Global stability of an age-structured epidemic model with general Lyapunov functional

Abstract

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number ${\cal R}_0$ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for ${\cal R}_0 \lt 1$ and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever ${\cal R}_0 \gt 1$. Several numerical simulations are given to illustrate our results.

Keywords

• sir epidemic model
• infection age
• nonlinear incidence
• persistence
• lyapunov function
• global stability