Threshold dynamics of a time-delayed nonlocal reaction-diffusion cholera model with temporary immunity

In this paper, a nonlocal and time-delayed cholera model is formulated that incorporates temporary immunity of the hosts, and incubation periods in the hosts and pathogen. The basic reproduction number Formula: see text for the model system is derived. It is shown that this number gives the threshold dynamics: if Formula: see text, there exists at least one positive steady state and that the disease persists; and if Formula: see text, the disease will die out, provided that cholera have no impact on host mobility or the immunity for the host is permanent. For the special homogeneous case, an explicit formula of Formula: see text is obtained, and there is a unique globally attractive positive constant steady state as Formula: see text. Numerical simulations are illustrated to investigate the influence of diffusion coefficients and spatial heterogeneity on the spread of the disease, and explore the sensitivity of Formula: see text on the model parameters.