一类肺结核传染病模型的分析

Analysis of a Tuberculosis Epidemic Model

建立了一类4-维肺结核传染病模型。首先,根据第二代矩阵理论我们得到了系统的基本再生数。其次,利用Lyapunov方法证明了当R0≤1时,系统的无病平衡点是全局稳定的。然后,利用Routh-Hurwitz判据证明了当R0>1时系统的地方病平衡点是局部稳定的。此外,在不考虑因病死亡的情况下,利用广义的Bendixson-Dulac定理,证明了地方病平衡点是全局渐近稳定的。最后,我们对数学结果做了生物解释。

In this paper,a 4- dimension tuberculosis epidemic model was constructed. Firstly,according to the next generation theory,the basic reproduction number were obtained. Secondly,by using of Lyapunov method,we proved that the disease- free equilibrium of the system is globally stable when R 0≤1. Thirdly,by using of Routh- Hurwitz criterion,we got that the endemic equilibrium of the system is locally stable when R 0 1. In addition,without considering the disease- related death rate,we also got the global stability of the endemic equilibrium by using Bendixson- Dulac theorem. Finally,we presented some biological interpretation for the mathematical results.