtained becomes(29)dVdt=μmβhT∗,R∗dIhdt+dImdt=μmβhT∗,R∗βhT,R∗ShIm−μh+δ+γhIh+βmT∗,R∗SmIh−μmIm=μmShIm−μmβhT∗,R∗μh+δ+γhIh+βmT∗,R∗SmIh−μmIm=βmT∗,R∗Sm−μmβhT∗,R∗μh+δ+γhIh−μm1−ShIm≤βmT∗,R∗Sm−μmβhT∗,R∗μh+δ+γhIh=βmT∗,R∗ΦT∗,R∗μm−μmβhT∗,R∗μh+δ+γhIh=μmμh+δ+γhβ0h+β2hμhΨR022−1Ih.
Consequently, we observe thatdV/dt<0 if R02<1 and dV/dt=0 if and only if Ih=0,Im=0. So, the dominant bounded invariant set Γ=Sh,Ih,Rh,Sm,Im∈Ω:dV/dt=0 is the singleton which is DFE in Ω. Therefore, by the well-known LaSalle’s invariant principle [21], every solution that begins in the domain bounded invariant set approaches DFE as time tends to infinity when R01<R02. Therefore, the DFE is globally asymptotically stable in Ω if R01<R02<1.


## 3.6. Malaria Present Equilibrium

Malaria endemic equilibrium point is a situation where malaria exists in the human population regardless of time and other factors. The endemic equilibrium pointE∗=Sh∗,Ih∗,Rh∗,Sm∗,Im∗,T∗,R∗ can be obtained by setting RHS of all model equations of system (1) equal to zero. From model (1), the malaria present equilibrium point at (T∗,R∗) = (T0,R0) is given by(30)Sh∗=1+z∗Imψ+ωhRh∗β0hIm∗+μh1+z∗Im,Rh∗=γhIh∗ωh+μh,Sm∗=Φ01+m∗Ihβ0mIh∗+μm1+m∗Ih,Im∗=β0mSh∗Ih∗μm1+m∗Ih.From equation (30), the endemic equilibrium easily satisfies the following polynomial and Ih∗ is computed from the equation: (31)C1Ih∗2+C2Ih∗=0,where(32)C1=β0mβ0hΦ0ωhδ+μhγh+ωh+δ+μh+μhωh+μhγh+δ+μhμm,C2=ωh+μhμhμm2μh+δ+γh1−R012.Hence,C1>0 and C2≥0 whenever R01≥1. Solving for Ih∗ in equation (31), we obtain Ih∗=−C2/C1<0. Thus, the model has no nonnegative malaria present equilibrium whenever R01<1. This guarantees that backward bifurcation does not exist in the model if R01<1.Similarly, the endemic equilibrium point at (T∗,R∗) = (Tmax,Rmax), and solving for Ih∗s as expression of parameters, we obtain(33)Sh∗=1+z∗Imψ+ωhRh∗β0h+β2hIm∗+μh1+z∗Im,Rh∗=γhIh∗ωh+μh,Sm∗=Φ01+m∗Ihβ0m+β2mIh∗+μm1+m∗Ih,Im∗=β0m+β2mSh∗Ih∗μm1+m∗Ih,where β2h=β1k,β2m=β1k,Φ2m=Φ1k, and k=1+mT−T0/Tmax.From equation (33), the endemic equilibrium satisfies the following polynomial and Ih∗ is computed from the equation(34)B1Ih∗2+B2Ih∗=0,where(35)B1=β3mβ3hΦ2mωhδ+μhγh+ωh+δ+μh+μhωh+μhγh+δ+μhμm,B2=ωh+μhμhμm2μh+δ+γh1−R022,where β3h=β0h+β2h,β3m=β0m+β2m, and Φ3m=Φ0+Φ2m.From equation (35), we see that if R02<1, then it immediately implies that R01<1 and DFE exists for both R01 and R02. However, R01<1 does not immediately imply R02<1 as the value of R02 will be larger than unity that shows that whereas R01 present DFE. Thus, R02 may become an endemic situation or exhibit backward bifurcation while R01 undergoes forward bifurcation only.


## 4. Sensitivity Analysis

Malaria control and eradication strategies should focus on key parameters that have a significant impact on the basic reproduction number. The purpose of carrying out the sensitivity analysis of the basic model parameter is to identify a parameter that affects the basic reproduction number. Essentially, the robustness of system predictions to parameter values can be expressed using the normalized sensitivity index of the basic reproduction number to the given basic parameters because the values of those parameters can increase or decrease a basic reproduction number and vice versa. This method relies on our knowledge of the parameters that have a large influence on the basic reproduction number (R02) in order to design the best disease control strategies. We used the technique described in [13, 22, 23] to perform the sensitivity analysis.Definition 1.
(See [13, 22, 24, 25]): The forward sensitivity index of R0, which is differentiable with respect to a given basic parameter D, is defined as(36)τkR0=∂R0∂D×kR0.
The sensitivity index ofR01 of model (1) with respect to parameter β0h, for instance, is obtained as(37)τβ0hR01=∂R01∂β0h×β0hR01=12β0hΨβ0mΦ0/μhμm2μh+δ+γh×Ψβ0mΦ0μhμm2μh+δ+γh×β0hR01=12>0.
Using the same approach with respect to the rest of the parameters,τβ0mR01,τΨR01,τΦ0R01,τμhR01,τμmR01,τδR01,τγhR01 are computed, and the sensitivity indices are presented in Table 2.
By the same procedure, the sensitivity index ofR02 of the model (1) with respect to Ψ is given as(38)ΠΨR02=∂R02∂Ψ×ΨR02=12β0h+β2hΨβ0m+β2mΦ0+Φ2m/μhμm2μh+δ+γh×β0h+β2hβ0m+β2mΦ0+Φ2mμhμm2μh+δ+γh×ΨR02=12>0.
Similarly, with respect to other basic parameters,Πβ0hR02,Πβ0mR02,ΠΨR02,ΠΦ0R02,Πβ1mR02,ΠΦ1mR02,ΠμhR02,ΠμmR02,ΠδR02,ΠγhR02 are computed, and the sensitivity indices are described in Table 3.Table 2
Sensitivity indices of parameter.
ParameterSensitivity indexΨ0.5Φ00.5β0h0.5β0m0.5μm1μh−0.087541δ−0.485158γh−0.024462Table 3
Sensitivity indices of parameter.
ParameterSensitivity indexΨ0.5Φ00.072438β0h0.281659β0m0.305714Φ1m0.079169β1h0.218341β1m0.224286μm−1μh−0.092541δ−0.475258γh−0.024461


### 4.1. Interpretation of the Sensitivity Indices

We described sensitivity indices of basic reproduction number (R02) with respect to eight basic parameters in Table 2. The output showed that the parameters with a positive sensitivity index increased the value of R01 as their values increased, while the rest of the parameters remained constant. And if the values of the parameters with negative indices are increased while the values of the other parameters remain constant, the value of R01 decreases. Similarly, the sensitivity indices of R02 with respect to eleven basic parameters are shown in Table 3. The parameters with positive sensitivity indices have a high impact on malaria transmission in the community as their values increase. The basic parameters with negative sensitivity indices increase the malaria disease if their values decrease while the other parameters remain constant. For example, ΠΨR02=0.5 shows that decreasing (increasing) the rate of the human recruitment by 10% decreases (increases) the basic reproduction number R02 by 5%; similarly, ΠμmR02=−1 indicates that decreasing (increasing) the rate of the mosquito death by 10% increases (decreases) the basic reproduction number R02 by 10%.


## 4.1. Interpretation of the Sensitivity Indices

We described sensitivity indices of basic reproduction number (R02) with respect to eight basic parameters in Table 2. The output showed that the parameters with a positive sensitivity index increased the value of R01 as their values increased, while the rest of the parameters remained constant. And if the values of the parameters with negative indices are increased while the values of the other parameters remain constant, the value of R01 decreases. Similarly, the sensitivity indices of R02 with respect to eleven basic parameters are shown in Table 3. The parameters with positive sensitivity indices have a high impact on malaria transmission in the community as their values increase. The basic parameters with negative sensitivity indices increase the malaria disease if their values decrease while the other parameters remain constant. For example, ΠΨR02=0.5 shows that decreasing (increasing) the rate of the human recruitment by 10% decreases (increases) the basic reproduction number R02 by 5%; similarly, ΠμmR02=−1 indicates that decreasing (increasing) the rate of the mosquito death by 10% increases (decreases) the basic reproduction number R02 by 10%.


## 5. Extension of the Model into Optimal Control

According to the necessity and severity of a disease, many intervention strategies are applied by public health officials to control the disease [25]. In this study, we extended the malaria transmission model (1) to an optimal control problem to determine control strategy decisions involving a mathematical model of biological situations [30]. Using this method, we hope to demonstrate the best malaria prevention strategy. After incorporating the controls into the malaria transmission model (1), the obtained state equations are as follows: (39)dShdt=Ψ1−u1−βhT,R1+z∗ImShIm−μhSh+ωhRh,dIhdt=1−u1βhT,R1+z∗ImShIm−μh+δ+ϒh+u2Ih,dRdt=ϒh+u1Ih−μh+ωhRh,dSmdt=ΦT−1−u1βmT,R1+m∗InSmIh−μm+u3Sm,dImdt=1−u1βmT,R1+m∗IhSmIh−μm+u3Im,dTdt=r1−TTmaxT−T0,dRdt=mdTdt⇒Rt=mTt−T0+ε,where βhT,R=β0h+β1hk,βmT,R=β0m+β1mk,фT,R=ф0+ф1m, andk=1+mT−T0/Tmax.The control functions represent that the use of a treated bed net to protect against mosquitoes isu1t, the treatment of infected humans with antimalarial drugs is u2t, and the use of indoor residual spraying to kill mosquitoes is u3t. The objective functional that we developed for the optimal control model (39) is given as(40)Ju1,u2,u3=min∫0tfD1Ih+D2Im+12C1u12+C2u22+C3u32dt,where tf denoted the terminal time, D1 and D2 are weight constants for the infected human and mosquito, respectively, and C1, C2, and C3 are weight constants for each control, respectively. The expression 1/2Biui2 represents the cost function that corresponds to the controls uit and is quadratic in the other pieces of literature [18, 31–34, 36–38]. The goal of the objective functional (40) is to reduce the total number of infected humans Iht, infected mosquitoes Imt, and control costs uit. The primary goal is to compute a triple optimal control u1∗,u2∗, and u3∗ such that(41)Ju1∗,u2∗,u3∗=minJu1,u2,u3:u1,u2,u3∈ϑ,where ϑ=u1,u2,u3: uit such that u1,u2, and u3 are Lebesgue measurable on t∈0,tf with 0≤uit≤1 is the control set. The obtained Hamiltonian (H) function of the optimal control problem that consists of equations (39) and (40) is represented as(42)H=A1Ih+A2Im+12∑i=13Biui2+λ1dShdt+λ2dIhdt+λ3dRhdt+λ4dSmdt+λ5dImdt+λ6dTdt+λ7dRdt.From equation (42), the minimize Hamiltonian function with respect to u1,u2,u3 is given by(43)H=A1Ih+A2Im+12B1u12+B2u22+B3u32+λ1Ψ−1−u1βhT,R1+z∗ImShIm−µhSh+ωhRh+λ21−u1βhT,R1+z∗ImShIm−µh+δ+γh+u2Ih+λ3γh+u2Ih−µh+ωhRh+λ4фT,R−1−u1βmT,R1+m∗IhSmIh−µm+u3Sm+λ51−u1βmT,R1+m∗IhSmIh−µm+u3Im+λ6r1−TTmaxT−T0+λ7mr1−TTmaxT−T0,where λ1,λ2,λ3,λ4,λ5,λ6, and λ7 are adjoint variables. Next, to obtain the costate variables by using Pontryagin’s maximum principle [32], with the existence result of [35], the following theorem is stated.Theorem 4.
Given optimal controlsu1∗,u2∗,u3∗ and a solution Sh∗,Ih∗,Rh∗,Sm∗,Im∗,T∗,R∗ of the corresponding state system that minimize J(u1,u2,u3) over ϑ subject to equation (39), then adjoint variables λ1,λ2,λ3,λ4,λ5, and λ6 hold the adjoint system(44)dλ1dt=1−u1βhT,R1+z∗ImImλ2−λ1+μhλ1,dλ2dt=1−u1βhT,R1+z∗ImSmλ5−λ4+λ2μh+δ+γh+u2−λ3γh+u2−D1,dλsdt=−ωhλ1+λ3μh+ωh,dλ4dt=−1−u1βmT,R1+m∗IhIhλ5−λ4+λ4μm+u3,dλ5dt=−1−u1βmT,R1+m∗IhShλ2−λ1+λ4μm+u3−D2,dλ6dt=rλ6−rλ6T0Tmax−TTmax,dλ7dt=mrλ7−rmλ7T0Tmax−TTmax.
With transversality conditions,(45)λ1tf=λ2tf=λ3tf=λ4tf=λ5tf=λ6tf=λ6tf=λ6tf=0.
Furthermore, the optimal controlsu1∗,u2∗,u3∗ are represented by(46)u1∗=max0,min1,λ2−λ1βhT,R/1+z∗ImSh∗Im∗+λ5−λ4βmT,R/1+m∗IhSm∗Ih∗C1,u2∗=max0,min1,λ2−λ3Ih∗C2,u3∗=max0,min1,λ4Sm∗+λ5Im∗C3.Proof.
To obtain the form of the costate equations, we compute the derivative of the Hamiltonian function (H) equation (42) with respect to Sh,Ih,Rh,Sm,Im, and T, respectively. Then, the adjoint or costate equation obtained is given by(47)dλ1dt=−1−u1βhT,R1+z∗ImImλ2−λ1+μhλ1,dλ2dt=−1−u1βhT,R1+z∗ImSmλ5−λ4+λ2μh+δ+γh+u2−λ3γh+u2−D1,dλsdt=−ωhλ1+λ3μh+ωh,dλ4dt=−1−u1βmT,R1+m∗IhIhλ5−λ4+λ4μm+u3,dλ5dt=−1−u1βmT,R1+m∗IhShλ2−λ1+λ4μm+u3−D2,dλ6dt=rλ6−rλ6T0Tmax−TTmax,dλ7dt=mrλ7−rmλ7T0Tmax−TTmax.
With transversality conditions,(48)λ1tf=λ2tf=λ3tf=λ4tf=λ5tf=λ6tf=λ7tf=0.
To obtain the control value, we compute the partial derivative of Hamiltonian given by(49)∂H∂ui=0fori=1,2,3.
Obviously, after a partial derivative of Hamiltonian with respect to the controls, the result becomes(50)0=∂H∂u1=λ2−λ1βhT,R/1+z∗ImSh∗Im∗+λ5−λ4βmT,R/1+m∗IhSm∗Ih∗+u1C1,0=∂H∂u2=λ3Ih∗−λ2Ih∗+u2C2,0=∂H∂u3=−λ4Sm∗+λ5Im∗+u3C3.
Moreover, solving for controls variables from equation (50), we obtain(51)u1∗=λ2−λ1βhT,R/1+z∗ImSh∗Im∗+λ5−λ4βmT,R/1+m∗IhSm∗Ih∗C1,u2∗=λ2−λ3Ih∗C2,u3∗=λ4Sm∗+λ5Im∗C3.
Rearranging the solution of (51) with the boundary condition of each control, we got(52)u1∗=max0,min1,λ2−λ1βhT,R/1+z∗ImSh∗Im∗+λ5−λ4βmT,R/1+m∗IhSm∗Ih∗C1,u2∗=max0,min1,λ2−λ3Ih∗C2,u3∗=max0,min1,λ4Sm∗+λ5Im∗C3.
Next, we will see the simulation of the optimality system to identify an optimal strategy that is most optimal to minimize the spread of malaria transmission.


## 6. Numerical Simulation

In this study, we solved the optimality system and used the forward-backward sweep to solve the state and adjoint systems in order to obtain the optimal strategy. We used the forward fourth-order Runge–Kutta to solve the state equations (39) due to the initial value of state variables. Similarly, the adjoint equations are solved using backward fourth-order Runge–Kutta due to the transversality condition (45) having the solution of state functions and the value of optimal controls. The controls are then updated using a convex combination of the previous controls and the value from the optimality conditions (46). This situation will continue until two consecutive iterations are close enough to each other [30]. The initial conditions that we used for numerical simulation of the optimality system are Sh0=120,Ih0=20,Rh0=10,Sm0=300,Im0=30,T(0) = 16°C, and R(0) = 11 mm, as well as the parameter values from Table 4. We used the following weight constant values for the states and controls: D1=60,D2=80,C1=40,C2=100, and C3=60. We used the following four strategies with different combinations of two controls at a time and three controls at a time to determine the impact of each control on malaria reduction.Table 4
Parameter description and taken values for model (1).
ParametersParameters’ descriptionValuesReferencesγhInfected human recover rate0.0035[19]ΨHuman population recruitment rate0.071[26]Φ0Mosquito population recruitment rate0.041[22]μmMosquito population natural death rate0.05[19]μhHuman population natural death rate0.00004[13]δHuman population induced death rate0.068[27]ωhImmunity loss rate of human population0.09[17]β1mIncreasing amount of vector breeding rate0.07[17]β1hIncreasing amount of human contact rate0.05[28]Φ1mIncreasing amount of vector contact rate0.09[28]β0hContact rate human with mosquito0.03[17]β0mContact rate of mosquito with human0.04[13]z∗Proportion of an antibody produced by human0.06Assumedm∗Proportion of an antibody produced by mosquito0.04AssumedmTemperature-dependent rate of precipitation0.08AssumedrTemperature growth rate0.007[28]T0Minimum values of temperature16℃[13]TmaxMaximum values of temperature28℃[29]


### 6.1. Strategy A: Combination of Use of Treated Bed Nets (u1) and Treatment of Infected Humans (u2)

We optimized the objective function (39) using the treated bed net u1 and the treatment of infected humans with antimalarial drugs u2 when the value of the indoor residual spraying u3 was set to zero. Also, as shown in Figure 2(a), if there are controls, the number of infected humans Ih decreases and then tends to zero, whereas the number of infected humans increases if there are no controls. Similarly, in Figure 2(b), we can see that the infected mosquito Im decreases as the control strategy is used, whereas the infected mosquito increases in the uncontrolled case. The control profiles shown in Figure 2(c) with this strategy suggest that controls on treated bed net u1 kept maximum level (100%) until the end of the implementation while infected human treatment using antimalarial drugs u2 retained its maximum bound for 6 days then gradually tended to a minimum after the 82nd day.Figure 2
Simulation results showing the use of (a) treated bed net (u1), (b) treatment of infectious human (u2), and (c) the control profile for u1,u2≠0.
(a)(b)(c)


### 6.2. Strategy B: Combination of Use of Treated Bed Net (u1) and Indoor Spraying (u3)

This strategy combined the treated bed net as a controlu1 and indoor residual spraying as a control u3 to reduce total infected populations and costs associated with the treatment of infected humans u2. We can see in Figure 3(a) that the amount of infected human Ih with controls decreases and reaches its lowest point. In contrast, when there are no controls, the number of infected humans increases to a certain point. According to Figure 3(b), an infected mosquito Im decreases in the control strategy and decreases. To its minimum point, an infected mosquito increases in the uncontrolled case. According to the control profiles depicted in Figure 3(c), control on treated bed net u1 is maintained at its maximum(100%) for 171 days, while indoor residual spraying u3 is maintained at its maximum for 8 days before dropping to its lowest value after the 120th day.Figure 3
Simulation results depicting the use of (a) treatment of infectives (u1), (b) indoor spray of insecticides (u3), and (c) the control profile u1,u3≠0.
(a)(b)(c)


### 6.3. Strategy C: Combination of Treatment of Infected Humans (u2) and Indoor Spraying (u3)

In this study, we used a combination treatment of infected humans as controlu2 and indoor residual spraying as control u3 to reduce total infected populations and save money. Figure 4(a) shows that when using this control strategy, the number of infected humans Ih decreases to its lowest value and becomes smaller when controls are used. In Figure 4(b), the infected mosquito Im decreases in the occurrence of control strategy and then drops to its minimum values, whereas in the uncontrolled case, an infected mosquito increase is observed. With this approach, the control profiles in Figure 4(c) conclude that controls on the treatment of infected human u2 maintain upper level (100 percent) for 9 days, while the insecticides spray u3 maintains maximum level for 7 days and then declines to lower bound at the end of the 80th day.Figure 4
Simulation results representing (a) treatment of malaria infectious (u2), (b) indoor spray of insecticides (u3), and (c) the control profile for u2,u3≠0.
(a)(b)(c)


### 6.4. Strategy D: Use of Treated Bed Net (u1), Treatment (u2), and Indoor Spraying (u3)

We used a combination of three continuous controls on the treated bed netu1; treatment of infected human u2; and indoor residual spraying u3 to reduce the objective function (39). By implementing those three control strategies, we discovered that, in Figure 5(a), the amount of infected human Ih decreases and decreases to a minimum level, whereas the amount of infected human increases to a certain point if no control is used. In Figure 5(b), the infected population of mosquitoes Im is decreasing and minimized to its lower bound when a control strategy is used, whereas in the absence of a control strategy, an increase in infected mosquitoes is observed. Using this strategy, the control profile in Figure 5(c) depicted that controls on treated bed net u1 and treatment u2 were kept at 100% coverage for 5 and 8 days, respectively. Then, treated bed net u1 and treatment u2 tend to their lowest bound after 80 days, whereas indoor spraying u3 retains its highest values (100 percent) for 162 days before dropping to the lowest level on the 100th day.Figure 5
Simulation results indicating treated bed net (u1), treatment (u2), and insecticides spray (u3).
(a)(b)(c)Figure6 shows the existence of bifurcation for malaria transmission model. Bifurcation occurs. This implies that, in equation (13), if R02<1, this automatically implies that R01<1 and DFE exists for both R01 and R02. However, R01<1 do not automatically describe R02<1, R02<1, because the value of R02 may be larger than unity that shows that DFE (E2) may depict backward bifurcation while DFE (E1) only depicts forward bifurcation.Figure 6
Diagram showing the existence of bifurcation whenT = T0 and T = Tmax for the malaria model problem.


## 6.1. Strategy A: Combination of Use of Treated Bed Nets (u1) and Treatment of Infected Humans (u2)

We optimized the objective function (39) using the treated bed net u1 and the treatment of infected humans with antimalarial drugs u2 when the value of the indoor residual spraying u3 was set to zero. Also, as shown in Figure 2(a), if there are controls, the number of infected humans Ih decreases and then tends to zero, whereas the number of infected humans increases if there are no controls. Similarly, in Figure 2(b), we can see that the infected mosquito Im decreases as the control strategy is used, whereas the infected mosquito increases in the uncontrolled case. The control profiles shown in Figure 2(c) with this strategy suggest that controls on treated bed net u1 kept maximum level (100%) until the end of the implementation while infected human treatment using antimalarial drugs u2 retained its maximum bound for 6 days then gradually tended to a minimum after the 82nd day.Figure 2
Simulation results showing the use of (a) treated bed net (u1), (b) treatment of infectious human (u2), and (c) the control profile for u1,u2≠0.
(a)(b)(c)


## 6.2. Strategy B: Combination of Use of Treated Bed Net (u1) and Indoor Spraying (u3)

This strategy combined the treated bed net as a controlu1 and indoor residual spraying as a control u3 to reduce total infected populations and costs associated with the treatment of infected humans u2. We can see in Figure 3(a) that the amount of infected human Ih with controls decreases and reaches its lowest point. In contrast, when there are no controls, the number of infected humans increases to a certain point. According to Figure 3(b), an infected mosquito Im decreases in the control strategy and decreases. To its minimum point, an infected mosquito increases in the uncontrolled case. According to the control profiles depicted in Figure 3(c), control on treated bed net u1 is maintained at its maximum(100%) for 171 days, while indoor residual spraying u3 is maintained at its maximum for 8 days before dropping to its lowest value after the 120th day.Figure 3
Simulation results depicting the use of (a) treatment of infectives (u1), (b) indoor spray of insecticides (u3), and (c) the control profile u1,u3≠0.
(a)(b)(c)


## 6.3. Strategy C: Combination of Treatment of Infected Humans (u2) and Indoor Spraying (u3)

In this study, we used a combination treatment of infected humans as controlu2 and indoor residual spraying as control u3 to reduce total infected populations and save money. Figure 4(a) shows that when using this control strategy, the number of infected humans Ih decreases to its lowest value and becomes smaller when controls are used. In Figure 4(b), the infected mosquito Im decreases in the occurrence of control strategy and then drops to its minimum values, whereas in the uncontrolled case, an infected mosquito increase is observed. With this approach, the control profiles in Figure 4(c) conclude that controls on the treatment of infected human u2 maintain upper level (100 percent) for 9 days, while the insecticides spray u3 maintains maximum level for 7 days and then declines to lower bound at the end of the 80th day.Figure 4
Simulation results representing (a) treatment of malaria infectious (u2), (b) indoor spray of insecticides (u3), and (c) the control profile for u2,u3≠0.
(a)(b)(c)


## 6.4. Strategy D: Use of Treated Bed Net (u1), Treatment (u2), and Indoor Spraying (u3)

We used a combination of three continuous controls on the treated bed netu1; treatment of infected human u2; and indoor residual spraying u3 to reduce the objective function (39). By implementing those three control strategies, we discovered that, in Figure 5(a), the amount of infected human Ih decreases and decreases to a minimum level, whereas the amount of infected human increases to a certain point if no control is used. In Figure 5(b), the infected population of mosquitoes Im is decreasing and minimized to its lower bound when a control strategy is used, whereas in the absence of a control strategy, an increase in infected mosquitoes is observed. Using this strategy, the control profile in Figure 5(c) depicted that controls on treated bed net u1 and treatment u2 were kept at 100% coverage for 5 and 8 days, respectively. Then, treated bed net u1 and treatment u2 tend to their lowest bound after 80 days, whereas indoor spraying u3 retains its highest values (100 percent) for 162 days before dropping to the lowest level on the 100th day.Figure 5
Simulation results indicating treated bed net (u1), treatment (u2), and insecticides spray (u3).
(a)(b)(c)Figure6 shows the existence of bifurcation for malaria transmission model. Bifurcation occurs. This implies that, in equation (13), if R02<1, this automatically implies that R01<1 and DFE exists for both R01 and R02. However, R01<1 do not automatically describe R02<1, R02<1, because the value of R02 may be larger than unity that shows that DFE (E2) may depict backward bifurcation while DFE (E1) only depicts forward bifurcation.Figure 6
Diagram showing the existence of bifurcation whenT = T0 and T = Tmax for the malaria model problem.


## 7. Cost-Effectiveness Analysis

In this study, we must determine the most effective and least expensive method of reducing disease transmission. We employed the incremental cost-effectiveness ratio method to arrive at this strategy (ICER). Using this technique, we incrementally compare more than one competing intervention; for example, one intervention could be compared to a second, less effective alternative. This approach was defined as the ratio of the difference in averted costs between two strategies to the difference in the total number of infections saved [23]. We calculated the total cost avoided and total infections saved from the numerical simulation of the optimal control problem, and the control strategy is ordered in increasing order based on the total infections saved, as shown in Table 5. The amount of total infection saved is calculated by subtracting the total number of humans infected with malaria using control from the total number of humans infected with malaria without control, while the cost averted of each strategy was obtained by using the cost function represented by 1/2C1u12, 1/2C2u22, and 1/2C3u32 over the time [17, 28]. The amount of the total infection saved and the total cost of all strategies with their ICER is given in Table 6.Table 5
Number of total infections saved and cost averted for all strategies.
StrategyDescriptionTotal infections avertedTotal cost ($)ATreated bed net and insecticides spray3794.796421.36CTreatment of infected and insecticides spray4092.551958.46BTreated bed net and treatment of infectious4116.466476.61DTreated bed net, treatment, and indoor spray4118.486492.28Table 6
Amount of the total infection averted and total cost used with ICER.
StrategyAmount of infections savedTotal cost ($)ICERB3794.796421.361.693C4092.551958.46−14.99A4116.466476.61188.96D4118.486492.287.76The value of the cost-effectiveness ratio (ICER) is computed from the total number of saved populations and total cost of averted for each strategy given in Table5, which is used to compare the differences between the two strategies as obtained and given by(53)ICERB=6421.363794.79=1.693,ICERC=1955.46−6420.364094.55−3792.79=−14.99,ICERA=6476.61−1958.464116.46−4092.55=188.96,ICERD=6492.28−6476.614118.48−4116.46=7.76.With the above results, the number of infections saved with ICER for four different strategies is shown in Table6.Table6 compares the interventions B and C. ICER(C) is less than ICER(B) as shown in the table.It implies that strategy B is costly and has a low likelihood of saving people. As a result, C saves more people than B. Then, from the competing strategies, we have removed B. Then, as shown in Table7, calculate the ICER for the remaining strategies C, A, and D.Table 7
Amount of the total infection averted and total cost used with ICER.
StrategyAmount of infections savedTotal cost ($)ICERC4092.551958.462.09A4116.466476.61188.96D4118.486492.287.76The ICER(A) is greater than the ICER(C) from the competing intervention strategies listed in Table7. This demonstrates that ICER(C) strategy outperforms ICER(A). As a result, ICER(A) is less effective and more expensive than ICER(C). As a result, we removed strategy A from the group of competing strategies and calculated the ICER as shown in Table 8.Table 8
Amount of the total infection averted and total cost used with ICER.
StrategyAmount of infections savedTotal cost ($)ICERC4092.551958.462.09D4118.486492.28181.35Table8 with intervention strategies C and D shows that ICER(C) is less than ICER(D). This indicates that strategy C strongly outperforms strategy D. As a result, the C strategy has the lowest total cost and is the most optimal. Based on the findings of the analysis, we believe that intervention C, which consists of treating infected humans and spraying indoors, is the best optimal and least expensive strategy for limiting disease spread.


## 8. Conclusion

The impact of climate variability on malaria epidemics is described in this paper using deterministic mathematical modeling. The model analysis revealed qualitatively that the model’s solution is both bounded and positive in the fixed domain. The method of the next-generation matrix is used to calculate the basic reproduction number with respect to the disease-free equilibrium. The Jacobian matrix and the Lyapunov method are used to demonstrate the local stability and global stability of the disease-free equilibrium. Furthermore, if the basic reproduction number is less than one, the disease-free equilibrium is locally and globally asymptotically stable; otherwise, a positive endemic equilibrium occurs. The model’s sensitivity has been described in detail, and the model demonstrates forward and backward bifurcation. According to the analytical findings, the best optimal way to prevent malaria epidemics is to reduce human-mosquito contact, increase mosquito death rates, and increase the treated rate of an infected human. Furthermore, we extended the model to an optimal control problem with three continuous controls, such as personal prevention with a treated bed net, drug treatment of infected humans, and indoor residual spraying for mosquito minimizing strategy. The maximum principle of Pontryagin is applied to obtain the necessary conditions for optimal control, and the cost-effectiveness analysis is described for all combinations of the controls considered in the study. Based on the numerical analysis, we propose that the combination of treatment and indoor residual spraying is the best strategy for effectively reducing malaria.



---

*Source: 5924549-2022-06-29.xml*