Modeling disease spread in two patches: Impact of asymptomatic mobility and environmental noise

The movement of asymptomatic infected individuals plays a crucial role in the spread of infectious diseases. This paper presents a two-patch SIAR model to examine the impact of asymptomatic infections on disease dynamics under environmental uncertainty. By incorporating white noise and Markov switching, the model captures the influence of environmental fluctuations on disease transmission. Sufficient conditions for disease extinction are derived by using appropriate Lyapunov functions. To address the degeneracy of the diffusion matrix, the weak Feller property and bounded average probability are employed to analyze the stationary distribution of the stochastic system. Numerical simulations validate the theoretical findings and provide additional insights into disease dynamics. Results reveal that large environmental disturbances can lead to disease extinction, whereas smaller disturbances may sustain disease persistence. Furthermore, the effects of unidirectional and bidirectional population movement are explored, demonstrating that infection-dependent movement strategies, especially those with nonlinear dynamics, can significantly reduce disease prevalence. These findings emphasize the importance of regulating population movement and managing environmental variability as essential strategies for controlling disease outbreaks and mitigating their impact.