Optimal control problem and backward bifurcation on malaria transmission with vector bias

Abstract

This article aims to apply a mathematical model to investigate the spread of malaria by considering vector bias, saturated treatment, and an optimal control approach. A mathematical analysis of the equilibrium points and an investigation of the basic reproduction number show that if the basic reproduction number (R0) is less than one, the disease-free equilibrium is locally asymptotically stable. Furthermore, the center-manifold theory is applied to analyze the stability of the endemic equilibrium when R0=1. We find that our model performs a backward bifurcation phenomenon when the saturated treatment or vector bias parameter is larger than the threshold. Interestingly, we found that uncontrolled fumigation could increase the chance of the appearance of backward bifurcation. From the sensitivity analysis of R0, we find that the fumigation and vector bias are the most influential parameters for determining the magnitude of R0. Using the Pontryagin maximum principle, the optimal control problem is constructed by treating fumigation and medical treatment parameters as the time-dependent variable. Our numerical results on the optimal control simulation suggest that time-dependent fumigation and medical treatment could suppress the spread of malaria more efficiently at minimum cost.