# Traveling waves for a diffusive SIR model with delay and nonlinear incidence

Volume 11, Issue 12, pp 1313--1330 Publication Date: September 13, 2018       Article History
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### Authors

Yanmei Wang - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China $\&$ School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030006, China
Guirong Liu - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
Aimin Zhao - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

### Abstract

This paper is concerned with the existence and non-existence of traveling wave solutions for a diffusive SIR model with delay and nonlinear incidence. First, we construct a pair of upper and lower solutions and a bounded cone. Then we prove the existence of traveling wave by using Schauder's fixed point theorem and constructing a suitable Lyapunov functional. The nonexistence of traveling wave is obtained by two-sided Laplace transform. Moreover, numerical simulations support the theoretical results. Finally, we also obtain that the minimal wave speed is decreasing with respect to the latent period and increasing with respect to the diffusion rate of infected individuals.

### Keywords

• SIR model
• traveling wave
• time delay
• nonlinear incidence

•  35K57
•  35C07
•  92D30

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