Global stability properties of renewal epidemic models

TL;DR

This paper analyzes the global stability of a general renewal epidemic model using Lyapunov methods, establishing that the basic reproduction number determines whether the disease-free or endemic equilibrium is globally stable.

Contribution

It introduces a comprehensive stability analysis for a broad class of renewal epidemic models with arbitrary infection age functions, extending previous results.

Findings

The infection-free equilibrium is globally stable when R0 ≤ 1.

The endemic equilibrium is globally stable when R0 > 1.

The basic reproduction number R0 is a sharp threshold parameter.

Abstract

We investigate the global dynamics of a general Kermack-McKendrick-type epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, $\tau$, and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, $R_0$, represents a sharp threshold parameter such that for $R_0\leq 1$, the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when $R_0 > 1$, i.e. when it exists.