## 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.

Original language | English |
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Pages (from-to) | 136-151 |

Number of pages | 16 |

Journal | Mathematics and Computers in Simulation |

Volume | 171 |

DOIs | |

Publication status | Published - 2020 May |

### Bibliographical note

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

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics