Abstract
The dynamics of an SIS epidemic patch model with asymmetric connectivity matrix is analyzed. It is shown that the basic reproduction number [Formula: see text] is strictly decreasing with respect to the dispersal rate of the infected individuals. When [Formula: see text], 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).
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