Global dynamics of an epidemiological model with age of infection and disease relapse

Rui Xu

doi:10.1080/17513758.2017.1408860

Abstract

ABSTRACTIn this paper, an epidemiological model with age of infection and disease relapse is investigated. The basic reproduction number for the model is identified, and it is shown to be a sharp threshold to completely determine the global dynamics of the model. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is verified that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, and hence the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease becomes endemic.

Highlights

It is well known that for some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class
By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is verified that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, and the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease becomes endemic
It has been shown that the global dynamics of system (2) is determined completely by the basic reproduction number R0

Summary

Introduction

It is well known that for some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class. Motivated by the works of Magal et al [18], Tudor [22] and van den Driessche et al [25], in the present paper, we are concerned with the joint effects of disease relapse and age of infection on the transmission dynamics of infectious diseases. A brief discussion is given in Section 8 to conclude this work

Boundedness of solutions

Asymptotic smoothness

Steady states and basic reproduction number

Local stability

Uniform persistence

Global stability

Numerical simulation

Discussion