Optimal Control of an SIR Model with Delay in State and Control Variables

Mohamed Elhia,Elhabib Benlahmar,Mostafa Rachik

doi:10.1155/2013/403549

Abstract

We will investigate the optimal control strategy of an SIR epidemic model with time delay in state and control variables. We use a vaccination program to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals. Existence for the optimal control is established; Pontryagin’s maximum principle is used to characterize this optimal control, and the optimality system is solved by a discretization method based on the forward and backward difference approximations. The numerical simulation is carried out using data regarding the course of influenza A (H1N1) in Morocco. The obtained results confirm the performance of the optimization strategy.

Highlights

For a long time, infectious diseases have caused several epidemics, leaving behind them millions of dead and infected individuals and severe socioeconomic consequences
The population dynamics is given by the following system of ordinary differential equations subject to nonnegative initial conditions: dS (t) dt
The work in this paper contributes to a growing literature on applying optimal control techniques to epidemiology

Summary

Introduction

Infectious diseases have caused several epidemics, leaving behind them millions of dead and infected individuals and severe socioeconomic consequences. To reflect the real behavior of some diseases, many researchers have proposed and analyzed more realistic models including delays to model different mechanisms in the dynamics of epidemics like latent period, temporary immunity and length of infection (see, e.g., [4,5,6,7,8] and the references therein). There have been some works (like [9, 10]) which study an optimal control problem with time delay but only in the state variable. We will investigate the effect of a vaccination program in the case of an SIR (susceptible-infected-recovered) epidemic model with time delay in the control and the state variables.

Mathematical Model

The Optimal Control Problem

Numerical Simulation

Findings

Conclusion