Dynamics of cholera epidemic models in fluctuating environments

Abstract

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 s is defined in terms of the basic reproduction number R0 for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When s 1, the cholera infection will extinct within finite periods of time almost surely. When 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 R0 for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that 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.