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

Junyoung Jang, Hee Dae Kwon, Jeehyun Lee

Research output: Contribution to journalArticle

### 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 Mathematics and Computers in Simulation https://doi.org/10.1016/j.matcom.2019.08.002 Accepted/In press - 2019 Jan 1

### Fingerprint

Optimality System
Reaction-diffusion Model
Vaccination
Vaccine
Inequality Constraints
Optimal Control Problem
Constrained Control
Vaccines
Influenza
Spatial Model
Penalty Function
Difference Method
Control Problem
Finite Difference
Optimal Control
Coverage
Finite Element Method
Vary
Restriction

### All Science Journal Classification (ASJC) codes

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

### Cite this

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title = "Optimal control problem of an SIR reaction–diffusion model with inequality constraints",
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.",
author = "Junyoung Jang and Kwon, {Hee Dae} and Jeehyun Lee",
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Optimal control problem of an SIR reaction–diffusion model with inequality constraints. / Jang, Junyoung; Kwon, Hee Dae; Lee, Jeehyun.

In: Mathematics and Computers in Simulation, 01.01.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Optimal control problem of an SIR reaction–diffusion model with inequality constraints

AU - Jang, Junyoung

AU - Kwon, Hee Dae

AU - Lee, Jeehyun

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Y1 - 2019/1/1

N2 - 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.

AB - 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.

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