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Optimal vaccination in a SIRS epidemic model

Author

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  • Salvatore Federico

    (Università di Genova)

  • Giorgio Ferrari

    (Bielefeld University)

  • Maria-Laura Torrente

    (Università di Genova)

Abstract

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton–Jacobi–Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.

Suggested Citation

  • Salvatore Federico & Giorgio Ferrari & Maria-Laura Torrente, 2024. "Optimal vaccination in a SIRS epidemic model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(1), pages 49-74, February.
    • Handle: RePEc:spr:joecth:v:77:y:2024:i:1:d:10.1007_s00199-022-01475-9
    DOI: 10.1007/s00199-022-01475-9
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    References listed on IDEAS

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    1. Garriga, Carlos & Manuelli, Rody & Sanghi, Siddhartha, 2022. "Optimal management of an epidemic: Lockdown, vaccine and value of life," Journal of Economic Dynamics and Control, Elsevier, vol. 140(C).
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    4. Glover, Andrew & Heathcote, Jonathan & Krueger, Dirk, 2022. "Optimal age-Based vaccination and economic mitigation policies for the second phase of the covid-19 pandemic," Journal of Economic Dynamics and Control, Elsevier, vol. 140(C).
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    6. Federico, Salvatore & Ferrari, Giorgio, 2021. "Taming the spread of an epidemic by lockdown policies," Journal of Mathematical Economics, Elsevier, vol. 93(C).
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    Cited by:

    1. Raouf Boucekkine & Ted Loch-Temzelides, 2024. "Introduction to the special issue on mathematical economic epidemiology models," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(1), pages 1-7, February.
    2. Alessandro Ramponi & Maria Elisabetta Tessitore, 2024. "Optimal Social and Vaccination Control in the SVIR Epidemic Model," Mathematics, MDPI, vol. 12(7), pages 1-17, March.
    3. 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.

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    More about this item

    Keywords

    SIRS model; Optimal control; Viscosity solution; Non-smooth verification theorem; Epidemic; Optimal vaccination;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • I12 - Health, Education, and Welfare - - Health - - - Health Behavior
    • I18 - Health, Education, and Welfare - - Health - - - Government Policy; Regulation; Public Health

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