Mathematical Modeling of the Impact of Temperature Variability on Malaria Epidemics with Optimal Control

Abstract

Malaria is an infectious disease that causes morbidity and mortality globally. In this study, we present two deterministic mathematical models of malaria transmission in the presence of temperature variability. The first is susceptible - infected - recovered - susceptible and susceptible - infected compartmental model. The model is both qualitatively and quantita tively analyzed. The local and global stability of the equilibrium points are shown using the Routh-Hurwitz criterion and Lyapunov function respectively. The sensitivity analysis of the model has described and the model exhibits forward and backward bifurcation. Moreover, we extended the model to the optimal control problem by incorporating three controls treated bed nets, treatment of infected and indoor insecticides spraying. The Pontraygin’s minimum principle is applied to obtain the necessary condition for the optimal control problem. Based on the numerical simulation of the optimality system and cost effectiveness analysis, the com bination of treatment and insecticide spraying is the most optimum and least cost strategy to minimize the disease. Secondly, we proposed and analyzed the susceptible - exposed - infected - recovered - susceptible and susceptible - exposed -infected compartmental model. The analysis of the model shows that the model is bounded and positive in certain domain. Applying the next-generation matrix method, the basic reproduction number with respect to the disease free equilibrium is computed. The local and global stability of the equilibrium points are obtained. The sensitivity analysis of the basic reproduction number with respect to all basic parameters was computed. The model has a forward and backward bifurcation. Using real data some parameters were estimated. Also the model is extended to optimal control problem. The Pontraygin’s minimum principle is introduced to obtain the neces sary condition for optimal control problem. Runge-Kutta forward-backward sweep method is used to solve the optimal control problem. Therefore, based on the numerical simulations of optimal control problem and cost-effectiveness analysis, the most optimal and least cost strategy to reduce the malaria is using treated bed nets and treatment.