Tuberculosis (TB) is a slowly progressive and chronic long-lasting communicable disease and several significant age-dependent heterogeneous factors are involved in TB transmission. To be specific, the individual susceptibility to TB varies with the ages during their life time, the current BCG vaccine effectiveness wanes since vaccination. And the rate of progression from latent TB infection to active TB, and the infectiousness of TB patients, the rate of treatment success for the patients and the risk of relapse for recovered individuals are also dependent on the sojourn time elapsed since the individuals enter into related compartmental classes. Based on the model framework developed by Li et al. (2022) a novel PDE TB model structured by ages of susceptibility, vaccination, latency, infection, treatment and relapse is formulated to evaluate the influences of the class-age structure on TB transmission. Global dynamical properties, including local stability, uniform persistence and global asymptotic stability, are systematically investigated and worked out by application of some new analytical techniques. Meanwhile, we develop an analytical method to determine kernel coefficients in Lyapunov functional. Our theoretical findings manifest that the age-structured PDE TB model remains global threshold stability characterized by the basic reproduction number. Some numerical simulations underscore the need for substantially improving vaccination rate for susceptible adolescents and adults population to effectively contain even end TB spread if new effective vaccines are available.

中文翻译：

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年龄结构结核病模型的全局渐近稳定性：确定李亚普诺夫函数核系数的分析方法

结核病 (TB) 是一种缓慢进展、慢性持久的传染病，结核病传播涉及多种重要的年龄依赖性异质因素。具体而言，个体对结核病的易感性随一生中年龄的变化而变化，目前的卡介苗疫苗接种后效果逐渐减弱。而从潜伏性结核感染进展为活动性结核的速度、结核病患者的传染性、患者的治疗成功率以及康复者的复发风险也取决于个体进入相关人群以来的逗留时间。隔室类。基于Li等人开发的模型框架。(2022) 制定了一种由易感年龄、疫苗接种、潜伏期、感染、治疗和复发构成的新型 PDE 结核病模型，以评估类别年龄结构对结核病传播的影响。通过应用一些新的分析技术，系统地研究和计算了全局动力学性质，包括局部稳定性、一致持久性和全局渐近稳定性。同时，我们开发了一种分析方法来确定李雅普诺夫泛函中的核系数。我们的理论研究结果表明，年龄结构的 PDE TB 模型仍然保持以基本再生数为特征的全局阈值稳定性。一些数值模拟强调，如果有新的有效疫苗可用，就需要大幅提高易感青少年和成年人的疫苗接种率，以有效遏制结核病的传播。