Stationary distribution and probability density function of a stochastic waterborne pathogen model with saturated direct and indirect transmissions

We investigate the dynamical behavior of deterministic and stochastic waterborne pathogen models with saturated incidence rates. In particular, the indirect transmission via person-water-person contact is half-saturated and the direct transmission by person-person contact takes the saturated form. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number R-0 of the model is obtained through the next-generation matrix method. It has been shown that the disease-free equilibrium is locally stable when R-0 < 1 and unstable for R-0 > 1. Furthermore, a stochastic Lyapunov method is adopted to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model, which reveals that the infection will persist if R-0(s) > 1. We also obtain an exact expression of the probability density function near the quasi-endemic equilibrium of stochastic system. Finally, numerical simulations are carried out to support our analytical findings. Results suggest that saturation constants in both direct and indirect transmissions play a positive role in controlling disease. Our results may provide some new insights for the elimination of waterborne disease.