预防接种情况下非线性饱和接触率SIR流行病模型动力学性态研究

研究了一类预防接种情况下具有一般非线性饱和接触率SIR流行病模型动力学性态。得到决定疾病灭绝和持续生存的基本再生数。当基本再生数小于等于1时，仅存在无病平衡态：当基本再生数大于1时，除存在无病平衡态外，还存在惟一的地方病平衡态。利用Hurwitz判据、Liapunov-Lasalle不变集原理得到各个平衡态局部渐近稳定及无病平衡态全局渐近稳定的条件。特别地。当传染率为双线性时，无病平衡态及地方病平衡态全局渐近稳定。

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.