Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 189, Issue 2, Pages 408-436Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-021-01830-1
Keywords
Optimal control; SIR epidemic model; Herd immunity; Lockdown policy; Epidemic final size
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This article explores how to apply partial or total containment to the SIR epidemic model during a given finite time interval in order to minimize the epidemic final size. The existence and uniqueness of an optimal strategy are proven, with the best policy being to apply maximal social distancing effort until the end of the interval. Theoretical results and numerical simulations demonstrate a significant decrease in the epidemic final size.
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.
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