Ergodicity of a stationary distribution for a stochastic cholera model with a general functional response and higher-order perturbation

A general stochastic compartment model for cholera with higher-order perturbation is proposed, which incorporates direct and indirect transmission by contaminated water. Nonlinear incidence, multiple stages of infection, multiple states of pathogen, and second-order white-noises perturbation are introduced into the model, which includes and extends the existing cholera model. The existence and ergodicity of the stationary distribution for the cholera system are obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $R_0^s$ $R_{0}^{s}$ corresponding to the basic productive number $R_0$ $R_{0}$ of the ordinary differential equation. The results show that, if $R_0^s>1$ $R_{0}^{s}>1$ , the system has a unique and ergodic stationary distribution, which implies the persistence of the diseases. Our general results are applied to a cholera system with a Holling type-II functional response.