The Study on Dynamic Behavior of SIR Epidemiological Model with Nonlinear Saturated Contact Rate Under Vaccination

The dynamic behavior of a kind of SIR epidemiological model with general nonlinear saturated contact rate is considered under vaccination. The basic reproductive number is found which determines the existence of the infection. When it is equal to or smaller than 1, there only exists disease free equilibrium, otherwise, two equilibria,the endemic equilibrium and the disease free equilibrium exist. By Hurwitz criterion and Liapunov-Lasalleinvariant theorem, the locally asymptotical stability of disease free equilibrium and the endemic equilibrium is proved and the condition under which the disease free equilibrium is globally asymptotically stable is found. Specially, if the infective rate is bilinear,both of the equilibria are globally asymptotically stable.