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.

In this paper, we deal with a Lévy noise-driven epidemic model reflecting the dynamics of measles infection subject to the effect of vaccination. After model formulation, the feasibility of the system was studied by using the underlying existence and uniqueness theory. Moreover, we discussed the behavior of solution around the infection-free and disease-present steady states. To check the persistence and extinction of the infection, we calculated the threshold parameter ${\mathbf{R}}_{\mathbf{s}}$ and it was determined that the disease vanishes whenever ${\mathbf{R}}_{\mathbf{s}}<1$. From January to October 2019, the reported measles cases in Pakistan wear used and the model was fitted against this data by using the well-known fitting techniques. The values of the parameter were estimated and future behavior of the infection was predicted by simulating the model. The model was further simulated and theoretical findings of the study were validated.

This research provides an improved theoretical framework of the Kermack-McKendrick system. By considering the general interference function and the polynomial perturbation, we give the sharp threshold between two situations: the disappearance of the illness and the ergodicity of the higher-order perturbed system. Obviously, the ergodic characteristic indicates the continuation of the infection in the population over time. Our study upgrades and enhances the work of Zhou et al. (2021) and suggests a new path of research that will serve as a basis for future investigations. As an illustrative application, we discuss some special cases of the polynomial perturbation to examine the precision of our outcomes. We deduce that higher order fluctuations positively affect the illness extinction time and lead to its rapid disappearance.