Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

Abstract

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.

Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

“…Since then, there has been extensive research with traveling wave solutions for delayed diffusive SIR models with various incidence rates, such as βSI 1+αI , βI p S q (p, q > 0) and general form f (S)g(I), see [3][4][5][6][7][8][9][10][11][12][13][14]. There have also been some papers about traveling wave solutions for delayed diffusive SIR models with external supplies [15,16] and delayed diffusive SIRS epidemic models [17,18].…”

Traveling Waves with Critical Speed in a Delayed Diffusive Epidemic Model

“…Usually, the infected individuals cannot be transmitted the epidemic to others immediately after being infected, and a certain amount of biological development is necessary before they can infect others. Thus, time delay has been incorporated into many epidemic models [3][4][5][6][7][8]. Travelling waves in these models describe the epidemic wave moving out from an initial disease-free equilibrium to the endemic equilibrium with a constant speed [9].…”

“…Until recently, much attention has been paid to the existence of super-critical travelling wave solutions for diffusive epidemic models with or without time delay [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For example, Zhan et al [19] investigated a spatially extended SI epidemic system with a nonlinear incidence rate.…”

“…For example, Zhan et al [19] investigated a spatially extended SI epidemic system with a nonlinear incidence rate. Using the method for the dynamical system, they obtained the existence of a heteroclinic orbit connecting two equilibrium points in R 3 which corresponds to a super-critical travelling wave solution connecting the disease-free and endemic equilibria for this SI epidemic system. Lotfi et al [7] derived the existence of supercritical travelling waves of a delayed diffusive epidemic model with specific nonlinear incidence rate by constructing a pair of upper and lower solutions and applying the Schauder's fixed point theorem.…”

“…Lotfi et al [7] derived the existence of supercritical travelling waves of a delayed diffusive epidemic model with specific nonlinear incidence rate by constructing a pair of upper and lower solutions and applying the Schauder's fixed point theorem. Very recently, Zhou et al [3] investigated the travelling wave solution of a delayed diffusive epidemic model (1.4). They also proved the nonexistence of travelling wave solutions.…”