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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 R 0 and the critical wave speed c*. Firstly, when R 0 > 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 R 0 > 1 and c = c*. Next, when R 0 ≤ 1 , the nonexistence of traveling wave solutions is established by contradiction. Besides, when R 0 > 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. [ABSTRACT FROM AUTHOR]

Cholera is a global epidemic infectious disease that seriously endangers human life. It is disturbed by random factors in the process of transmission. Therefore, in this paper, a class of stochastic SIRB cholera model with Ornstein–Uhlenbeck process is established. On the basis of verifying that the model exists a unique global solution to any initial value, a sufficient criterion for the existence of a stationary distribution of the positive solution of the random model is established by constructing an appropriate random Lyapunov function. Furthermore, under the same condition that there is a stationary distribution, the specific expression of the probability density function of the random model around the positive equilibrium point is calculated. Finally, the theoretical results are verified by numerical model. [ABSTRACT FROM AUTHOR]