The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this infinite-horizon problem, and a full characterization of the solution is provided. The best policy consists in applying the maximal allowed social distancing effort until the end of the interval, starting at a date that is not always the closest date and may be found by a simple algorithm. Both theoretical results and numerical simulations demonstrate that it leads to a significant decrease in the epidemic final size. We show that in any case the optimal intervention has to begin before the number of susceptible cases has crossed the herd immunity level, and that its limit is always smaller than this threshold. This problem is also shown to be equivalent to the minimum containment time necessary to stop at a given distance after this threshold value.

本文的目的是了解如何在给定的有限时间间隔内对 SIR 流行病模型应用部分或全部遏制，以最小化流行病的最终规模，即在整个流行病过程中的累计感染病例数。证明了该无限范围问题的最优策略的存在性和唯一性，并提供了解决方案的完整表征。最好的策略是在间隔结束之前应用最大允许的社交距离努力，从一个并不总是最接近的日期开始，并且可以通过简单的算法找到。理论结果和数值模拟都表明，它导致流行病最终规模显着减小。我们表明，无论如何，最佳干预必须在易感病例数量超过群体免疫水平之前开始，并且其限制始终小于该阈值。这个问题也被证明等同于在该阈值之后在给定距离处停止所需的最小遏制时间。