Abstract

In this paper, we present a nonlinear deterministic mathematical model for malaria transmission dynamics incorporating climatic variability as a factor. First, we showed the limited region and nonnegativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Furthermore, the fundamental reproduction number was determined using the next-generation matrix approach, and the sensitivity of model parameters was investigated to determine the most affecting parameter. The Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the fundamental reproduction number is smaller than one, a disease-free equilibrium point is both locally and globally asymptotically stable, but endemic equilibrium occurs otherwise. The model exhibits forward and backward bifurcation. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates three controls, namely, using treated bed net, treatment of infected with antimalaria drugs, and indoor residual spraying strategy. The Pontryagin’s maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system and cost-effectiveness analysis reveals that the combination of treated bed net and treatment is the most optimal and least-cost strategy to minimize the malaria.

1. Introduction

Malaria is a vector-borne disease caused by a parasite called Plasmodium and transmitted between humans and mosquitos via bites of infected female Anopheles mosquitoes. It is a major public health problem, particularly on the African continent [1]. This disease can also be transmitted by blood transfusion or congenital. According to the world malaria report published in December 2021, an estimated 241 million cases and 627000 deaths occurred worldwide, with the African Region accounting for 95 percent of all cases in 2020 [2]. Climate variability is recognized to have a significant impact on the malaria vector’s life cycle, and the life of the mosquito is governed by temperature and rainfall [3]. The most effective way to avoid the malaria dynamics are treated bed net, treatment with antimalaria drugs, and indoor residual spraying [4].

The mathematical model for malaria dynamics is the most useful in comprehending the disease’s presence in the human population. Ross devised the first mathematical model for malaria dynamics transmission [5]. Various mathematical models based on Ross’s basic malaria model were described by different scholars with different factors such as including the exposed class in mosquitoes and humans [68] and the impact of climate variations on malaria epidemics in terms of death rate, birth rate, and prevalence of mosquito population [911].

Many investigations of malaria dynamics models were presented using optimum control problems employing optimal control theory. For example, Agusto et al. [12] investigated optimum solutions for decreasing the dynamics of malaria using a malaria transmission model with an optimal control problem. The authors proposed that the combination of the treated bed net, therapy, and indoor residual spraying is the optimum technique for illness reduction. Okosun et al. [13] created a malaria model that includes optimum control techniques of malaria disease transmission with two measures that regulate treatment infectious and immunization. The authors determined that the combination of therapy and immunization is the most effective and least expensive way to control a malaria condition. Leiton et al. [14] presented an SEIRS-SI optimal control model for malaria transmission in Colombia considering three optimal control strategies. The authors concluded that integrated of control measures treated bed net, intermittent prophylactic treatment in pregnancy and effective case management is the best strategy to prevent the malaria dynamics. Olaniyi et al. [15] formulated a malaria transmission dynamic model with an optimal control analysis using four time dependent continuous controls. The authors concluded that the combination of all these controls is the best measure to minimize the spread of malaria. Keno et al. [16] formulated an optimal control and cost effectiveness strategies of malaria transmission with temperature variability factor by considering three optimal control strategies. The model analysis provides that the combination of treatment and indoor residual spraying is the most efficient and less costly strategy to minimize the malaria. Olaniyi et al. [17] proposed SEIRS for the human population and SEI malaria model with optimal control and cost-effectiveness analysis in the presence of reinfection and relapse in malaria dynamics. The authors suggested that the strategy with a combination of treated bed net and indoor residual spraying is the most effective and least cost to eradicate the malaria.

To the best of our knowledge, 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 mosquitoes breeding and malaria infection was not considered in their models. In the malaria transmission model [11], we considered impact of temperature variability on malaria epidemics. In this paper, we extended model [11] to the impact of climate variability (temperature and rainfall) with respect to mosquitoes breeding rate and malaria infection with optimal control and cost effectiveness analysis.

The following is the format of this paper: in Section 2, we develop a malaria transmission model that incorporates the impact of climate variability on malaria epidemics. The model’s qualitative analysis is described in Section 3. We perform sensitivity analysis in Section 4. The optimal control analysis of the malaria transmission model is described in Section 5. In Section 6, we compared numerical simulation results to analytical results. The cost-effectiveness analysis is depicted in Section 7. Finally, Section 8 discusses the study’s conclusion.

2. Model Description and Formulation

In this section, we proposed and developed a malaria transmission model that takes into account the human and vector populations, with the total human population denoted by and the vector population denoted by . The SEIRS model describes the human population and divides it at time into the following subpopulations: susceptible human , exposed human , infected human , and recovered human . The SEI model describes the mosquito population, and the total mosquito population at time , denoted by , is subdivided into susceptible mosquito , exposed mosquito , and infected mosquito . The total human and mosquito populations are then given by and , respectively. Assume that all of the parameters in the model are positive. The new recruits (assumed to be susceptible) enter the human population by birth (migration) at the rate An infectious mosquito transmits malaria to the susceptible human, and the susceptible human moves to the exposed human at a climate dependent rate of , is the contact rate of humans with mosquitos when there is no variation in climate, and is the incremental contact rate of humans with mosquitos due to climate variation. The progression of exposed humans to infected humans is , the natural death rate of all humans is , and induced death is . The infected human recovered due to the use of antimalarial drugs through a treatment rate of . The recovered populations of humans have short period immunity that can be lost and become susceptible to reinfection rate of . The recruitment of mosquito populations at a rate of is climate-dependent; is the birth rate of mosquitos when there is no variation in climate, and is the incremental birth rate of mosquitos when there is variation in climate. Mosquito population contact with an infected human with malaria and moved to exposed class at a climate dependent rate of , is mosquito contact rate with human when there is no variation in climate, and is incremental mosquito contact rate with a human due to climate variation. The exposed mosquito moved at a rate of to an infected mosquito. Every mosquito dies at the same rate Mosquitoes do not recover from malaria because an infected mosquito is infectious until it dies. The temperature growth rate follows a logistic function, is the temperature dependent rate of precipitation, is the maximum temperature for the mosquito to be most active, and is the minimum temperature for the mosquito to be less active. All the description of parameters are given in Table 1, and the diagram of malaria disease transmission is shown in Figure 1.

The transmission dynamics of malaria is described by the following system of nonlinear differential equations, based on the flow diagram depicted in Figure 1: where and .

Base on the fact that as average temperatures at the earth’s surface rise 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 that is the temperature dependent rate of precipitation, and is the amount of precipitation at In essence,

3. Qualitative Analysis of the Model

3.1. Invariant Region

The invariant region is used to determine where the model’s solution is bounded. The model (1) is divided into two parts: the host population and the vector population. The total human population is represented by Then, by differentiating both sides with respect to time and combining the first four equations from the model (1), we get

Then, equation (4) becomes

By solving the equation (5), we can see that Thus, for the human population, the bounded region of the system (1) is given by

Similarly, the equation gives the total mosquito population in the system (1) is

Obviously, the result we obtain after differentiating with respect to time is

By solving the equation (8), we get . As just a result, for the mosquito population, the invariant region of the system (1) is given by is a positive invariant. As a result, the system’s invariant region (1) is given by is a positive invariant set, and all of the solution set of system (1) is bounded in within the region.

3.2. Positivity of the Solution

The purpose of this subsection is to demonstrate that all solutions of the model (1) will remain nonnegative in the future if their initial data is nonnegative.

Theorem 1. The model solution (1) given by and with nonnegative initial conditions and remain nonnegative for all time .

Proof. First, take the begin equation from system (1) as given by Consequently, this indicates that We obtain by integrating equation (12) with respect to time and solving it using the technique of variable separation with initial condition. The other state variables are nonnegative for all time by the same procedure. As a result, the malaria transmission model stated in system (1) is both epidemiologically significant and mathematically well posed in

3.3. Basic Reproduction Number

To find the steady-state solution of the model (1), we equated the left-hand side of the system (1) to zero with the value of , and obtained the disease free-equilibrium of system (1) denoted by or where

The basic reproduction number is defined as the average amount of secondary infectious caused by a primary infectious over a given time period [18], and it has been calculated using the next-generation matrix method [19]. Then, to get and at and , respectively, we rewrite the model (1) beginning with newly infective classes of human and mosquito population as

The right hand side of system (15) can then be written as , where

The Jacobian matrices of and at the disease-free equilibrium points give and , respectively, where

The next-generation matrix from the product of equation (17) calculated by is obtained as

Thus, the fundamental reproduction number where is the dominant eigenvalue of the product and the and at disease free equilibrium and are obtained, respectively, in equations (19) and (20) as given by where , and

As a result, the basic reproduction number () at maximum temperature can be written in terms of in the equation (21), as described below: where denotes the basic reproduction number at for the mosquito to be less active in breeding.

3.4. Local Stability of Disease-Free Equilibrium

Theorem 2. If ; then, the disease free equilibrium point(s) of the system (1) is locally asymptotically stable in

Proof. The Jacobian matrix of equation (1) at the disease-free equilibrium point is given as where
Setting and from the equation (22) the Jacobian matrix obtained as the polynomial function given by where From equation (23), we obtain and we get from the final polynomial equation, Using the Routh-Hurwitz criteria [20, 21], we can see that all of the equation’s eigenvalues (29) have negative roots or imaginary roots with a negative real part if Clearly, we have seen that and because they are a sum of positive parameters and at the value of is described by However, for to be positive, must also be positive, resulting in Furthermore, when , then, , implying that disease-free equilibrium (DFE) is unstable. As a result, since , the DFE is locally asymptotically stable in if

3.5. Global Stability of Disease-Free Equilibrium

Theorem 3. If ; then, the disease free-equilibrium point(s) of the system (1) is globally asymptotically stable in

Proof. We used the technique implemented by Lyapunov theorem [19]; first, we developed the following Lyapunov function defined as Then differentiating the Lyapunov function with respect to time the obtained result is given by where , and
Obviously, if and iff This demonstrates that the dominant compact invariant set in represents the singleton set DFE in As a result of LaSalle’s invariant principle [22], every solution which begins in the region approaches DFE as (time) tends to infinity. Since , the DFE is globally asymptotically stable in if

3.6. Malaria Present Equilibrium

The endemic equilibrium point is the condition in which malaria is found in the human population, which can be obtained by equating all the model equations of system (1) to zero. At this point, let and be force of infection of human and mosquito, respectively, and from the system (1), the endemic equilibrium point at is provided by

By substituting and from equation (34) into and , respectively, and is obtained by solving the equation where

Therefore, and whenever so that As a result, positive endemic equilibrium does not occur when This shows that endemic equilibrium is found if is less than zero which means that there is endemic equilibrium for the model. Similarly, applying the same procedure, the endemic equilibrium point the model (1) with is described in the following as obtain:

Similarly, substituting and from equation (44) into and , respectively, and is obtained by solving the equation where

As a result of equation (21), if , it follows that exists, and DFE exists for both and . However, does not automatically imply because the value of can be greater than 1, implying that may present backward bifurcation while can only present forward bifurcation.

4. Model Parameter Sensitivity Analysis

Malaria eradication strategies should focus on key parameters that have a significant impact on the basic reproduction number. The parameters found in the model (1) influence the basic reproduction number. Because those parameters can increase or decrease a basic reproduction number as their values change, and vice versa. The sensitivity analysis is used to identify the parameters that have a large influence on the basic reproduction number . To perform the sensitivity analysis, the method described in [2325] is used.

Definition 4 (see [23, 24]). The normalized forward sensitivity index of that differentiable with respect to a given parameter is defined as for describes the basic parameters.

For example, the sensitivity index of with respect to basic parameter is obtained as

Using a similar procedure for the remaining parameters, , are obtained, and their sensitivity indices are shown in Table 2 as follows.

The sensitivity index of with respect to parameters is obtained using a similar method, and their sensitivity indices are written in Table 3.

4.1. Expression of the Sensitivity Indices

The sensitivity analysis description of a basic reproductive number with respect to ten basic parameters was stated in the Table 2. We can conclude that the parameters , and with positive sensitivity indices will increase the value of if their values increase while the other parameters remain constant. This demonstrates that the population’s infection rate is increasing as a result of secondary cases. In contrast, the basic parameters , and with negative indices decrease the value of if their value increases while retaining the constant of rest parameters. For example, shows that decreasing (increasing) the mosquito contact rate by 10% reduces (increases) the by Similarly, and indicates that decreasing (increasing) the mosquito death rate by increases (decreases) the basic reproduction number by Also, the sensitivity analysis expression of the basic reproductive number with respect to thirteen basic parameters was shown in Table 3. As a result, the parameters , and have positive sensitivity indices. They will have a large influence on the spread of the malaria disease in the population. Because the transmission of malaria disease increases as their value increases, so does the number of infections as secondary cases in the population. The sensitivity indices for the parameters , and are all negative. As their values increase, those parameters have the potential to reduce malaria disease transmission while remaining constant. For example, indicates that decreasing (increasing) the mosquito contact rate by decreases (increases) the by In contrast, demonstrates that decreasing (increasing) the mosquito death rate by increases (decreases) the basic reproduction number by

5. Optimal Control Model

In this section, we provide a thorough qualitative examination of the time-dependent malaria dynamic model (1). The Pontryagin’s maximum principle [28] is used to describe this analysis, with the goal of lowering the exposed human , infected human , infected mosquito , and control costs . In the case of time-dependent control, we employ Pontryagin’s maximum principle to derive the necessary conditions for disease control. After incorporating the controls into the malaria transmission model (1), the optimal control problem is as follows: where represents protective control using treated bed net, represents antimalaria drug treatment control, and denotes mosquito control via indoor residual spraying. The objective functional (57) of the optimal control problem (56) is given by where represents the final time of control implementation, and quantities , and are weights constants of the exposed human population, infected human population, and infected mosquito population, respectively, while , and are weight constants used for treated bed net, treatment control using antimalaria drugs, and indoor residual spraying. In this paper, we make the expression for the cost-control functions to be quadratic in order to obtain a unique optimal expression for each of the control variables from the optimality condition as in other studies [2934]. The main goal is to achieve the best control triple , and so that where such that , and are Lebesgue measurable on with for is the control set. We express the Hamiltonian (H), which consists of the state equations (56) and integrand of the objective functional (57), by using the Pontryagin’s maximum principle [28] to obtain the necessary condition.

By plugging the given equations (56) and (57) into a minimizing problem of a Hamiltonian function (H) with respect to controls. The Hamiltonian (59) becomes where for is adjoint variables. Next, we present the adjoint system and control characterizations using Pontryagin’s maximum principle [30], in conjunction with the existence of the optimal control problem [35], the following result can be obtained.

Theorem 5. Suppose we have an optimal control set and solutions of the respective state system (56) that minimize over ; then, there exist costate variables , and such that