School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, 730070, Gansu, People's Republic of China.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People's Republic of China.
J Math Biol. 2023 Oct 5;87(5):66. doi: 10.1007/s00285-023-01999-1.
Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor [Formula: see text] within an infinite dimensional space [Formula: see text]. Then we define the basic reproduction number [Formula: see text] and prove that it determines the global dynamics of the model. If [Formula: see text], the global attractor [Formula: see text] reduces to the disease-free equilibrium state, indicating that the disease-free equilibrium state is globally asymptotically stable. When [Formula: see text], the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor [Formula: see text] for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle's Invariance Principle, we show that the global attractor [Formula: see text] is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of [Formula: see text] to the parameters and find the most sensitive parameters to [Formula: see text]. Finally, we present an improved strategy to achieve the WHO's goal of reducing the incidence of tuberculosis by 90% by 2035 compared to 2015.
由于结核病的潜伏期和治疗与康复之间存在时滞，存在异质性。在这项研究中，我们开发了一个考虑年龄相关潜伏期和治疗时间延迟的结核病模型，以描述结核病的传播。我们首先表明，模型的解半流是适定的，并且在无限维空间[Formula: see text]内具有全局吸引子[Formula: see text]。然后，我们定义基本繁殖数[Formula: see text]并证明它决定了模型的全局动力学。如果[Formula: see text]，全局吸引子[Formula: see text]减少到无病平衡点，表明无病平衡点是全局渐近稳定的。当[Formula: see text]时，模型生成的半流是一致持久的，并且这个一致持久的模型存在一个内部全局吸引子[Formula: see text]。通过构造合适的李雅普诺夫函数并应用拉塞尔不变原理，我们表明全局吸引子[Formula: see text]减少到地方病平衡点，这意味着地方病平衡点是全局渐近稳定的。基于中国 2007 年至 2020 年的结核病数据，我们模拟了所提出模型的参数和初始值。此外，我们计算了[Formula: see text]对参数的敏感性，并找到了对[Formula: see text]最敏感的参数。最后，我们提出了一种改进的策略，以实现世界卫生组织 2035 年比 2015 年将结核病发病率降低 90%的目标。
