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Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model

Author

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  • Pierre-Alexandre Bliman

    (Inria, Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, équipe Mamba)

  • Michel Duprez

    (Inria, équipe MIMESiS, Université de Strasbourg, ICUBE, équipe MLMS)

  • Yannick Privat

    (Université de Strasbourg, CNRS UMR 7501, INRIA, Institut de Recherche Mathématique Avancée (IRMA))

  • Nicolas Vauchelet

    (LAGA, UMR 7539, CNRS, Université Sorbonne Paris Nord)

Abstract

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.

Suggested Citation

  • Pierre-Alexandre Bliman & Michel Duprez & Yannick Privat & Nicolas Vauchelet, 2021. "Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 408-436, May.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:2:d:10.1007_s10957-021-01830-1
    DOI: 10.1007/s10957-021-01830-1
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    References listed on IDEAS

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    1. T Déirdre Hollingsworth & Don Klinkenberg & Hans Heesterbeek & Roy M Anderson, 2011. "Mitigation Strategies for Pandemic Influenza A: Balancing Conflicting Policy Objectives," PLOS Computational Biology, Public Library of Science, vol. 7(2), pages 1-11, February.
    2. Facundo Piguillem & Liyan Shi, 2022. "Optimal Covid-19 Quarantine and Testing Policies," The Economic Journal, Royal Economic Society, vol. 132(647), pages 2534-2562.
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    Cited by:

    1. Michael Freiberger & Dieter Grass & Michael Kuhn & Andrea Seidl & Stefan Wrzaczek, 2022. "Chasing up and locking down the virus: Optimal pandemic interventions within a network," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 24(5), pages 1182-1217, October.
    2. Cacace, Simone & Oliviero, Alessio, 2024. "Reliable optimal controls for SEIR models in epidemiology," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 523-542.
    3. Choudhury, Nishat Alam & Ramkumar, M. & Schoenherr, Tobias & Singh, Shalabh, 2023. "The role of operations and supply chain management during epidemics and pandemics: Potential and future research opportunities," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 175(C).
    4. Huberts, Nick F.D. & Thijssen, Jacco J.J., 2023. "Optimal timing of non-pharmaceutical interventions during an epidemic," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1366-1389.

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