Abstract：We are concerned with qualitative analysis to two SIS epidemic models in a patchy environment, without/with linear recruitment. For the model with total population, proposed by Allen et al. in 2007, we first establish the global stability of the endemic equilibrium in some special cases, which partially solves an open problem left there. Then we investigate the asymptotic behavior of the unique endemic equilibrium as the mobility of infected individuals tends to zero, including the scenario that the movement of susceptible individuals may shrink simultaneously. In particular, our theoretical results indicate that, for the model with constant total population, the optimal strategy to eliminate the infectious disease is to restrict the movement of the susceptible population rather than that of the infected. Finally, our findings for the model with linear recruitment implies that the disease becomes more threatening and difficult to control, due tovarying total population.