Global stability and periodicity on SIS epidemic models with backward bifurcation

In this paper we study SIS epidemic models with the vaccine efficacy and waning. First, continuous vaccination to both newborns and susceptible individuals is considered. In this case, a backward bifurcation leading to bistability possibly occurs, and global dynamics are shown by compound matrices and geometric approaches. Second, we consider the impulsive vaccination to susceptible individuals, which is more realistic. The global stability of positive periodic infection-free solution is proved, further, by bifurcation theory; we obtain a supercritical bifurcation at the threshold for the period of pulsing. Lastly, we change constant incidence rate to a general periodic contact rate due to seasonal variation; besides the global stability of disease-free equilibrium, we also show the existence of positive periodic solution with the help of the continuation theorem based on coincidence degree.