Feedback control problem of an SIR epidemic model based on the Hamilton-Jacobi-Bellman equation

Yoon-gu Hwang; Hee-Dae Kwon; Jeehyun Lee; Yoon-gu Hwang; Hee-Dae Kwon; Jeehyun Lee

doi:10.3934/mbe.2020121

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2284-2301. doi: 10.3934/mbe.2020121

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Research article

Feedback control problem of an SIR epidemic model based on the Hamilton-Jacobi-Bellman equation

1.
Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea
2.
Department of Mathematics, Inha University, 100 Inha-ro, Michuhol-gu, Incheon 22212, Republic of Korea
3.
Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea

Received: 10 November 2019 Accepted: 07 January 2020 Published: 16 January 2020

We consider a feedback control problem of a susceptible-infective-recovered (SIR) model to design an efficient vaccination strategy for influenza outbreaks. We formulate an optimal control problem that minimizes the number of people who become infected, as well as the costs of vaccination. A feedback methodology based on the Hamilton-Jacobi-Bellman (HJB) equation is introduced to derive the control function. We describe the viscosity solution, which is an approximation solution of the HJB equation. A successive approximation method combined with the upwind finite difference method is discussed to find the viscosity solution. The numerical simulations show that feedback control can help determine the vaccine policy for any combination of susceptible individuals and infectious individuals. We also verify that feedback control can immediately reflect changes in the number of susceptible and infectious individuals.
- SIR model,
- feedback control problem,
- Hamilton-Jacobi-Bellman (HJB) equation,
- upwind finite difference method
Citation: Yoon-gu Hwang, Hee-Dae Kwon, Jeehyun Lee. Feedback control problem of an SIR epidemic model based on the Hamilton-Jacobi-Bellman equation[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2284-2301. doi: 10.3934/mbe.2020121

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Abstract

We consider a feedback control problem of a susceptible-infective-recovered (SIR) model to design an efficient vaccination strategy for influenza outbreaks. We formulate an optimal control problem that minimizes the number of people who become infected, as well as the costs of vaccination. A feedback methodology based on the Hamilton-Jacobi-Bellman (HJB) equation is introduced to derive the control function. We describe the viscosity solution, which is an approximation solution of the HJB equation. A successive approximation method combined with the upwind finite difference method is discussed to find the viscosity solution. The numerical simulations show that feedback control can help determine the vaccine policy for any combination of susceptible individuals and infectious individuals. We also verify that feedback control can immediately reflect changes in the number of susceptible and infectious individuals.

References

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