The stability analysis of an epidemic model with age-structure in the exposed and infectious classes

Yuji Li ,Rui-Hua Xu

doi:10.28919/cmbn/3337

Abstract

In this paper, we propose an epidemic model with age-structure in the exposed and infectious classes for a disease like hepatitis-B. Asymptotic smoothness of semi-flow generated by the model is studied. By calculating the basic reproduction number and analyzing the characteristic equation, we study the local stability of disease-free and endemic steady states. By using Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable. https://doi.org/10.28919/cmbn/3337

Highlights

Hepatitis B is a worldwide disease and it has become a serious threat to human health
By calculating the basic reproduction number and analyzing the characteristic equation, we study the local stability of disease-free and endemic steady states
By using Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable

Summary

Introduction

Hepatitis B is a worldwide disease and it has become a serious threat to human health. To study the transmission of Hepatitis B, several epidemic models for the infection of Hepatitis B. In [1], Liu considered an HBV infection epidemic model as follows: S(t) = Λ − (μ + p)S(t) − β1S(t)I1(t) − β3S(t)I2(t), V (t) = pS(t) − (μ + ρ)V (t) − β2V (t)I1(t) − β4V (t)I2(t),. Several medical studies show that the scaled probability of Hepatitis B virus infection is in connection with age of infection and the risk per unit time of activation appears to be higher in the early stages of infection than in later stages. Motivated by the above works, in this paper, we propose an SVEI1I2R epidemic model with continuous age-dependent latency, acute hepatitis B infection and chronic hepatitis B infection as follows:. I(t, a), j(t, a) represent the density of patients with acute hepatitis B and chronic hepatitis B with age of infection a at time t, respectively. More details concerning the global stability analysis of epidemic model approach, we refer readers to [9,10,11,12,13,14,15,16,17,18,19]

Preliminaries

State space

Boundedness

Asymptotic smoothness

The existence of equilibria

Local stability

Global stability