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This paper examines an epidemic patch model with mass-action transmissionmechanism and asymmetric connectivity matrix. Results on the global dynamics ofsolutions and the spatial structures of endemic solutions are obtained. Inparticular, we show that when the basic reproduction number $\mathcal{R}_0$ isless than one and the dispersal rate of the susceptible population $d_S$ islarge, the population would eventually stabilize at the disease-freeequilibrium. However, the disease may persist if $d_S$ is small, even if$\mathcal{R}_0<1$. In such a scenario, explicit conditions on the modelparameters that lead to the existence of multiple endemic equilibria areidentified. These results provide new insights into the dynamics of infectiousdiseases in multi-patch environments. Moreover, results in [27], which is forthe same model but with symmetric connectivity matrix, are generalized andimproved.

On the dynamics of an epidemic patch model with mass-action transmission mechanism and asymmetric dispersal patterns