Asymptotic profile of the positive steady state for an SIS epidemic reaction–diffusion model: Effects of epidemic risk and population movement

• Study an SIS model in a more complicated heterogeneous environment.
• Apply PDE theory to conduct rigorous mathematical analysis.
• Show how epidemic risk and population movement affect disease transmission.
• Suggest important implications for optimal disease control strategies.
• Perform numerical simulations to support theoretical results.

Identifying the epidemic risk for infectious disease is crucial in order to effectively perform control measures. In a series of our work, from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious disease via an SIS epidemic reaction–diffusion model proposed by Allen et al. (2008) in  [36]. In Allen et al. (2008)  [36], Peng (2009)  [37], it was assumed that the habitat of the populations consists of only the low and high risk areas. The present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs, and deals with two cases: (i) only the moderate and high risk areas exist; (ii) the low, moderate and high risk areas coexist. In each case, we rigorously determine the asymptotic profile of the positive steady state (i.e., the endemic equilibrium) as the migration rate of either the susceptible or infected population tends to zero. Our results show how epidemic risk and population movement affect the spatial distribution of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies. Numerical simulations are carried out to support the theoretical results.