Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix

The dynamics of an SIS epidemic patch model with asymmetric connectivity matrix is analyzed. It is shown that the basic reproduction number R is strictly decreasing with respect to the dispersal rate of the infected individuals. When R> 1 , the model admits a unique endemic equilibrium, and its asymptotic profiles are characterized for small dispersal rates. Specifically, the endemic equilibrium converges to a limiting disease-free equilibrium as the dispersal rate of susceptible individuals tends to zero, and the limiting disease-free equilibrium has a positive number of susceptible individuals on each low-risk patch. Furthermore, a sufficient and necessary condition is provided to characterize that the limiting disease-free equilibrium has no positive number of susceptible individuals on each high-risk patch. Our results extend earlier results for symmetric connectivity matrix, providing a positive answer to an open problem in Allen et al. (SIAM J Appl Math 67(5):1283–1309, 2007).