Stochastic Environmental Impacts on Two-Patch Cholera Model: Threshold Analysis and Ergodic Stationary Distribution

In-depth analysis of epidemic models, particularly for cholera, is crucial because they serve as significant tools for disease transmission prediction, evaluation of control strategies, and optimization of healthcare resource management. The stochastic models provide increased realism by incorporating environmental uncertainty such as variability in water quality, disparities in access to sanitation, and population mobility. The present work generalizes a deterministic two-patch cholera model to a stochastic framework. We first prove the existence and uniqueness of global solutions, then establish the extinction condition R0*<1 for disease eradication in the long term. A key contribution lies in proving the existence of a unique ergodic stationary distribution when R0(1)>1 and R0(2)>1. Furthermore, we derive the stochastic threshold R0=max{R0(1),R0(2)}, which corresponds to the basic reproduction number R0=max{R0(1),R0(2)}. Lastly, numerical simulations are employed to confirm theoretical results.