Global stability of a network-based SIS epidemic model with a saturated treatment function

Abstract

We study global stability of a network-based SIS epidemic model with a saturated treatment function. The model was proposed by Huang and Li (2019). They obtained a threshold R0, depending on the network structure and some parameters (except the parameter α used to measure the extent of the effect of the infected being delayed for treatment), and some sufficient conditions on the stability of the equilibria. The aim of the present paper is to conduct a further analysis on the global stability of the equilibria by means of an iterative technique. For the case when R0<1, it is proved that if the disease-free equilibrium is the unique one, then it is globally asymptotically stable. In addition, we give some new conditions guaranteeing that the disease-free equilibrium is the unique one, and show the existence of two endemic equilibria if α is sufficiently large, which verifies partially their numerical observation. For the case when R0>1, it is proved that if the model admits a unique endemic equilibrium, then the unique endemic equilibrium is globally attractive, and is globally asymptotically stable if α is sufficiently large or small. In particular, we present a condition such that the threshold R0 determines the global stability of the disease-free equilibrium and the endemic equilibrium. Numerical experiment is also performed to illustrate our theoretical results.