In this paper, we perform qualitative analysis to two SIS epidemic models in a patchy environment, without and with linear recruitment. The model without linear recruitment was proposed and studied by Allen et al. (SIAM J Appl Math 67(5):1283-1309, 2007). This model possesses a conserved total population number, whereas the model with linear recruitment has a varying total population. However, both models have the same basic reproduction number. For both models, we establish the global stability of endemic equilibrium in a special case, which partially solves an open problem. Then we investigate the asymptotic behavior of endemic equilibrium as the mobility of infected and/or susceptible population tends to zero. Though the basic reproduction number is a well-known critical index, our theoretical results strongly suggest that other factors such as the variation of total population number and individual movement may also play vital roles in disease prediction and control. In particular, our results imply that the variation of total population number can cause infectious disease to become more threatening and difficult to control.
Keywords: Asymptotic profile; Basic reproduction number; Disease-free equilibrium; Endemic equilibrium; Global dynamics; SIS epidemic patch model.