Narrow Results

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value ${\mathcal{ℛ}}_s$ is defined in terms of the basic reproduction number $R_0$ for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When ${\mathcal{ℛ}}_s<1$, the cholera infection will extinct within finite periods of time almost surely. When ${\mathcal{ℛ}}_s>1$, the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number $R_0$ for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that ${\mathcal{ℛ}}_s<1$, the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate our analysis, and to compare with the corresponding deterministic model. Biological implications are pointed out.