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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 [Formula: see text]. 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.

The transmission of production-limiting disease in farm, such as Neosporosis and Johne's disease, has brought a huge loss worldwide due to reproductive failure. This paper aims to provide a modeling framework for controlling the disease and investigating the spread dynamics of Neospora caninum-infected dairy as a case study. In particular, a dynamic model for production-limiting disease transmission in the farm is proposed. It incorporates the vertical and horizontal transmission routes and two vaccines. The threshold parameter, basic reproduction number R 0 , is derived and qualitatively used to explore the stability of the equilibria. Global stability of the disease-free and endemic equilibria is investigated using the comparison theorem or geometric approach. On the case study of Neospora caninum-infected dairy in Switzerland, sensitivity analysis of all involved parameters with respect to the basic reproduction number R 0 has been performed. Through Pontryagin's maximum principle, the optimal control problem is discussed to determine the optimal vaccination coverage rate while minimizing the number of infected individuals and control cost at the same time. Moreover, numerical simulations are performed to support the analytical findings. The present study provides useful information on the understanding of production-limiting disease prevention on a farm.