Global Stability of SIR Epidemiological Model with Vaccinal Immunity and Bilinear Incidence Rates

The global stability of SIR epidemiological model with vaccinal immunty and bilinear incidence rates is (studied). The basic reproductive number R_0 is found, which determines the existence of the infective disease. When R_0≤1, there only exists disease free (equilibrium) E_0. While R_01, two equilibriums, the endemic equilibria E~* and the disease free (equilibrium) E_0 exist. By Hurwitz criterion and Liapunov-Lasalle invarient theorem, we have proved the disease free (equilibrium) E_0 is globally asymptotically stable if R_01, while the disease free equilibrium E_0 is unstable and the endemic (equilibrium) E~* is globally asymptotically stable if R_01. Specially, if the basic reproductive number R_0=1, the computer numerical value simulation implies the disease free equilibrium E_0 is possiblly stable.