Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infected-susceptible) epidemic model with n patches in which susceptible and infected individuals can both move between patches. 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. The basic reproduction number ${\cal R}_0$ for the model is determined. It is then shown that when the disease-free equilibrium is stable (${\cal R}_0 < 1$) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (${\cal R}_0 > 1$) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (${\cal R}_0 < 1$) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (${\cal R}_0 > 1$). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.