Travelling Waves of a Delayed SIR Epidemic Model with Nonlinear Incidence Rate and Spatial Diffusion

Jing Yang,Yi Zhang,Siyang Liang,Mike B Gravenor

doi:10.1371/journal.pone.0021128

Jing Yang, Yi Zhang + Show 2 more

Open Access

https://doi.org/10.1371/journal.pone.0021128

Copy DOI

Abstract

This paper is concerned with the existence of travlelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.

Highlights

Let S(t) represent the number of individuals who are susceptible to the disease, that is, who are not yet infected at time t; I(t) represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and R(t) represent the number of individuals who have been infected and removed from the possibility of being infected again or of spreading at time t
We have dealt with the existence of travelling wave solutions for an SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay
By analyzing the corresponding characteristic equations, we discussed the local stability of a disease-free steady state and an endemic steady state to system (0.3) under homogeneous Neumann boundary conditions

Summary

Introduction

Let S(t) represent the number of individuals who are susceptible to the disease, that is, who are not yet infected at time t; I(t) represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and R(t) represent the number of individuals who have been infected and removed from the possibility of being infected again or of spreading at time t. In [1], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time t is given by bS(t)I(t{t), where b is the average number of contacts per infective per day and tw0 is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. >: I_ (t)~bS(t)I(t{t){(m2 R_ (t)~cI(t){m3R(t), zc)I (t), ð0:1Þ where parameters m1, m2, m3 are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. The parameters B and c are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [2,3,4,5])

Methods

Results

Conclusion