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

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 a 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 - a 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(& lowast;) is established. Firstly, when R-0 > 1 and the wave speed c > c(& lowast;), 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(& lowast;). Secondly, when R-0 > 1 and 0 < c