Optimal one-round interventions for an SIR epidemic under two distinct time horizons

Jiaqi Yu; Yihong Li; Weiqiang Li; Jiaqi Yu; Yihong Li; Weiqiang Li

doi:10.3934/era.2026304

Electronic Research Archive

2026, Volume 34, Issue 10: 7027-7045. doi: 10.3934/era.2026304

Previous Article Next Article

Research article Special Issues

Optimal one-round interventions for an SIR epidemic under two distinct time horizons

School of Mathematics, North University of China, Taiyuan 030051, China

Received: 26 January 2026 Revised: 22 July 2026 Accepted: 11 August 2026 Published: 17 August 2026

Non-pharmaceutical interventions play a vital role in mitigating the outbreak of infectious diseases. During the COVID-19 pandemic, numerous countries and regions imposed multiple rounds of intervention measures to curb the spread of the disease. To get insights into optimal implementation of one-round interventions, two optimal control problems were examined under distinct time horizons, based on whether the herd immunity was reached. Here, one-round interventions were defined as interventions that operated under a budget constraint. Both of optimal control policies were proved to be purely bang-bang with a single switch, but their timings of onset were different. These results may provide some theoretical support for developing intervention strategies.
- non-pharmaceutical interventions,
- one-round interventions,
- SIR model,
- time horizon
Citation: Jiaqi Yu, Yihong Li, Weiqiang Li. Optimal one-round interventions for an SIR epidemic under two distinct time horizons[J]. Electronic Research Archive, 2026, 34(10): 7027-7045. doi: 10.3934/era.2026304

Related Papers:

Abstract

Non-pharmaceutical interventions play a vital role in mitigating the outbreak of infectious diseases. During the COVID-19 pandemic, numerous countries and regions imposed multiple rounds of intervention measures to curb the spread of the disease. To get insights into optimal implementation of one-round interventions, two optimal control problems were examined under distinct time horizons, based on whether the herd immunity was reached. Here, one-round interventions were defined as interventions that operated under a budget constraint. Both of optimal control policies were proved to be purely bang-bang with a single switch, but their timings of onset were different. These results may provide some theoretical support for developing intervention strategies.

References

[1]	L. Bolzoni, E. Bonacini, R. D. Marca, M. Groppi, Optimal control of epidemic size and duration with limited resources, Math. Biosci., 315 (2019), 108232. https://doi.org/10.1016/j.mbs.2019.108232 doi: 10.1016/j.mbs.2019.108232
[2]	L. Bolzoni, R. D. Marca, On the optimal vaccination control of SIR model with Erlang-distributed infectious period, J. Optim. Theory Appl., 205 (2025), 39. https://doi.org/10.1007/s10957-025-02645-0 doi: 10.1007/s10957-025-02645-0
[3]	S. Dong, L. Xu, Y. A, Z. Z. Lan, D. Xiao, B. Gao, Application of a time-delay SIR model with vaccination in COVID-19 prediction and its optimal control strategy, Nonlinear Dyn., 111 (2023), 10677–10692. https://doi.org/10.1007/s11071-023-08308-x doi: 10.1007/s11071-023-08308-x
[4]	L. M. Moschen, M. S. Aronna, Bang-bang optimal control of vaccination in metapopulation epidemics with linear cost structures, IEEE Control Syst. Lett., 9 (2025), 492–497. https://doi.org/10.1109/LCSYS.2025.3575227 doi: 10.1109/LCSYS.2025.3575227
[5]	F. Saldaña, Vaccination strategies in a pair formation model for human papillomavirus infection: an optimal control approach, J. Theor. Biol., 597 (2025), 111994. https://doi.org/10.1016/j.jtbi.2024.111994 doi: 10.1016/j.jtbi.2024.111994
[6]	M. D. Rossa, L. Freddi, D. Goreac, Optimality of vaccination for an SIR epidemic with an ICU constraint, J. Optim. Theory Appl., 204 (2025), 8. https://doi.org/10.1007/s10957-024-02598-w doi: 10.1007/s10957-024-02598-w
[7]	S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland, et al., Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe, Nature, 584 (2020), 257–261. https://doi.org/10.1038/s41586-020-2405-7 doi: 10.1038/s41586-020-2405-7
[8]	S. Lai, N. W. Ruktanonchai, L. Zhou, O. Prosper, W. Luo, J. R. Floyd, et al., Effect of non-pharmaceutical interventions to contain COVID-19 in China, Nature, 585 (2020), 410–413. https://doi.org/10.1038/s41586-020-2293-x doi: 10.1038/s41586-020-2293-x
[9]	P. Bliman, M. Duprez, Y. Privat, N. Vauchelet, Optimal immunity control and final size minimization by social distancing for the SIR epidemic model, J. Optim. Theory Appl., 189 (2021), 408–436. https://doi.org/10.1007/s10957-021-01830-1 doi: 10.1007/s10957-021-01830-1
[10]	D. I. Ketcheson, Optimal control of an SIR epidemic through finite-time non-pharmaceutical intervention, J. Math. Biol., 83 (2021), 7. https://doi.org/10.1007/s00285-021-01628-9 doi: 10.1007/s00285-021-01628-9
[11]	P. Bliman, M. Duprez, How best can finite-time social distancing reduce epidemic final size?, J. Theor. Biol., 511 (2021), 110557. https://doi.org/10.1016/j.jtbi.2020.110557 doi: 10.1016/j.jtbi.2020.110557
[12]	Y. Feng, G. Iyer, L. Li, Scheduling fixed length quarantines to minimize the total number of fatalities during an epidemic, J. Math. Biol., 82 (2021), 69. https://doi.org/10.1007/s00285-021-01615-0 doi: 10.1007/s00285-021-01615-0
[13]	P. Bliman, A. Rapaport, On the problem of minimizing the epidemic final size for SIR model by social distancing, IFAC-PapersOnLine, 56 (2023), 4049–4054. https://doi.org/10.1016/j.ifacol.2023.10.1726 doi: 10.1016/j.ifacol.2023.10.1726
[14]	R. Balderrama, M. I. Prieto, C. S. de la Vega, F. Vázquez, Optimal control for an SIR model with limited hospitalised patients, Math. Biosci., 378 (2024), 109317. https://doi.org/10.1016/j.mbs.2024.109317 doi: 10.1016/j.mbs.2024.109317
[15]	J. M. Greene, E. D. Sontag, Minimizing the infected peak utilizing a single lockdown: a technical result regarding equal peaks, in 2022 American Control Conference (ACC), IEEE, Atlanta, GA, USA, (2022), 3640–3647. https://doi.org/10.23919/ACC53348.2022.9867521
[16]	T. Britton, L. Leskelä, Optimal intervention strategies for minimizing total incidence during an epidemic, SIAM J. Appl. Math., 83 (2023), 354–373. https://doi.org/10.1137/22M1504433 doi: 10.1137/22M1504433
[17]	E. Molina, A. Rapaport, An optimal feedback control that minimizes the epidemic peak in the SIR model under a budget constraint, Automatica, 146 (2022), 110596. https://doi.org/10.1016/j.automatica.2022.110596 doi: 10.1016/j.automatica.2022.110596
[18]	M. T. Angulo, F. Castaños, R. Moreno-Morton, J. X. Velasco-Hernández, J. A. Moreno, A simple criterion to design optimal non-pharmaceutical interventions for mitigating epidemic outbreaks, J. R. Soc. Interface, 18 (2021), 20200803. https://doi.org/10.1098/rsif.2020.0803 doi: 10.1098/rsif.2020.0803
[19]	L. Miclo, D. Spiro, J. Weibull, Optimal epidemic suppression under an ICU constraint: an analytical solution, J. Math. Econ., 101 (2022), 102669. https://doi.org/10.1016/j.jmateco.2022.102669 doi: 10.1016/j.jmateco.2022.102669
[20]	F. Avram, L. Freddi, D. Goreac, Optimal control of a SIR epidemic with ICU constraints and target objectives, Appl. Math. Comput., 418 (2022), 126816. https://doi.org/10.1016/j.amc.2021.126816 doi: 10.1016/j.amc.2021.126816
[21]	D. Breda, M. D. Rossa, L. Freddi, Viability and control of a delayed SIR epidemic with an ICU state constraint, ESAIM: COCV, 31 (2025), 9. https://doi.org/10.1051/cocv/2024083 doi: 10.1051/cocv/2024083
[22]	F. D. Lauro, I. Z. Kiss, J. C. Miller, Optimal timing of one-shot interventions for epidemic control, PLoS Comput. Biol., 17 (2021), e1008763. https://doi.org/10.1371/journal.pcbi.1008763 doi: 10.1371/journal.pcbi.1008763
[23]	F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer London, (2013), 435–501. https://doi.org/10.1007/978-1-4471-4820-3
[24]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London, Ser. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[25]	M. Y. Li, An Introduction to Mathematical Modeling of Infectious Diseases, Springer Cham, (2018), 37–41. https://doi.org/10.1007/978-3-319-72122-4
[26]	S. P. Sethi, Optimal Control Theory: Applications to Management Science and Economics, Springer Cham, (2021), 119–121. https://doi.org/10.1007/978-3-030-91745-6
[27]	E. Hansen, T. Day, Optimal control of epidemics with limited resources, J. Math. Biol., 62 (2011), 423–451. https://doi.org/10.1007/s00285-010-0341-0 doi: 10.1007/s00285-010-0341-0

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)