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Analysis of a Reaction-Diffusion Susceptible-Infected-Susceptible Epidemic Patch Model Incorporating Movement Inside and Among Patches

In this paper, we propose and analyze a reaction-diffusion susceptible-infected-susceptible (SIS) epidemic patch model. The individuals are assumed to reside in different patches, where they are able to move inside and among the patches. The movement of individuals inside the patches is descried by diffusion terms, and the movement pattern among patches is modeled by an essentially nonnegative matrix. We define a basic reproduction number $\mathcal{R}_0$ for the model and show that it is a threshold value for disease extinction versus persistence. The monotone dependence of $\mathcal{R}_0$ on the movement rates of infected individuals is proved when the dispersal pattern is symmetric or non-symmetric. Numerical simulations are performed to illustrate the impact of the movement of individuals inside and among patches on the transmission of the disease.