GLOBAL stability for SIRS epidemic models with general incidence rate and tranfer from infectious to susceptible

TL;DR

This paper analyzes the global stability of SIRS epidemic models with a broad class of incidence functions, establishing conditions under which disease-free or endemic equilibria are globally stable.

Contribution

It introduces a comprehensive stability analysis for SIRS models with general incidence rates, including non-monotonic and concave cases, using Lyapunov and LaSalle's invariance principles.

Findings

Disease-free equilibrium is globally stable if R0 ≤ 1.

Endemic equilibrium is globally stable if R0 > 1.

Results apply to a wide range of incidence functions.

Abstract

We study a class of SIRS epidemic dynamical models with a general non-linear incidence rate and transfer from infectious to susceptible. The incidence rate includes a wide range of monotonic, con- cave incidence rates and some non-monotonic or concave cases. We apply LaSalle's invariance principle and Lyapunov's direct method to prove that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number R0 lesser or equal to 1, and the endemic equilibrium is globally asymptotically stable if R0 > 1, under some conditions imposed on the incidence function f(S; I).