The stability analysis of the SIR epidemic models with vertical infection and continuous vaccination

An SIR epidemic model with vertical infection,continuous vertical and bilinear incidence rates,is considered. The threshold which determines the existence of the infective disease is found. When it is smaller than 1,there only exists disease free equilibrium;when it is bigger than 1,the endemic equilibrium and the disease free equilibrium exist. By Hurwitz criterion,the locally asymptotical stability of the disease free equilibrium is proved. By Lasalle invariant theorem and Liapunov function,the global asymptotical stability of disease free equilibrium and the endemic equilibrium is proved.