## 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.

Original language | English |
---|---|

Pages (from-to) | 2284-2301 |

Number of pages | 18 |

Journal | Mathematical Biosciences and Engineering |

Volume | 17 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 Jan 16 |

### Bibliographical note

Funding Information:The work of Hee-Dae Kwon was supported by NRF-2016R1D1A1B04931897 and the NST grant by the Korea government (MSIP) (No. CRC-16-01-KRICT). The work of Jeehyun Lee was supported by NRF-2015R1A5A1009350 and NRF-2016R1A2B4014178.

Publisher Copyright:

© 2020 Y. Hwang, H. Kwon, and J. Lee, 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)

## All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Agricultural and Biological Sciences(all)
- Computational Mathematics
- Applied Mathematics