Stability Analysis of an SIS Epidemic Model in Heterogeneous Environment

This paper studies an susceptible-infected-susceptible reaction-diffusion model in spatially heterogeneous environment proposed in [Allen et al., Discrete Contin. Dyn. Syst., 21, 2008], where the existence and uniqueness of the endemic equilibrium are established and its stability is proposed as an open problem. However, till now, there is no progress in the stability analysis except for special cases with either equal diffusion coefficients or constant endemic equilibrium. In this paper, we demonstrate the first criterion in determining the stability of the non-constant endemic equilibrium with different diffusion coefficients. Thanks to this criterion, when one of the diffusion rates is small or large, the impact of spatial heterogeneity on the stability can be characterized based on the asymptotic behavior of the endemic equilibrium.