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The dynamics of an SIS epidemic patch model with asymmetric connectivitymatrix is analyzed. It is shown that the basic reproduction number $R_0$ isstrictly decreasing with respect to the dispersal rate of the infectedindividuals, and the model has a unique endemic equilibrium if $R_0>1$. Theasymptotic profiles of the endemic equilibrium for small dispersal rates arecharacterized. In particular, it is shown that the endemic equilibriumconverges to a limiting disease-free equilibrium as the dispersal rate ofsusceptible individuals tends to zero, and the limiting disease-freeequilibrium has a positive number of susceptible individuals on each low-riskpatch. Moreover a sufficient and necessary condition is found to guarantee thatthe limiting disease-free equilibrium has no positive number of susceptibleindividuals on each high-risk patch. Our results extend earlier results forsymmetric connectivity matrix, and we also partially solve an open problem byAllen et al. (SIAM J. Appl. Math., 67: 1283-1309, 2007).

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