Complex Systems Research Center, Shanxi University, Taiyuan, People's Republic of China.
Institute of Applied Mathematics, Army Engineering University, Shijiazhuang, People's Republic of China.
J Biol Dyn. 2019 Dec;13(1):675-705. doi: 10.1080/17513758.2019.1683628.
In this paper, a mathematical model describing tuberculosis transmission with fast and slow progression and age-dependent latency and infection is investigated. It is assumed in the model that infected individuals can develop tuberculosis by either of two pathogenic mechanisms: direct progression or endogenous reactivation. It is shown that the transmission dynamics of the disease is fully determined by the basic reproduction number. By analyzing corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By using the persistence theory for infinite dimensional system, it is proved that the system is uniformly persistent when the basic reproduction number is greater than unity. By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, it is verified that the global dynamics of the system is completely determined by the basic reproduction number.
本文研究了一个描述具有快速和缓慢进展以及年龄相关潜伏期和感染的结核病传播的数学模型。该模型假设感染个体可以通过两种致病机制之一发展为结核病：直接进展或内源性再激活。结果表明，疾病的传播动态完全由基本繁殖数决定。通过分析相应的特征方程，建立了模型无病平衡点和地方病平衡点的局部稳定性。利用无穷维系统的持久性理论，证明了当基本繁殖数大于 1 时，系统是一致持久的。通过构造合适的李雅普诺夫泛函并利用拉塞尔不变原理，验证了系统的全局动力学完全由基本繁殖数决定。
