Global stability of two epidemic models

In this paper we study the global stability of two epidemic models by ruling out the presence of periodic orbits, homoclinic orbits and heteroclinic cycles. One model incorporates exponential growth, horizontal transmission, vertical transmission and standard incidence. The other one incorporates constant recruitment, disease-induced death, stage progression and bilinear incidence. For the first model, it is shown that the global dynamics is completely determined by the basic reproduction number $R_0$. If $R_0\leq1$, the disease free equilibrium is globally asymptotically stable, whereas the unique endemic equilibrium is globally asymptotically stable if $R_0>1$. For the second model, it is shown that the disease-free equilibrium is globally stable if $R_0\leq1$, and the disease is persistent if $R_0>1$. Sufficient conditions for the global stability of an endemic equilibrium of the model are also presented.