Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment

Chengxia Lei,Jiang Liu,Fujun Li

doi:10.3934/dcdsb.2018173

Chengxia Lei, Jiang Liu + Show 1 more

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https://doi.org/10.3934/dcdsb.2018173

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Journal: Discrete & Continuous Dynamical Systems - B	Publication Date: Jan 1, 2018
Citations: 13	License type: cc-by

Abstract

In this paper, we deal with a diffusive SIR epidemic model with nonlinear incidence of the form $I^pS^q$ for $0<p≤1$ in a heterogeneous environment. We establish the boundedness and uniform persistence of solutions to the system, and the global stability of the constant endemic equilibrium in the case of homogeneous environment. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibrium if the diffusion rate of the susceptible or infected population is small. Our theoretical analysis shows that, different from the studies of [1,28,38,44] for the SIS models, restricting the movement of the susceptible or infected population can not lead to the extinction of infectious disease for such an SIR system.

Highlights

Kermack and McKendrick [24], according to the principle of mass action, the bilinear incidence βIS was used to describe the spread of an infection between susceptible and infected individuals
In certain situations a nonlinear incidence rate is used to govern the spread of infectious disease
When the spatial environment becomes homogeneous, Theorem 2.2 below asserts that the unique constant EE is globally attractive in cases (i) and (ii)

Summary

Introduction

In order to model disease dynamics, in the pioneering work of Kermack and McKendrick [24], according to the principle of mass action, the bilinear incidence βIS was used to describe the spread of an infection between susceptible and infected individuals. When the spatial environment becomes homogeneous (that is, b, β, μ and δ are positive constants), Theorem 2.2 below asserts that the unique constant EE is globally attractive in cases (i) and (ii). We are concerned with the uniform persistence property of solutions to (2) and the global attractivity of the constant EE in the case that all the parameters in (2) are assumed to be positive constants. By the standard theory for parabolic equations, given continuous and nonnegative initial data (S0, I0), (2) admits a unique classical solution (S(x, t), I(x, t)) which exists for all positive time, and S(x, t) > 0 and I(x, t) ≥ 0 for all x ∈ Ω and t > 0.

Some straightforward computation yields

Assume that w

This clearly indicates that