In this paper, mathematical analysis is carried out for a mathematical model of Tuberculosis (TB) with age-dependent latency and active infection. The model divides latent TB infection into two stages: an early stage of high risk of developing active TB and a late stage of lower risk for developing active TB. Infected persons initially progress through the early latent TB stage and then can either progress to active TB infection or progress to late latent TB infection. The model is formulated by incorporating the duration that an individual has spent in the stages of the early latent TB, the late latent TB and the active TB infection as variables. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, it is shown that the global dynamics of the disease is completely determined by the basic reproduction number: if the basic reproduction number is less than unity, the TB always dies out; if the basic reproduction number is greater than unity, a unique endemic steady state exists and is globally asymptotically stable in the interior of the feasible region and therefore the TB becomes endemic. Numerical simulations are carried out to illustrate the theoretical results.

中文翻译：

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具有年龄相关潜伏期和活动性感染的结核病数学模型的全球动态

在本文中，对具有年龄依赖性潜伏期和活动性感染的结核病（TB）的数学模型进行了数学分析。该模型将潜伏性结核感染分为两个阶段：发展为活动性结核病的高风险早期阶段和发展为活动性结核病风险较低的晚期阶段。感染者最初会发展为早期潜伏性结核病阶段，然后可以发展为活动性结核病感染或发展为潜伏性结核病感染。该模型是通过将个体在早期潜伏性结核病、晚期潜伏性结核病和活动性结核病感染阶段所花费的持续时间作为变量来制定的。通过构建合适的 Lyapunov 泛函并利用 LaSalle 不变原理，表明疾病的全局动态完全由基本再生数决定：如果基本再生数小于 1，则 TB 总是消失；如果基本再生数大于 1，则存在唯一的地方性稳定状态，并且在可行区域内部是全局渐近稳定的，因此 TB 成为地方性。进行了数值模拟来说明理论结果。