  This paper examines an epidemic patch model with mass-action transmission
mechanism and asymmetric connectivity matrix. Results on the global dynamics of
solutions and the spatial structures of endemic solutions are obtained. In
particular, we show that when the basic reproduction number $\mathcal{R}_0$ is
less than one and the dispersal rate of the susceptible population $d_S$ is
large, the population would eventually stabilize at the disease-free
equilibrium. However, the disease may persist if $d_S$ is small, even if
$\mathcal{R}_0<1$. In such a scenario, explicit conditions on the model
parameters that lead to the existence of multiple endemic equilibria are
identified. These results provide new insights into the dynamics of infectious
diseases in multi-patch environments. Moreover, results in [27], which is for
the same model but with symmetric connectivity matrix, are generalized and
improved.
