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.

Fractional optimal control problem for an age-structured model of COVID-19 transmission.

The aim of this study is to model the transmission of COVID-19 and investigate the impact of some control strategies on its spread. We propose an extension of the classical SEIR model, which takes into account the age structure and uses fractional-order derivatives to have a more realistic model. For each age group j the population is divided into seven classes namely susceptible S j , exposed E j , infected with high risk I h j , infected with low risk I l j , hospitalized H j , recovered with and without psychological complications R 1 j and R 2 j , respectively. In our model, we incorporate three control variables which represent: awareness campaigns, diagnosis and psychological follow-up. The purpose of our control strategies is protecting susceptible individuals from being infected, minimizing the number of infected individuals with high and low risk within a given age group j , as well as reducing the number of recovered individuals with psychological complications. Pontryagin's maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Numerical simulations performed using Matlab, are provided to show the effectiveness of three control strategies and the effect of the order of fractional derivative on the efficiency of these control strategies. Using a cost-effectiveness analysis method, our results show that combining awareness with diagnosis is the most effective strategy. To the best of our knowledge, this work is the first that propose a framework on the control of COVID-19 transmission based on a multi-age model with Caputo time-fractional derivative.

A novel control set-valued approach with application to epidemic models.

This work is considered in the framework of studies dedicated to the control problems, especially in epidemiology where the scientist are concerned to develop effective control strategies to minimize the number of infected individuals. In this paper, we set this problem as an asymptotic target control problem under mixed state-control constraints, for a general class of ordinary differential equations that model the temporal evolution of disease spread. The set of initial data, from which the number of infected people decrease to zero, is generated by a new type of Lyapunov functions defined in the sense of viability theory. The associated controls are provided via selections of adequately designed feedback map. The existence of such selections is improved by using Micheal selection theorem. Finally, an application to the SIRS epidemic model, with numerical simulations, is given to show the efficiency of our approach. To the best of our knowledge, our work is the first one that used a set-valued approach based on the viability theory to deal with an epidemic control problem.