Travelling Waves of a Diffusive Epidemic Model with Latency and Relapse

Zhiping Wang,Rui Xu

doi:10.1155/2013/869603

Abstract

An SEIR epidemic model with relapse and spatial diffusion is studied. By analyzing the corresponding characteristic equations, the local stability of each of the feasible steady states to this model is discussed. The existence of a travelling wave solution is established by using the technique of upper and lower solutions and Schauder's fixed point theorem. Numerical simulations are carried out to illustrate the main results.

Highlights

In most epidemic models, individuals are often divided into several classes such as susceptible, infective, and recovered classes
Incomplete treatment can lead to relapse, but relapse can occur in patients who took a full course of treatment and were declared cured [3,4,5]
Motivated by the work mentioned above, the main purpose of this paper is to study the effect of the spatial diffusion of the individuals on the dynamics of an epidemic model with latency and relapse

Summary

Introduction

Individuals are often divided into several classes such as susceptible, infective, and recovered classes. For some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class. The environment in which an individual lives is often heterogeneous making it necessary to distinguish the locations, and due to the large mobility of individuals within an area or even worldwide, spatially uniform models are not sufficient to give a realistic picture of a disease’s transmission For this reason, the spatial effects cannot be neglected in studying the spread of epidemics. Motivated by the work mentioned above, the main purpose of this paper is to study the effect of the spatial diffusion of the individuals on the dynamics of an epidemic model with latency and relapse.

Local Stability

Existence of Travelling Waves

Numerical Simulations

Conclusions