Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence

C. Connell Mccluskey

doi:10.3934/mbe.2010.7.837

Abstract

An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0 is less than 1 and globally attracting if R0=1; if R0 is larger than 1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function.

Highlights

The prevalence of disease in a population is often described by an SIR model where the population is subdivided into three classes: susceptibles, infecteds and recovereds
The delay appears in the incidence term which is typically the only nonlinearity, and is the “cause” of all “interesting behaviour”
Various forms have been used for the incidence term, both for ordinary differential equations (ODEs) and for delay equations

Summary

Introduction

The prevalence of disease in a population is often described by an SIR model where the population is subdivided into three classes: susceptibles, infecteds and recovereds (or removeds). The simplest forms of these models are ordinary differential equations (ODEs) [10, 11]. Various forms have been used for the incidence term, both for ODEs and for delay equations. Common forms include mass action βSI [2, 18, 23], saturating incidence βS. Changing the form of the incidence function can potentially change the behaviour of the system. In [14] a system of ODEs with a general incidence term f (S, I) is studied. The goal of this paper is to present a similar analysis for equations with a bounded distributed delay and a general nonlinear incidence function.

CONNELL MCCLUSKEY

2: Saturating