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

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 development of a vaccination law. Two approaches are used. The first one is based on a discretization 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.