Traveling waves in a nonlocal delayed epidemic model with diffusion

Abstract

In this paper, we are concerned with traveling waves in a nonlocal delayed epidemic model with diffusion. Firstly, by considering a six‐dimensional nondelayed system with the help of variable transformation, we establish the existence of monotone traveling waves by means of the abstract theory, which implies the existence of traveling waves connecting the disease‐free equilibrium and the epidemic coexistence equilibrium. Secondly, we prove the global stability of traveling waves based on spectral analysis method, which reveals that the solutions of the initial values and the traveling waves are exponentially close. Thirdly, we obtain that the wave speed is unique by choosing suitable parameters to construct new upper and lower solutions, which shows that the bistable waves keep the uniqueness of the wave speeds in the case of nonlocal delays. As an application of our results, we give a special example.