Threshold dynamics and bifurcation analysis of an SIS patch model with delayed media impact

In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number $R_0$ is defined, and the threshold dynamics are studied. It is shown that the disease-free equilibrium is globally asymptotically stable if $R_0<1$ and the disease is uniformly persistent if $R_0>1$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed.