Malaria Model with Optimal Control

Abstract: An SIRS (Susceptible–Infected–Removed-Susceptible) mathematical model for the 9 transmission dynamics of the Transfusion–Transmitted Malaria (TTM) model with optimal control 10 pair ) ( 1 t u and ) ( 2 t u was developed and studied in this research work. The model Transfusion– 11 Transmitted Malaria disease–free equilibrium and endemic equilibriums points were determined. 12 The model exhibited two equilibriums; disease-free and endemic equilibrium. It is shown that the 13 disease–free equilibrium was locally asymptotically stable if the associated basic reproduction 14 numbers 0 R is less than unity while the disease persists if 0 R is greater than unity. The global 15 stability of the Transfusion–Transmitted Malaria model at the disease-free equilibrium was 16 established using the comparison method. The optimality system was derived and an optimal 17 control model of blood screening and drug treatment for the Transfusion–Transmitted Malaria 18 model was investigated. Conditions for the optimal control were considered using Pontryagin’s 19 Maximum Principle and solved numerically using the Forward and Backward Finite Difference 20 Method (FBDM). Numerical results obtained are in perfect agreement with our analytical results. 21


INTRODUCTION
Transfusion -transmitted malaria (TTM) was first documented in 1911 [5]. The global incidence and occurrence of TTM based on available data indicates that over hundred cases are reported annually, mostly restricted to endemic countries [2]. The chances of TTM due to donor blood in Sub Saharan African countries is increased due to malaria prevalence in donor blood sample [8]. In countries where malaria is endemic, differentiating cases of TTM from natural infection still remain a challenge as malaria infection occurrence after transfusion may be as a result of either natural infection (infection through bites from an infected female anopheles mosquito) or transfusion transmitted (TT). This explains the reason the number of TTM cases in endemic countries is under-reported. Acquisition of malaria parasite due to donor exposure is an increasing problem as a result in global travelling and immigration. Thus, it is more challenging to develop an optimal strategy to reduce the risk of TTM in endemic countries without unnecessary exclusion of blood donation which remain a subject of debate. In [10,11], a general overview of current strategies in non-endemic countries was considered. The strict donor deferral system which is based on travel history of individual has been adopted by most countries, However, this strategy is not optimal due to many healthy donors are differed which may result in donation loss because lengthy deferrals may discourage the donors from coming back [5]. Consequently, the optimal control strategy for a given country or location may vary according to the background level of malaria risk faced by the donor and the recipient population viz-a-viz the resources available. Thus, we aim to study in this work mathematical analysis of transfusiontransmitted malaria TTM) model with optimal control

Model Formation
We formulate the mathematical model for the Transfusion -Transmitted Malaria by considering the dynamical system of equation with optimal control analysis for human population only. The human population is divided into three sub-groups: Susceptible-Infected -Removal. Thus, we assume that the population of human is constant. Our model also includes the rate of transfusion of infected blood and transmitted rate of the disease with malaria induced death rate For our dynamical equations, we define the following variables and parameters as follows:

Parameters and Variables Description
Susceptible human population With the following assumptions: (1) Transmission of the plasmodium is via transfusion of blood to blood contact of infected blood with plasmodium to a susceptible individual (2) Due to the assumption made in (1), the Vector population is excluded (3) Susceptible individual become infected upon blood to blood contact of infected blood with plasmodium The flow diagram for the model is given in Figure 1: 3.0 MODEL ANALYSIS System (1) is resolved by non-dimensionalizing the variables as follow by setting : Differentiating to get Substituting equation (2) and (3) into (1) yields

The Population Dynamics of the Model
Thus as   t , the total human population reach a constant value given as

Positivity of solution
For the Transfusion -Transmitted Malaria model of equation (4) to be epidemiologically well posed, we need to show that all solution with non-negative initial conditions will remain non -negative, for all 0  t .
Theorem 3.1: Let: Proof: From the first differential equation of system (1), Similar reasoning can be used for other differential equations of equation (4) hence, it follows that the Transfusion -Transmitted Malaria model is positive and bounded with a unique solution.

3.3
The Local Stability of Disease-free Equilibrium, 0 P System (4) has a disease-free equilibrium (DFE) obtained by setting the right-hand side of equation (4) to zero, given by The Jacobian Matrix of equation (4) about (8) is So that the eigenvalues  are real and given by ) The expression in (9) was obtained using the next generation matrix approach by finding the dominant eigenvalues of the matrix , then the eigenvalues are all negative then 0 P is locally asymptotically stable.
, then two eigenvalues are negative and one is positive, then 0 P is unstable.
The above result is summarized in the following theorem Theorem 3.2: The disease -free equilibrium (DFE) 0 P of equation (4) is locally asymptotically stable if 1 0  R and unstable if 1 0  R .

Global Stability of Disease -Free Equilibrium (DFE)
There are conditions for global asymptotic stability (GAS) of the disease -free equilibrium to be established, one of such condition is maintaining a constant population size. Observe that the model (4) will maintain a constant population size with time given by The global stability could be proved by several methods. The Lyapunov method has been used by several researchers, but here, the comparison approach as described in Lashmkantham et.al [6] will be used. The following theorem proves the global stability of the (DFE). Theorem 3.3: Assuming that the system of equation (4) describes a human population, then the (DFE) 0 P of (2) is globally asymptotically stable (GAS) if 1 0  R , otherwise unstable.

Proof:
Using the comparison approach, the rate of change of the infected and recovered compartments of equation (4) can be written as Where F and V retain their original meaning, according to Castillo-Chaves and song [3], all eigenvalues of . It follows that 2  is real and negative . Hence, the linearized differential inequality (10) is stable whenever . Hence, the disease free equilibrium 0 P is globally asymptotically stable (GAS) if 1 0  R .

3.5
The local asymptotical stability of the endemic equilibrium Observe that equation (4) have the endemic equilibrium point * P defined as ) , , ( The Jacobian matrix of equation (4) The eigenvalues of (11) are given as will have two real negative roots if 1 0  R by Descartes rule of sign, hence all eigenvalues of (8) are real and negative if 1 0  R .which implies that the endemic equilibrium point * P is locally asymptotically stable. The foregoing discussion is summarized as follows: Theorem 3.4: The endemic equilibrium point * P of system (4) is locally asymptotically stable if 1 0  R otherwise unstable.

Impact of Transfusion Rate    on Malaria Transmission
To analyze the effect of transfusion rate on malaria, we begin by expressing 0 R in terms of  as follows: If equation (14) is greater than zero, then an increase in transfusion rate result in an increase in the number of malaria cases. However, if equation (14) is equal to zero, then the transfusion rate  does not have any significant effect on the transmission dynamics of malaria.

Analysis of Optimal Control
This section focus on the optimal control analysis of model equation (4), using the Pontryagin's Maximum Principle [9] to analyze and determine the necessary conditions for the optimal control of transfusion -transmitted malaria. Time dependent preventive and treatment controls are introduced into the model (4) to determine the optimal strategy for controlling the disease. Thus, we have Our aim is to minimize the number of susceptible humans to malaria due to TT, infected humans with malaria and the cost of applying preventive and treatment controls ) . Thus, we consider the objective functional We then seek to find an optimal control, ) Considering the conditions that an optimal solution must satisfy as it was given by Pontryagin Maximum Principle [9], this principle helps to convert (15) and (16) Proof: Corollary 4.1 of Fleming and Rishel [4] establish the existence of an optimal control due to the convexity of the integrand of J with respect to ) ( 1 t u and ) ( 2 t u , a priori boundedness of the state variable solutions and the Lipschitz property of the state system with respect to the state variables. The differential equations governing the adjoint variables are obtained as follows:

NUMERICAL SIMULATIONS AND DISCUSSION OF RESULTS
The numerical solutions are illustrated using MAPEL 18 program with computation times of 3.52s on a windows 7 operating system. The optimality system, consist of the state system, adjoint system, initial conditions for the state system and the transversality conditions for the adjoint system. The state systems are solved by the forward finite difference scheme using the current iterations solutions of the state equations. The adjoint systems are solved by the backward finite difference scheme using the current iterations solutions of the state equations because of the transversality conditions. Then the controls are updated by using a convex combination of the previous controls and the value from the characterization (20) and (21). Thus, the process is repeated and the iterations are stopped at the final time f t . The table of parameter descriptions and values used in the numerical simulation of the model are given in Table 2.
One of the ways of controlling the spread of malaria disease is through blood screening of donors; however, lack of information or ignorance may affect the impact blood screening can have on malaria transmission. The behaviour of the total human population was investigated over time in Figure 2. It was observed for threshold parameter 1 0  R , the asymptotic nature of the population was established. The number of susceptible individuals increases with time and infected humans recovered while the infected humans' decreases asymptotically over time. However, for 1 0  R , the unstable nature of the population became evident as depicted in Figure 3.
Consequently, optimal control strategies using the combination of screening donor's blood   t u 1 and treatment for those infected   t u 2 were used on the model to control the transmission of malaria. The following scenarios were considered:

Parameters
Baseline value Source 0.1 Assumed 0.5 [7] 0.01 [7] 1.0 Assumed 1/(70x365) [1] 0.01 [1]  In this case, the two controls are used to optimize the objective function J . It was observed in Figure 4 that the combination of both controls resulted in significant decrease in the number of infected humans (green solid line) as against the drastic increase observed in the uncontrolled case (red dotted line). Here, the objective function J was optimized using control   t u 2 while the control on blood screening was set to zero. It was observed that number of infected humans showed significant reduction while there is an increase in the number of infected humans in the uncontrolled case as shown in Figure 5 .

(c) Optimal control using Screening of donor's blood   t u 1 only
The objective functional J was optimized in this case by setting the control on treatment   t u 2 to zero. The result of this strategy clearly underline that screening of blood before transfusion is carried out is important as the number of infected humans that would have been infected with malaria reduces as a result of blood screening while the number of infected individuals increases as a result of no blood screening as depicted in Figure 6.  R with respect to rate of transfusion of infected blood  revealed that more individual is likely to become infected as it has a positive impact on 0 R , this led to the introduction of controls   t u 1 and   t u 2 in the optimal control model. The control model was analyzed using Pontryagin's Maximum Principle. The result of the analysis revealed that the combination of using both controls yielded the best result. Conclusively, models must be developed to include information on human nature and behavior in order to give realistic estimates on malaria dynamics. Lack of screening donor's blood may affect the eradication of malaria especially in malaria endemic regions.