This work is considered in the framework of studies dedicated to the control problems, especially in epidemiology where the scientist are concerned to develop effective control strategies to minimize the number of infected individuals. In this paper, we set this problem as an asymptotic target control problem under mixed state-control constraints, for a general class of ordinary differential equations that model the temporal evolution of disease spread. The set of initial data, from which the number of infected people decrease to zero, is generated by a new type of Lyapunov functions defined in the sense of viability theory. The associated controls are provided via selections of adequately designed feedback map. The existence of such selections is improved by using Micheal selection theorem. Finally, an application to the SIRS epidemic model, with numerical simulations, is given to show the efficiency of our approach. To the best of our knowledge, our work is the first one that used a set-valued approach based on the viability theory to deal with an epidemic control problem.

这项工作是在致力于控制问题的研究框架内考虑的，特别是在流行病学领域，科学家们关注制定有效的控制策略以尽量减少感染个体的数量。在本文中，我们将此问题设置为混合状态控制约束下的渐近目标控制问题，用于模拟疾病传播的时间演化的一般常微分方程。感染人数减少到零的初始数据集是由生存力理论定义的新型李雅普诺夫函数生成的。通过选择适当设计的反馈图来提供相关的控制。使用迈克尔选择定理改进了这种选择的存在性。最后，应用到 SIRS 流行病模型，通过数值模拟，给出了我们方法的效率。据我们所知，我们的工作是第一个使用基于可行性理论的集值方法来处理流行病控制问题的工作。