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.

In this paper, we formulate an optimal control problem based on a tuberculosis model with multiple infectious compartments and time delays. In order to have a more realistic model that allows highlighting the role of detection, loss to follow-up and treatment in TB transmission, we propose an extension of the classical SEIR model by dividing infectious patients in the compartment (I) into three categories: undiagnosed infected (I), diagnosed patients who are under treatment (T) and diagnosed patients who are lost to follow-up (L). We incorporate in our model delays representing the incubation period and the time needed for treatment. We also introduce three control variables in our delayed system which represent prevention, detection and the efforts that prevent the failure of treatment. The purpose of our control strategies is to minimize the number of infected individuals and the cost of intervention. The existence of the optimal controls is investigated, and a characterization of the three controls is given using the Pontryagin's maximum principle with delays. To solve numerically the optimality system with delays, we present an adapted iterative method based on the iterative Forward-Backward Sweep Method (FBSM). Numerical simulations performed using Matlab are also provided. They indicate that the prevention control is the most effective one. To the best of our knowledge, it is the first work to apply optimal control theory to a TB model which considers infectious patients diagnosis, loss to follow-up phenomenon and multiple time delays.

The purpose of this paper is modelling and controlling the spread of COVID-19 disease in Morocco. A nonlinear mathematical model with two subclasses of infectious individuals is proposed. The population is divided into five classes, namely, susceptible (S), exposed (E), undiagnosed infectious ( I n c ), diagnosed patients ( I c ), and removed individuals. To reflect the real dynamic of the COVID-19 transmission in Morocco, the real reported data are used for estimating model parameters. Two controls representing screening effort and limited treatment are considered. Based on viability theory and set-valued analysis, a Lyapunov function is constructed such that both exposed and infected populations are decreased to zero asymptotically. The corresponding controls are derived via a continuous selection of adequately designed feedback map. Numerical simulations are presented with three scenarios (cases when each control is used alone and the case when two controls are combined). Our results show that when only one control is to be applied, screening is the most effective in decreasing the number of people in the three infected compartments, whereas combining both controls is found to be highly effective and leads to a significant improvement in the epidemiological situation of Morocco. To the best of our knowledge, this work is the first one that applies the set-valued approach to a controlled COVID-19 model which agrees with the observed cases in Morocco.

In this paper, we propose a new fractional-order model of alcohol drinking involving the Caputo derivative and six groups of individuals. We introduce road accidents and violence related to alcohol consumption as separate classes to highlight the role of alcoholism in the aggressive and risky behaviour of heavy drinkers. We show the existence and uniqueness of the non-negative solutions, and we determine the basic reproduction number R0. The sensitivity analysis of the model parameters is performed to characterize the important parameters that have the most effects on the reproduction number. Furthermore, the stability analysis of the model shows that the system is locally and globally asymptotically stable at drinking-free equilibrium E0 when R0<1, and the drinking present equilibrium E∗ exists. The system is locally and globally asymptotically stable at E∗ when R0>1. Finally, numerical simulations are carried out to illustrate the theoretical results for different values of the order of the fractional derivative.

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