Modeling and optimal control analysis of malaria epidemic in the presence of temperature variability

In this paper, we propose and analyze a nonlinear deterministic malaria disease model for the impact of temperature variability on malaria epidemics. Firstly, we analyzed the invariant region and the positivity solution of the model. The basic reproduction number with respect to disease free-equilibrium is calculated by the next-generation matrix method. The local stability and global stability of the equilibrium points are shown using the Routh-Hurwitz criterion and the Lyapunov function, respectively. A disease-free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one and endemic equilibrium exists otherwise. Moreover, we have shown the sensitivity analysis of the basic reproduction number and the model exhibits forward and backward bifurcation. Secondly, we apply the optimal control theory to describe the model with incorporates three controls, namely using treated bed nets, treatment of infected with anti-malaria drugs and for vector killing using insecticide spray strategy. Pontraygin's maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the simulation result of optimal control problem and analysis of cost-effectiveness show that a combination of using treated bed nets and treatment is the most effective and least-cost strategy to prevent the malaria disease.