State feedback control law design for an age-dependent SIR model

TL;DR

This paper designs a state-feedback vaccination control law for an age-dependent SIR model, ensuring disease eradication by analyzing stability and positivity through linearization and semigroup theory, supported by numerical simulations.

Contribution

It introduces two novel feedback control laws for an age-structured SIR model, extending existing methods to include age-dependent dynamics and stability analysis.

Findings

The disease-free equilibrium is unstable if the basic reproduction number exceeds 1.

The proposed feedback laws guarantee stability and positivity of the controlled system.

Numerical simulations confirm the effectiveness of the control strategies.

Abstract

An age-dependent SIR model is considered with the aim to develop a state-feedback vaccination law in order to eradicate a disease. A dynamical analysis of the system is performed using the principle of linearized stability and shows that, if the basic reproduction number is larger than 1, the disease free equilibrium is unstable. This result justifies the developement of a vaccination law. Two approaches are used. The first one is based on a dicretization of the partial integro-differential equations (PIDE) model according to the age. In this case a linearizing feedback law is found using Isidori's theory. Conditions guaranteeing stability and positivity are established. The second approach yields a linearizing feedback law developed for the PIDE model. This law is deduced from the one obtained for the ODE case. Using semigroup theory, stability conditions are also obtained. Finally, numerical simulations are presented to reinforce the theoretical arguments.