具有接种与治疗的肺结核模型稳定性分析

The stability analysis for a tuberculosis model with vaccination and treatment

建立具有接种和不完全治疗的肺结核模型,讨论其平衡点的存在性和稳定性,定义了模型的基本再生数,得到了疾病绝灭与否的阈值R0,并通过运用LaSalle不变原理和构造Lyapunov函数,证明了无病平衡点和地方病平衡点的全局稳定性,即当R0≤1时,该肺结核模型仅存在无病平衡点E0,且E0全局渐近稳定;当R0>1时,该模型存在无病平衡点E0和地方病平衡点E*,且E*是全局渐近稳定的.

A tuberculosis model is established with vaccination and incomplete treatment.The existence and stability of the model equilibrium point are discussed,and the threshold value R_0 is obtained,which determines the disease extinction or not.By LaSalle′invariant principle and Lyapunov functions,the global stability of disease-free equilibrium and epidemic equilibrium are proved.For this TB model,it is proved that the unique disease-free equilibrium E_0 is global asymptotic stability if R_0≤1;the tuberculosis model has disease-free equilibrium E_0 and endemic equilibrium E＊,and the endemic equilibrium E＊is global asymptotic stability if R_01.