This paper aims to model the transmission of tungiasis disease and assess the optimal control schemes to stop its occurrence. Based on the development stage of fleas and propagation process of diseases, we propose a human-flea model without control, in which the susceptible-infected in latent stage-infectious populations and the egg-larva-pupa-adult stage of fleas are all in involved. In the light of the Lyapunov function method, we prove global stability of equilibria. The model is extended by reformulating it as an optimal control problem, with the use of four time-dependent controls, to assess the impact of individual protection, treatment and two flea control strategies (killing adult fleas and reduction of eggs and larvae). By using Pontryagin’s maximum principle, we characterize the optimal control. Using the data of human and flea in Brazil and Nigeri, numerical simulations are performed. The numerical results show that enhancing the protection and treatment of people and increasing the killing efficacy of flea adults would contribute to prevent and control the spread of the disease appreciably.

中文翻译：

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软骨病传播的动力学与优化控制

本文旨在模拟穿衣病的传播，并评估阻止其发生的最佳控制方案。基于跳蚤的发育阶段和疾病的传播过程，我们提出了一个不受控制的人跳蚤模型，其中潜伏感染人群中的易感感染和跳蚤的卵-幼虫-蛹-成虫阶段都处于涉及。根据李雅普诺夫函数方法，我们证明了平衡的全局稳定性。该模型通过将其重新表述为最佳控制问题进行扩展，使用四个时间相关控制来评估个体保护、治疗和两种跳蚤控制策略（杀死成年跳蚤和减少卵和幼虫）的影响。通过使用庞特里亚金的极大原理，我们描述了最优控制。利用巴西和尼日利亚的人类和跳蚤数据，进行了数值模拟。数值结果表明，加强对人的保护和治疗，提高对成虫的杀灭效果，将有助于预防和控制该病的传播。