MATHEMATICAL MODELING OF MALARIA DISEASE WITH CONTROL STRATEGY

1Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Pretoria Gauteng, 0028, South Africa 2Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS 66045, USA 3Department of Mathematics and Statistics, Lagos State Polytechnic, Ikorodu, Lagos, Nigeria 4Department of Mathematical Sciences, University of Zululand, Private Bag X1001, KwaDlangezwa, 3886, South Africa


INTRODUCTION
Malaria, a life-threatening disease which happens to be a vector borne, and is one of the major deadly infectious diseases worldwide. [1,2,3,4]. The disease is caused by the protozoan Plasmodium and it is transmitted in humans by an effective bite of an infected adult female Anopheles mosquito (the malaria vector) [1,5,6]. The species causing agents include; (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, and Plasmodium malariae).These aforementioned species differ in microscopic appearance, geographical distribution, and clinical characteristics. By clinical characteristics, we refer to the infection potential, severity, and the ability to cause relapse. Among all the species, P. falciparum has been recognized as the most dangerous to humans [6]. The WHO African Region continues to carry a disproportionately high share of the global malaria burden. In 2018, the region was home to 93% of malaria cases and 94% of malaria deaths. The prevalent of malaria is in over 100 countries, with approximately 216 millions cases, and 655,000 deaths in 2010. [7,8]. In various parts of the world, malaria has been widespread for many decades [5,9], yet it still remains a major public health burden in affected areas, predominantly the tropical and subtropical areas in Africa, Eastern Mediterranean Regions, Asia and South America. In addition, the high risk groups include pregnant women, nonimmune travelers.
Health related problems aside, malaria constitutes a great socioeconomic threat to malariaendemic nations. For instance, as documented by [2,5,10], in Africa alone, $8 billion was estimated to be the annual economic burden of malaria. These reports have made it essential to explore the possibility of formulating numerous intervention strategies to alleviate the influence of malaria in many countries. As cited by [5], there is currently no perfect vaccine against malaria in humans (although collective global efforts are under way to develop such a vaccine [11,12,13,14,15,16,17]. In order to control the spread of malaria, preventive measures include mosquito-reduction strategies and self-protection against mosquito bites (via the use of insecticide treated bed nets (ITNs); intermittent preventive treatment (IPT), and the reduction of vector population through the destruction of their breeding sites [2,18,19,20]. Studies also revealed other intervention strategies, such as the use of indoor residual spraying (IRS) for killing infected indoors mosquitoes through the use of sterile insect technique [2,21], and the use of anti-malaria drugs to regulate malaria (see, for instance, [6,22,23,24,25]).
Over the years, several numbers of mathematical models on the transmission dynamics of malaria have been examined. Following the simple S-I-R malaria model of Ross [5,26] and Macdonald [5,27], many researchers have elaborated these models by incorporating different features associated to malaria transmission dynamics and its control. This includes modeling the effect of age groups on the transmission of malaria, as shown by [5,28,29]; the use of preventive and therapeutic strategies [25]; repeated exposure [30]; impact of climate variables such as temperature and rainfall [31,1,32,33].
In addition to these, a number of papers has been published on the use of optimal control theory in investigating optimal strategies for malaria control in the community. The influence of three control variables (vector-reduction, blood screening strategies and personal protection) on malaria dynamics with indirect and direct transmissions was presented by [34], while bed net and optimal vaccination control efforts in the populations with diverse ranks of naturally acquired immunity were described by [35]. [36] used optimal control theory to explore optimal approaches in controlling malaria; using insecticide treated bed nets (ITNS), spray of mosquito insecticide and treatment as the system control variables, while [37] examined the dynamics of vector-borne disease with nonlinear prevalence by employing similar control variables as in [36]. The optimal control and transmission dynamics of malaria in Kenya was investigated in [38],by incorporating four-time dependent control strategies, namely: insecticide treated bed nets (ITNs), indoor residual spray (IRS), treatment and intermittent preventive treatment of malaria in pregnancy (IPTp).
Motivated by the above studies, we design and develop a dynamical mathematical model to examine the transmission dynamics of malaria. We further explore the best control strategies to curb the disease, through the use of optimal control theory. Specifically, we apply Pontryagin's Maximum Principle (PMP) to ascertain the required conditions for the optimal approaches to control the spread of malaria. The organization of the paper is as follows: section 2 discusses the model formulation, the basic quantitative properties of the model will be the focus in section 3, while the stability analysis of the system equilibrium points will be considered in section 4. Optimal control analysis is presented in section 5, and the discussion of results follows in sections 6 and 7.

MODEL FORMULATION
The model sub-divides the total human population at time t, denoted by N h (t), into susceptible humans (S h ), infected humans (I h ), and recovered humans (R h ). Thus, the total human population is given by Furthermore, the model sub-divides the total vector population at time t, denoted by N v (t), into immature mosquitoes (A v ) susceptible mosquitoes (S v ), and infected mosquitoes (I v ). Thus, the total vector population is given by As the disease progresses based on the disease status, individuals move from one class to the other. Individuals are recruited into the susceptible human population by either via birth or immigration at the rate π h , and also by the loss of immunity of recovered humans at a rate ψ. The humans population susceptible is depopulated by infection subsequent to contact with infectious vectors at a rate λ h (b), defined as The parameter β hv is the probability of effective transmission from human to mosquitoes, following the contact rate of mosquito to human by the rate ε(b). Furthermore, the population of susceptible humans are decreased by natural death, at a rate µ. Thus, the rate of change of the population of susceptible human is given by The population of infected humans individuals is generated by the infection of susceptible humans (at the rate λ h (b)) and is decreased by recovery of infected individuals (at a rate τ h ), natural death (at a rate µ h ) and disease induced death (at a rate δ h ), so that Finally on human population, the population of recovered humans is generated by the recovery of infected humans (at a rate τ h ). It is decreased by loss of immunity and natural death (at a rate ψ and µ h respectively). Thus, the recovered humans population is given by The immature mosquitoes are populated by mosquitoes egg deposition (at a rate π v (b)), and reduced by maturation of immature mosquitoes (at a rate γ v ) and natural mortality (at a rate µ v (q)). Thus, the immature mosquitoes population is given by The population of susceptible mosquitoes is generated by the maturation of immature mosquitoes (at a rate γ v ). It is further depopulated by infection following effective contact with infectious humans at a rate λ v (b), defined as The parameter β vh is the probability of effective transmission from mosquitoes to humans, following the contact rate of mosquito to human by the rate ε(b). Furthermore, the population of susceptible vectors are decreased by natural death, at a rate µ v . Thus, the rate of change of the population of susceptible vectors is given by Finally, the population of infected vectors is generated by the infection of susceptible vectors (at a rate λ v (b)) and is decreased by natural death of mosquitoes(at a rate µ v ). The rate of change of the population of infected vectors is given by Hence, following the above descriptions, the transmission dynamics of malaria in the population is given by the following nonlinear system of ordinary differential equations Mortality rate of mosqitoes due to treated net µ max (q) Mortality rate of mosqitoes due to insecticide spray

Positivity and Boundedness of solutions.
Here the basic properties of the malaria model (11) will be explored. Holding on to the reality of human and vector population, for the malaria model (11) to be epidemiologically meaningful, it is important to establish that all its state variables are non-negative for all time t ≥ 0. In other words, solutions of the model system (11) with non-negative initial condition will remain non-negative for all time t > 0.
Theorem 3.1. Let the initial conditions for the malaria model of the model with positive initial conditions, will remain positive for all time t > 0. Proof Thus, t 1 > 0. It follows from the first equation of the system (11), that using the integrating factor, this can be written as: Hence, Similarly, it can be shown that Therefore, all the solutions of the model (11) remain positive for all non-negative initial conditions.
Consider the biologically feasible region consisting of .
Proof. It follows from the model (11) using (1) and (2) it gives: Since the term δ h I h (t) and µ v asypmtotically. Hence, the region D attracts all solution in R 6 + .

LOCAL STABILITY OF THE DISEASE FREE EQUILIBRIUM (DFE)
The malaria model (11) has a disease free equilibrium obtained by setting the right-hand sides of the equations in the model to zero and solving at I h = I v = 0. Thus, the disease free equilibrium is given by , , 0 The local stability of the DFE, φ 0 will be explored using the next generation operator method [39,40]. The matrices F of the new infection terms of the model (11), and V , of the transition terms of the model (11), are given respectively by It follows that, the reproduction number is given by R 0 = ρ(FV −1 ), where ρ is the spectral radius of the matrix. Hence, Substituting S * h and S * v into (16) as defined in (14) yields Hence, The result below follows from Theorem 2 of [39].
The basic reproduction number R 0 measures the average number of new infections generated by a single infected person during his or her infectious period in a population that is fully susceptible [41,42]. Thus, the epidemiological implication of theorem 3.1 above is that malaria will be controlled in the population whenever R 0 < 1 if the initial sizes of the population of the model are in the basin of attraction of the DFE (E 0 ).

Existence of endemic equilibrium point (EEP).
Here, we will explore the possible equilibra of the model and establish the conditions for the existence of an equilibrium for which malaria is endemic in the population. The infection equilibria of the malaria model (11) rep- is obtained by equating the right hand side of the equations in (11) to be zero and solve simultaneouly. Hence, the endemic equilibria is given below as , The total population N * * h is obtained by summing up the sub-population as given below Hence, perturbing the two force of infection below The endemic equilibria of the model (11), satisfies the following polynomial The quadratic equation (21) above can be analyzed for the possibility of multiple endemic equilibria whenever R 0 < 1. It should be noted that the coefficient a 0 , of the quadratic equation (21) is always positive; and the constant term c 0 , is negative (positive) whenever R 0 is greater (less) than unity. Hence, the following result is established. Thus, it is clear from case (i) of Theorem 3.2 that the malaria model (11) has a unique EEP (of the form φ 1 ) whenever R 0 > 1. Furthermore, case (iii) of Theorem 3.2 indicates the possibility of backward bifurcation, where a LAS DFE co-exists with a LAS endemic equilibrium when the associated reproduction number R 0 is less than unity (R 0 < 1). The epidemiological importance of the phenomenon of backward bifurcation is that the requirement of having R 0 < 1 is although necessary, but not sufficient for disease elimination. In this case, disease elimination will depend upon the initial sizes of the sub-populations of the model.
To check for the possibility of backward bifurcation in the malaria model (11), the discrimi- (21), is set to zero and the result solved for the critical value of R 0 (denoted by R C < 1). This result to for which the following theorem holds.   [48,46] µ max (q) Assumed

OPTIMAL CONTROL ANALYSIS
This section seeks to explore the application of Pontryagin's Maximum Principle (PMP) to determine the necessary conditions for the optimal control of the Malaria disease. We endeavor to incorporate time dependent controls into the proposed system (14), in order to determine the best optimal strategy which can be used to control the disease. Thus, we have We therefore define the control problem as per the above discussion for control policies and costs incurred, as follows: t ∈ [0, T f ] and u 1 , u 2 and u 3 are Lebesgue measurable subject to the model system (1): Following the initial conditions The objective function J represents the total cost incurred as a result of the application of control plans and the burden of the disease. Theorem 3.3.Given that optimal control variables u * 1 , u * 2 , &u * 3 and S * h , I * h , R * h , A * v , S * v &I * v are corresponding optimal state variables of the control system (14) and (15), there exists an adjoint variable λ = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 ) ∈ ℜ 6 + that satisfies the following equations.
S v , I v and with transversality conditions The corresponding optimal controls u * 1 , u * 2 & u * 3 are given as, where; Proof. Corrollary 4.1 of Fleming and Rishel [49] gives the existence of an optimal control due to the convexity of the integrand of J with respect to u 1 , u 2 and u 3 , a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables.
The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then the adjoint equation can be written as Solving for u * 1 , u * 2 and u * 3 subject to the constraints, the characterization (16 -19) can be derived and we have with transversality conditions The corresponding optimal controls u * 1 , u * 2 & u * 3 are given as Hence, we obtain (21, 22 & 23) by using Lenhart and Workman [50,51] (33) By standard control arguments involving the bounds on the controls, we conclude that where; Hence, we discuss the numerical solutions of the optimality system and the corresponding results of varying the optimal controls u 1 , u 2 & u 3 the parameter choices, and the interpretations from various cases.
Hence, we discuss the numerical solutions of the optimality system and the corresponding results of varying the optimal controls u 1 , u 2 & u 3 the parameter choices, and the interpretations from various cases.

Descriptions of Control.
The impact of these control on malaria disease progression is investigated. We also optimize the cost incurred, in order to implement the controls. The description of the three controls is done in this section and the corresponding cost incurred is determined.
i Provision of treated pesticide Bed nets to susceptible population. Preventive measure such as provision of pesticide treated bed nets will reduce the susceptibility of susceptible individuals to malaria infection. u 1 represent time preventive control through treated bed net use such that 0 ≤ u 1 (t) ≤ 1. u 1 = 0 represents no use of treated bed nets while u 1 = 1 terms the full use of treated bed nets. In order to eliminate the uses of treated bed-net, we include cost of providing and distribution of the nets to individuals. We shall find the optimal response of susceptible individuals via treated bed nets ii Providing treatment to infected population. Treatment of infected individual does not only reduce the disease prevalence but also affects its progression. We assume that the treatment is available to infected individual. We also include treatment as a saturation rate function εu 2 I h 1+αI h , since the availability of medical related resources such as diagnosis, treatment e.t.c can not be unlimited . ε is the treatment rate with intensity u 2 and α is the saturation constant. The treatment intensity u 2 is considered as a control variable such that 0 ≤ u 2 (t) ≤ 1. The cost of providing treatment will include diagnosis, medicines, health-care, hospitalized and other related cost.
iii Providing pesticides/insecticides to destroy larva population. As we know, spraying insecticide/ pesticides on larva site which net as breeding ground for mosquitoes will not only reduce the susceptible mosquito populations but also increase the fatality on both adult susceptible mosquitoes and infected mosquitoes. Suppose the pesticides are available for spraying on the larva breeding site, then the intensity of spraying u 3 pesticides on the larva is taken as control such that 0 ≤ u 3 (t) ≤ 1. The cost of actualizing this control includes purchase of spraying equipments, cost of chemicals, clearing of sites, e.t.c

Numerical Simulation.
In this section, we study the impact of these control on malaria disease progression and also optimize the cost incurred in their implementation numerically. on medication and the control on (u 3 ) insecticide spraying are both used to optimize the objective function J, while the control on treated bed-nets (u 1 ) are set to zero. The results in Figure   1 shows a significant difference in the Infected human (I h ) and Infected mosquitoes (I v ) with optimal control strategy compared to (I h ) and (I v ) without control. We noticed in Figure 1(a) that the control strategies resulted in decrease in the number of symptomatic humans (I h ) as against increases in the uncontrolled case. Similarly, in Figure 1 Simulations of the malaria model showing the effect of all the control strategies on each of the compartment 8.5. Optimal treated bed-net, treatment medication and spray insecticide. In this strategy, all three controls (u 1 (t), u 2 (t) & u 3 (t)) are used to optimize the objective function J. It is obvious in Figure 4(a) that there is a significant difference between the number of infected mosquitoes (I v ) under control the number of infected mosquitoes were initially controlled and after 20 days it was more significant that the control strategy employed is effective in controlling the infected mosquitoes. In Figure 4(b), there is significant difference between the presence of control and without control cases. In Figure 4(c), there is significant difference between the number of infected humans (I h ) were initially rising but after 15 days it starts reducing. It is shows that the strategy is very effective in controlling (I h ). It was observed that that the number of susceptible mosquitoes (S v ) which was initially increase has been reduced. There is significant difference between Figure 4(d) and Figure 4(e). In Figure 4 8.6. Conclusion. In this paper, we presented a malaria model using system of ordinary differential equations and established that the model is locally asymptotically stable when the associated reproduction number is less than unity. In the optimal control problem considered, we use the combination of two controls at a time, while setting the other to zero, to investigate and compare the effects of the control strategies on malaria progression eradication. The analysis made possible by Pontryagin's Maximum Principle coupled with numerical simulations reveal that the combination of the three control strategies may be adopted in controlling malaria progression disease among the human and mosquito interacting populations . Our numerical results shows that the combination of the three (3) controls, treated bed-nets, medication and insecticides spray, has the highest impact on the control of the disease (malaria). This is followed by the combination of medication and insecticide among the human population; and lastly by the combination involving the use of treated bed-nets and insecticide use. In communities where resources are scarce, we suggest that the combination of treatment and treated bed-nets should be adopted, having observed from the comparison of all three control strategies in Figure 4, that there is significant difference between this strategy and the combination of the all three (3) controls.
of DSI-NRF Centre of Excellence in Mathematical & Statistical Sciences (CoE-MaSS) for the postdoc position at University of Pretoria, South Africa.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.