Analysis of a stochastic epidemic model for cholera disease based on probability density function with standard incidence rate

Acute diarrhea caused by consuming unclean water or food is known as the epidemic cholera. A model for the epidemic cholera is formulated by considering the instants at which a person contracts the disease and the instant at which the individual exhibits symptoms after consuming the poisoned food and water. Initially, the model is formulated from the deterministic point of view, and then it is converted to a system of stochastic differential equations. In addition to the biological interpretation of the stochastic model, we proved the existence of the possible equilibria of the associated deterministic model, and accordingly, stability theorems are presented. It is demonstrated that the proposed stochastic model has a unique global solution, and adequate criteria are constructed by using the Lyapunov function theory, which guarantees that the system has persistence in the mean whenever ${\mathbf{R}}_{\mathbf{s}}^{\mathbf{0}}>1$. For the case of $R_s<1$, we proved that the disease will tend to be eliminated from the community. Some graphical solutions were produced in order to better validate the analytical results that were acquired. This research can offer a solid theoretical foundation for comprehensive knowledge of other chronic communicable diseases. Additionally, our approach seeks to offer a technique for creating Lyapunov functions that may be utilized to investigate the stationary distributions of models with non-linear stochastic perturbations.