Wave propagation of a reaction-diffusion cholera model with hyperinfectious vibrios and spatio-temporal delay

In this paper, we consider a reaction-diffusion cholera model with hyperinfectious vibrios and spatio-temporal delay. In the model, it is assumed that cholera has a fixed latent period and the latent individuals can diffuse, and a non-local term is incorporated to describe the mobility of individuals during the latent period. It is shown that the existence and nonexistence of traveling wave solutions are fully determined by the basic reproduction number R0 and the critical wave speed c*. Firstly, when R0>1 and the wave speed c > c*, the existence of strong traveling waves is obtained by using Schauder’s fixed point theorem and Lyapunov functional approach. By employing a limiting argument, the existence of strong traveling waves is established when R0>1 and c = c*. Next, when R0≤1, the nonexistence of traveling wave solutions is established by contradiction. Besides, when R0>1 and c < c*, the nonexistence of traveling wave solutions is obtained by means of two-sided Laplace transform. This indicates that c* is indeed the minimal wave speed. Numerical simulations are carried out to illustrate the theoretical results.