Spatial patterns in a discrete-time SIS patch model.

How do spatial heterogeneity, habitat connectivity, and different movement rates among subpopulations combine to influence the observed spatial patterns of an infectious disease? To find out, we formulated and analyzed a discrete-time SIS patch model. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease persists only when the mobility of infected individuals lies below some threshold value, but for high-risk habitats, the disease always persists. When the disease does persist, then there exists an endemic equilibrium (EE) which is unique and positive everywhere. This EE tends to a spatially inhomogeneous disease-free equilibrium (DFE) as the mobility of susceptible individuals tends to zero. The limiting DFE is nonempty on all low-risk patches and it is empty on at least one high-risk patch. Sufficient conditions for the limiting DFE to be empty on other high-risk patches are given in terms of disease transmission and recovery rates, habitat connectivity, and the infected movement rate. These conditions are also illustrated using numerical examples.