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Optimality of Vaccination for an SIR Epidemic with an ICU Constraint

Author

Listed:
  • Matteo Della Rossa

    (Università di Udine)

  • Lorenzo Freddi

    (Università di Udine)

  • Dan Goreac

    (Université Laval
    Shandong University, Weihai
    LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS)

Abstract

This paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the intensive care unit (ICU) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affecting the epidemic dynamic. After a preliminary viability analysis, the existence of optimal controls is established, and their structure is characterized by using a state-constrained version of Pontryagin’s theorem. The resulting optimal controls necessarily have a bang-bang regime with at most one switch. More precisely, the optimal strategies impose the maximum-allowed vaccination effort in an initial period of time, which can cease only once the ICU constraint can be satisfied without further vaccination. The switching times are characterized in order to identify conditions under which vaccination should be implemented or halted. The uniqueness of the optimal control is also discussed. Numerical examples illustrate our theoretical results and the corresponding optimal strategies. The analysis is eventually extended to the infinite horizon by $$\Gamma $$ Γ -convergence arguments.

Suggested Citation

  • Matteo Della Rossa & Lorenzo Freddi & Dan Goreac, 2025. "Optimality of Vaccination for an SIR Epidemic with an ICU Constraint," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-35, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02598-w
    DOI: 10.1007/s10957-024-02598-w
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    References listed on IDEAS

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    1. J. Frédéric Bonnans & Constanza Vega & Xavier Dupuis, 2013. "First- and Second-Order Optimality Conditions for Optimal Control Problems of State Constrained Integral Equations," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 1-40, October.
    2. Miclo, Laurent & Spiro, Daniel & Weibull, Jörgen, 2022. "Optimal epidemic suppression under an ICU constraint: An analytical solution," Journal of Mathematical Economics, Elsevier, vol. 101(C).
    3. Avram, Florin & Freddi, Lorenzo & Goreac, Dan, 2022. "Optimal control of a SIR epidemic with ICU constraints and target objectives," Applied Mathematics and Computation, Elsevier, vol. 418(C).
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