Traveling waves of a delayed epidemic model with spatial diffusion

Abstract

In this paper, we study the existence and non-existence of traveling waves for a delayed epidemic model with spatial diffusion. That is, by using Schauder's fixed-point theorem and the construction of Lyapunov functional, we prove that when the basic reproduction number $R_0>1$, there exists a critical number $c^*>0$ such that for all $c>c^*$, the model admits a non-trivial and positive traveling wave solution with wave speed $c$. And for $c<c^*$, by the theory of asymptotic spreading, we further show that the model admits no non-trivial and non-negative traveling wave solution. And also, some numerical simulations are performed to illustrate our analytic results.

Keywords

• traveling wave solutions
• delayed epidemic model
• schauder fixed point theorem
• lyapunov functional