Set-Valued Control to COVID-19 Spread with Treatment and Limitation of Vaccination Resources

In this paper, we consider an SEIR model that describes the dynamics of the COVID-19 pandemic. Subject to this model with vaccination and treatment as controls, we formulate a control problem that aims to reduce the number of infectious individuals to zero. The novelty of this work consists of considering a more realistic control problem by adding mixed constraints to take into account the limited vaccines supply. Furthermore, to solve this problem, we use a set-valued approach combining a Lyapunov function defined in the sense of viability theory with some results from the set-valued analysis. The expressions of the control variables are given via continuous selection of an adequately designed feedback map. The main result of our study shows that even though there are limits of vaccination resources, the combination of treatment and vaccination strategies can significantly reduce the number of exposed and infectious individuals. Some numerical simulations are proposed to show the efficiency of our set-valued approach and to validate our theoretical results.