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Optimal control problem of an SIR reaction–diffusion model with inequality constraints

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  • Jang, Junyoung
  • Kwon, Hee-Dae
  • Lee, Jeehyun

Abstract

This paper studies an optimal control problem of a susceptible–infected–recovered (SIR) reaction–diffusion model to derive an efficient vaccination strategy for influenza outbreaks. The control problem reflects realistic restrictions associated with limited total vaccination coverage and the maximum daily vaccine administration using state variable inequality constraints. We prove the existence of the optimal control solution and also investigate an optimality system by introducing a penalty function to deal with the constrained optimal control problem. A gradient-based algorithm is discussed to solve the optimality system. The spatial SIR model is solved by using the finite difference method (FDM) in time and the finite element method (FEM) in space. The results of numerical simulations show that the optimal vaccine strategy varies regionally according to the spreading rate of the disease.

Suggested Citation

  • Jang, Junyoung & Kwon, Hee-Dae & Lee, Jeehyun, 2020. "Optimal control problem of an SIR reaction–diffusion model with inequality constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 136-151.
    • Handle: RePEc:eee:matcom:v:171:y:2020:i:c:p:136-151
    DOI: 10.1016/j.matcom.2019.08.002
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    References listed on IDEAS

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    1. Wang, Yi & Cao, Jinde & Sun, Gui-Quan & Li, Jing, 2014. "Effect of time delay on pattern dynamics in a spatial epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 412(C), pages 137-148.
    2. David G. Luenberger & Yinyu Ye, 2008. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 0, number 978-0-387-74503-9, September.
    3. Lahrouz, A. & El Mahjour, H. & Settati, A. & Bernoussi, A., 2018. "Dynamics and optimal control of a non-linear epidemic model with relapse and cure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 299-317.
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    Cited by:

    1. Masoud Saade & Sebastian Aniţa & Vitaly Volpert, 2023. "Dynamics of Persistent Epidemic and Optimal Control of Vaccination," Mathematics, MDPI, vol. 11(17), pages 1-15, September.
    2. Turkyilmazoglu, Mustafa, 2022. "A restricted epidemic SIR model with elementary solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    3. Han, Lili & Song, Sha & Pan, Qiuhui & He, Mingfeng, 2023. "The impact of multiple population-wide testing and social distancing on the transmission of an infectious disease," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    4. Aníbal Coronel & Fernando Huancas & Esperanza Lozada & Marko Rojas-Medar, 2021. "The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System," Mathematics, MDPI, vol. 9(5), pages 1-17, February.
    5. Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    6. Pan, QiuHui & Song, Sha & He, MingFeng, 2021. "The effect of quarantine measures for close contacts on the transmission of emerging infectious diseases with infectivity in incubation period," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 574(C).

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