Traveling wave solutions of a cholera transmission model with nonlocal diffusion and spatio-temporal delay

Abstract

In this paper, we consider the traveling wave solutions of a cholera transmission model with nonlocal diffusion and spatio-temporal delay, in which the discrete delay $\tau$ represents the latent period of cholera and a nonlocal infection term is introduced to describe the impact of infections at all possible locations at time $t-\tau$ on the current location at time $t$. The basic reproduction number $R_0$ is calculated by using the method of next generation matrix. In addition, the critical wave speed $c^{\ast }$ is established. Firstly, when $R_0>1$ and the wave speed $c>c^{\ast }$, the existence of traveling waves connecting the disease-free steady state and endemic steady state is obtained by using Schauder’s fixed point theorem, the prior estimate, limit theory and suitable Lyapunov functional. By employing a limiting argument, the existence of traveling waves is established when $R_0>1$ and $c=c^{\ast }$. Secondly, when $R_0>1$ and $0<c<c^{\ast }$, the nonexistence of traveling wave solution is proved by means of two-sided Laplace transform. It is shown that $c^{\ast }$ is indeed the minimal wave speed. Numerical simulations are carried out to illustrate the theoretical results. Finally, the impacts of nonlocal diffusion and latent period on minimal wave speed are addressed.