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

Jiazhe Lin,Rui Xu,Yuji Li

doi:10.1186/s13662-018-1635-6

Abstract

In this paper, an HBV epidemic model that incorporates saturating incidence and age-structure in the exposed and infectious classes is proposed. We study the asymptotic smoothness of semi-flow generated by the model. By calculating the basic reproduction number and analyzing the characteristic equations, the local stability of disease-free and endemic steady states is studied. We investigate the global dynamics of this model by using Lyapunov functionals and LaSalle’s invariance principle and prove 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.

Highlights

Hepatitis B is very contagious and it is hard to control its transmission
The model’s dynamics is determined by the basic reproduction number
When the basic reproduction number is less than unity, the diseasefree steady state is globally asymptotically stable and all hepatitis B patients will recover; when the basic reproduction number is larger than unity, there exists a unique endemic steady state and it is globally asymptotically stable

Summary

Introduction

Hepatitis B is very contagious and it is hard to control its transmission. studying the law of its infection has become the focus of attention. Researchers have presented an HBV epidemic model to study hepatitis B transmission and some results of the investigation have been given. There exist many epidemic models, some are not appropriate to show the hepatitis B transmission rules. In [1], Liu divided hepatitis B patients into acute and chronic patients and presented an ordinary differential model. Considering the real transmission rules of hepatitis B, it is necessary to introduce saturating incidence to describe the law of hepatitis B’s infection. 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. We study the existence of equilibria and obtain the expression of the basic reproduction number R0 in Sect. |θ2(a) – ξ (a)| ≥ μ0, θ3(a) ≥ μ0, for all a > 0

State space Define the space of functions X as

S0 – n

Discussion