A new mathematical model of tuberculosis (TB) featuring exogenous re‐infection and incomplete treatment is presented and analyzed. The model divides total population into susceptible, latently infected, actively infected (uninformed and enlightened), and treatment classes. The model with or without incomplete treatment exhibits backward bifurcation phenomenon, which is caused by the presence of exogenous re‐infection. However, further investigation reveals that the absence of incomplete treatment has the potential to reduce the backward bifurcation range significantly. The global dynamics of the TB model without exogenous re‐infection is completely determined by the basic reproduction number under certain modifications on parameters. Furthermore, the model is extended to include three time‐dependent control functions, such as public awareness campaign, treatment effort, and preventive control against incomplete treatment. The existence of the optimal control for the nonautonomous model is proven and the three controls are characterized using Pontryagin's maximum principle. Numerical simulations are performed to show the significance of singular implementation of each of the controls and combination of the three controls in minimizing the TB burden in the population.

中文翻译：

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外源性再感染和治疗不完全的结核病模型的最优控制分析

提出并分析了一种具有外源性再感染和不完全治疗的新型结核病数学模型。该模型将总人口分为易感人群，潜伏感染人群，主动感染人群（不知情和开明）和治疗类别。不论是否进行不完全处理的模型都会出现后向分叉现象，这是由于存在外源性再感染引起的。但是，进一步的研究表明，不完整的治疗可能会显着减小后向分叉范围。没有参数外源性再感染的结核病模型的整体动力学完全取决于在参数上进行某些修改后的基本繁殖数。此外，该模型已扩展为包括三个与时间有关的控制功能，例如公众意识运动，治疗工作以及对不完全治疗的预防控制。证明了非自治模型最优控制的存在，并使用庞特里亚金的最大原理对三个控制进行了表征。进行了数值模拟，显示了每个控制措施的单一实施以及三个控制措施的组合在最大程度减少人群结核病负担方面的重要性。