We consider the problem of controlling an SIR-model epidemic by temporarily reducing the rate of contact within a population. The control takes the form of a multiplicative reduction in the contact rate of infectious individuals. The control is allowed to be applied only over a finite time interval, while the objective is to minimize the total number of individuals infected in the long-time limit, subject to some cost function for the control. We first consider the no-cost scenario and analytically determine the optimal control and solution. We then study solutions when a cost of intervention is included, as well as a cost associated with overwhelming the available medical resources. Examples are studied through the numerical solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we provide some examples related directly to the current pandemic.

我们考虑通过暂时降低人群中的接触率来控制 SIR 模型流行病的问题。控制采取成倍减少感染个体接触率的形式。该控制只允许在有限的时间间隔内应用，而目标是在长期限制内最小化被感染的个体总数，这取决于控制的一些成本函数。我们首先考虑无成本情景并分析确定最优控制和解决方案。然后，当包括干预成本以及与压倒可用医疗资源相关的成本时，我们研究解决方案。通过相关的 Hamilton-Jacobi-Bellman 方程的数值解来研究示例。最后，