Stationary distribution and probability density function of a stochastic waterborne pathogen model with logistic growth

Waterborne disease threatens public health globally. Previous studies mainly consider that the birth of pathogens in water sources arises solely by the shedding of infected individuals. However, for free-living pathogens, intrinsic growth without the presence of hosts in environment could be possible. In this paper, a stochastic waterborne disease model with a logistic growth of pathogens is investigated. We obtain the sufficient conditions for the extinction of disease and also the existence and uniqueness of an ergodic stationary distribution if the threshold ℜ^s₀ > 1. By solving the Fokker-Planck equation, an exact expression of probability density function near the quasi-endemic equilibrium is obtained. Results suggest that the intrinsic growth in bacteria population induces a large reproduction number to determine the disease dynamics. Finally, theoretical results are validated by numerical examples.