In this paper, we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection, in which the treatment is effective for a number of infectious individuals and it fails for some other infectious individuals who are being treated. We show that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity. Also, it is shown that under certain conditions the model cannot exhibit backward bifurcation. Furthermore, it is shown in the absence of re-infection, the backward bifurcation phenomenon does not exist, in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. The global asymptotic stability of the endemic equilibrium, when the associated reproduction number is greater than unity, is established using the geometric approach. Numerical simulations are presented to illustrate our main results.

中文翻译：

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两种治疗外源性再感染结核病流行模型的数学分析

在本文中，我们提出了一种结核病的数学模型，即两种治疗和外源性再感染，其中治疗对一些感染者有效，而对其他一些正在接受治疗的感染者则无效。我们表明，该模型表现出后向分叉现象，当相关的基本繁殖数小于 1 时，稳定的无病平衡与稳定的地方性平衡共存。此外，它表明在某些条件下，模型不能表现出向后分叉。此外，表明在没有再感染的情况下，不存在后向分叉现象，其中当相关再生数小于1时，模型的无病平衡是全局渐近稳定的。当相关的繁殖数大于一时，地方性平衡的全局渐近稳定性是使用几何方法建立的。数值模拟用于说明我们的主要结果。