Global stability of an SIS epidemic model with general incidence function in a patchy environment

Abstract

We investigate some analytical results for an SIS compartmental epidemic model that describes the propagation of a disease in a population of individuals who can travel among $n$ patches. The model is formulated as a system of ordinary differential equations, with terms accounting for general incidence, recovery, birth, death, and travel between cities. The basic reproduction number, $\mathcal{R}_0$ of the model is computed; and we show that it is the threshold dynamics between the persistence and the extinction of the epidemic. Hence, we show that if $\mathcal{R}_0<1$, the disease-free equilibrium is locally and globally attractive while, the system is uniformly persistent and admits a unique endemic equilibrium which is globally asymptotically stable if $\mathcal{R}_0>1$. An example of two patches are studied, and we prove that the choice of incidence function can strongly impact the spread of the disease. Finally, numerical examples are performed to illustrate some theoretical results.