Dynamics of SIR epidemic models with horizontal and vertical transmissions and constant treatment rates

We investigate the dynamics and bifurcations of SIR epidemic model with horizontal and vertical transmissions and constant treatment rates. It is proved that such SIR epidemic model have up to two positive epidemic equilibria and has no positive disease-free equilibria. We find all the ranges of the parameters involved in the model under which the equilibria of the model are positive. By using the qualitative theory of planar systems and the normal form theory, the phase portraits of each equilibria are obtained. We show that the equilibria of the epidemic system can be saddles, stable nodes, stable or unstable focuses, weak centers or cusps. We prove that the system has the Bogdanov-Takens bifurcations, which exhibit saddle-node bifurcations, Hopf bifurcations and homoclinic bifurcations.