In this paper, we propose a fractional-order and two-patch model of tuberculosis (TB) epidemic, in which susceptible, slow latent, fast latent and infectious individuals can travel freely between the patches, but not under treatment infected individuals, due to medical reasons. We obtain the basic reproduction number [Formula: see text] for the model and extend the classical LaSalle’s invariance principle for fractional differential equations. We show that if [Formula: see text], the disease-free equilibrium (DFE) is locally and globally asymptotically stable. If [Formula: see text] we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We extend the model by inclusion the time-dependent controls (effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches), and formulate a fractional optimal control problem to reduce the spread of the disease. The numerical results show that the use of all controls has the most impact on disease control, and decreases the size of all infected compartments, but increases the size of susceptible compartment in both patches. We, also, investigate the impact of the fractional derivative order [Formula: see text] on the values of the controls ([Formula: see text]). The results show that the maximum levels of effective treatment controls in both patches increase when [Formula: see text] is reduced from 1, while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when [Formula: see text] limits to 1.

中文翻译：

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使用分数阶导数的两补丁结核病流行模型的全局稳定性和最优控制

在本文中，我们提出了结核病 (TB) 流行的分数阶和两斑块模型，其中易感、慢潜伏、快速潜伏和传染性个体可以在斑块之间自由移动，但不能在治疗中的受感染个体，由于医疗原因。我们获得了模型的基本再生数[公式：见正文]，并扩展了分数微分方程的经典拉萨尔不变原理。我们证明，如果 [公式：见正文]，无病平衡 (DFE) 是局部和全局渐近稳定的。如果 [公式：见正文] 我们获得了充分条件，在该条件下，地方性平衡是唯一的并且全局渐近稳定。我们通过包含时间相关控制（补丁中的有效治疗控制和补丁之间传染性个体旅行的筛查控制）来扩展模型，并制定分数最优控制问题以减少疾病的传播。数值结果表明，所有对照的使用对疾病控制的影响最大，并且减小了所有受感染区室的大小，但增加了两个斑块中易感区室的大小。我们还研究了分数导数阶 [公式：参见文本] 对控件值（[公式：参见文本]）的影响。结果表明，当 [公式：见正文] 从 1 减少时，两个补丁中有效治疗控制的最大水平增加，