In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease
free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.