In this paper, we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals, in which only susceptible individuals can travel freely between the patches. The model has multiple equilibria. We determine conditions that lead to the appearance of a backward bifurcation. The results show that the TB model can have exogenous reinfection among the treated individuals and, at the same time, does not exhibit backward bifurcation. Also, conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained. In case without reinfection, the model has four equilibria. In this case, the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations (FDEs). Numerical simulations confirm the validity of the theoretical results.

中文翻译：

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不完全治疗结核病流行的分数阶和两块模型的后向分叉

在本文中，我们考虑了一个治疗不完全的结核病模型，并将该模型扩展为 Caputo 分数阶和两补丁版本，在治疗个体之间存在外源性再感染，其中只有易感个体可以在补丁之间自由移动。该模型具有多重均衡。我们确定导致后向分叉出现的条件。结果表明，结核病模型可以在治疗个体中发生外源性再感染，同时不表现出后向分叉。此外，获得了导致无病平衡的全局渐近稳定性的条件。在没有再感染的情况下，该模型有四个平衡点。在这种情况下，使用 Lyapunov 函数理论和分数微分方程 (FDE) 的 LaSalle 不变原理建立了平衡的全局渐近稳定性。数值模拟证实了理论结果的有效性。