IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v205y2025i2d10.1007_s10957-025-02645-0.html
   My bibliography  Save this article

On the Optimal Vaccination Control of SIR Model with Erlang-Distributed Infectious Period

Author

Listed:
  • Luca Bolzoni

    (Istituto Zooprofilattico Sperimentale della Lombardia e dell’Emilia Romagna)

  • Rossella Della Marca

    (SISSA – International School for Advanced Studies)

Abstract

In this work, we investigate optimal vaccination strategies in SIR models with Erlang-distributed infectious period (the Erlang distribution is a gamma distribution with integer shape parameter). We consider the problem of minimizing the epidemic burden under either limited vaccination resources or linear costs of vaccination. By applying the Pontryagin’s minimum principle, we prove that both the optimal control problems admit only bang–bang solutions with at most one switch, the one-switch control being the case of reactive vaccination. We show that different assumptions on the infectious period distribution may lead to qualitatively and quantitatively different shapes of the optimal control. The main results are the following. We prove that in the absence of control the epidemic burden in SIR models is the same irrespectively of the infectious period distribution. However, when the vaccination resources are limiting, the optimal control problem leads to solutions where the epidemic burden is larger (and the time needed to complete vaccination campaigns is longer) in models with Erlang-distributed infectious periods with respect to exponential ones.

Suggested Citation

  • Luca Bolzoni & Rossella Della Marca, 2025. "On the Optimal Vaccination Control of SIR Model with Erlang-Distributed Infectious Period," Journal of Optimization Theory and Applications, Springer, vol. 205(2), pages 1-31, May.
    • Handle: RePEc:spr:joptap:v:205:y:2025:i:2:d:10.1007_s10957-025-02645-0
    DOI: 10.1007/s10957-025-02645-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02645-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-025-02645-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Oluwaseun Sharomi & Tufail Malik, 2017. "Optimal control in epidemiology," Annals of Operations Research, Springer, vol. 251(1), pages 55-71, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michela Baccini & Giulia Cereda & Cecilia Viscardi, 2021. "The first wave of the SARS-CoV-2 epidemic in Tuscany (Italy): A SI2R2D compartmental model with uncertainty evaluation," PLOS ONE, Public Library of Science, vol. 16(4), pages 1-23, April.
    2. Imane Abouelkheir & Fadwa El Kihal & Mostafa Rachik & Ilias Elmouki, 2019. "Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity," Mathematics, MDPI, vol. 7(5), pages 1-21, May.
    3. Ozgur M. Araz & Mayteé Cruz-Aponte & Fernando A. Wilson & Brock W. Hanisch & Ruth S. Margalit, 2022. "An Analytic Framework for Effective Public Health Program Design Using Correctional Facilities," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 113-128, January.
    4. Kılıç, Hakan & Güneş, Evrim Didem, 2024. "Patient adherence in healthcare operations: A narrative review," Socio-Economic Planning Sciences, Elsevier, vol. 91(C).
    5. Kumar, Arjun & Dubey, Uma S. & Dubey, Balram, 2024. "The impact of social media advertisements and treatments on the dynamics of infectious diseases with optimal control strategies," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 50-86.
    6. Mustafa Akan, 2019. "Optimal Control Theoretic Approach To Investment In Doctors," Copernican Journal of Finance & Accounting, Uniwersytet Mikolaja Kopernika, vol. 8(4), pages 91-111.
    7. Zhang, Jingwen & Wang, Xinwei & Rong, Lili & Pan, Qiuwei & Bao, Chunbing & Zheng, Qinyue, 2024. "Planning for the optimal vaccination sequence in the context of a population-stratified model," Socio-Economic Planning Sciences, Elsevier, vol. 92(C).
    8. Sharbayta, Sileshi Sintayehu & Buonomo, Bruno & d'Onofrio, Alberto & Abdi, Tadesse, 2022. "‘Period doubling’ induced by optimal control in a behavioral SIR epidemic model," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    9. Juliano Marçal Lopes & Coralys Colon Morales & Michelle Alvarado & Vidal Augusto Z. C. Melo & Leonardo Batista Paiva & Eduardo Mario Dias & Panos M. Pardalos, 2022. "Optimization methods for large-scale vaccine supply chains: a rapid review," Annals of Operations Research, Springer, vol. 316(1), pages 699-721, September.
    10. Mira Baude & Alf Kimms, 2024. "Modelling pandemic behavior with a network-SIRD approach," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 32(2), pages 521-541, June.
    11. László Á. Kóczy, 2022. "Core-stability over networks with widespread externalities," Annals of Operations Research, Springer, vol. 318(2), pages 1001-1027, November.
    12. Fouad El Ouardighi & Eugene Khmelnitsky & Suresh P. Sethi, 2022. "Epidemic control with endogenous treatment capability under popular discontent and social fatigue," Production and Operations Management, Production and Operations Management Society, vol. 31(4), pages 1734-1752, April.
    13. 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).
    14. Chohan, Usman W., 2022. "Analyzing sound COVID-19 policy responses in developing countries: the case study of Pakistan," Studia z Polityki Publicznej / Public Policy Studies, Warsaw School of Economics, vol. 9(2), pages 1-22, August.
    15. Singh, Rajdeep & Wrzaczek, Stefan & Freiberger, Michael, 2025. "The inoculation dilemma: Partial vs Full immunization during the early rollout in a pandemic," Omega, Elsevier, vol. 132(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:205:y:2025:i:2:d:10.1007_s10957-025-02645-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.