Bliman Pierre-Alexandre, Duprez Michel, Privat Yannick, Vauchelet Nicolas
Inria, Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, équipe Mamba, Paris, France.
Inria, équipe MIMESiS, Université de Strasbourg, ICUBE, équipe MLMS, Strasbourg, France.
J Optim Theory Appl. 2021;189(2):408-436. doi: 10.1007/s10957-021-01830-1. Epub 2021 Mar 1.
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传染病模型应用部分或完全遏制措施，以最小化疫情最终规模，即疫情全过程中累计感染病例数。针对这个无限期问题，证明了最优策略的存在性和唯一性，并给出了解的完整特征。最佳策略是从一个并非总是最近日期且可通过简单算法找到的日期开始，在区间结束前一直应用最大允许的社交距离措施。理论结果和数值模拟均表明，这会导致疫情最终规模显著下降。我们表明，在任何情况下，最优干预都必须在易感病例数超过群体免疫水平之前开始，且其极限始终小于该阈值。这个问题还被证明等同于在超过该阈值后在给定距离处停止所需的最短遏制时间。
