On the optimal vaccination and travel-restriction controls with a discrete multi-region SIRS epidemic model

Abstract

Malaria is one of the many kinds of vector-borne diseases which threaten many developing countries around the world every year. Malaria is caused by more than one type of Plasmodium, which allows a superinfection between two kinds of Plasmodium in the human body. This article presents a mathematical model that describes the superinfection between Plasmodium Falciparum and Plasmodium Vivax. The model is developed as a system of a nonlinear ordinary differential equation which accommodates several essential factors, such as birth and death rate, infection process, superinfection phenomenon, recovery rate, etc. Mathematical analysis regarding the existence and stability of fixed points is discussed followed by the construction of the respective ”local” basic reproduction numbers and the ”invasion” basic reproduction numbers between Plasmodium. We found that the malaria-free equilibrium point will be locally stable if both local basic reproduction numbers are less than unity. Our results also indicate that although the ”local” basic reproduction number exhibits the existence of a single Plasmodium equilibrium, it is still possible that this equilibrium is not stable if the invasion basic reproduction number is not larger than unity. Some numerical experiments were conducted to obtain a visual interpretation of the analytical results.