mathematics Article Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability Haiyue Chen † , Benedikt Haus † and Paolo Mercorelli * ,†   Citation: Chen, H.; Haus, B.; Mercorelli, P. Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability. Mathematics 2021, 9, 2076. https://doi.org/10.3390/ math9172076 Academic Editor: Eugene Eugene Postnikov Received: 13 July 2021 Accepted: 24 August 2021 Published: 27 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Institute of Product and Process Innovation, Leuphana University Lüneburg, Universitätsallee 1, D-21335 Lüneburg, Germany; haiyue.chen@stud.leuphana.de (H.C.); haus@leuphana.de (B.H.) * Correspondence: mercorelli@uni.leuphana.de † These authors contributed equally to this work. Abstract: Due to the worldwide outbreak of COVID-19, many strategies and models have been put forward by researchers who intend to control the current situation with the given means. In particular, compartmental models are being used to model and analyze the COVID-19 dynamics of different considered populations as Susceptible, Exposed, Infected and Recovered compartments (SEIR). This study derives control-oriented compartmental models of the pandemic, together with constructive control laws based on the Lyapunov theory. The paper presents the derivation of new vaccination and quarantining strategies, found using compartmental models and design methods from the field of Lyapunov theory. TheLyapunov theory offers the possibility to track desired trajectories, guaran- teeing the stability of the controlled system. Computer simulations aid to demonstrate the efficacy of the results. Stabilizing control laws are obtained and analyzed for multiple variants of the model. The stability, constructivity, and feasibility are proven for each Lyapunov-like function. Obtaining the proof of practical stability for the controlled system, several interesting system properties such as herd immunity are shown. On the basis of a generalized SEIR model and an extended variant with additional Protected and Quarantined compartments, control strategies are conceived by using two fundamental system inputs, vaccination and quarantine, whose influence on the system is a crucial part of the model. Simulation results prove that Lyapunov-based approaches yield effective control of the disease transmission. Keywords: COVID-19; compartmental models; Lyapunov approach; practical stability 1. Introduction The COVID-19 pandemic, among other pandemics from the past, has attracted great attention not only from mathematicians but researchers from numerous fields. This is due to the fact that the exponential growth in the number of cases of infection has made the recent situation very worrying. Hence, various measures are taken for the purpose of limiting the spread of infection. Concerning COVID-19, since the extent and duration of it has lasted much longer than expected, solving the overwhelming chaos is recognized as being the most important issue in recent months. The outbreak of the pandemic has been affecting almost all countries in the world, changing people’s daily lives and causing heavy casualties. The situation calls for a dynamic model of the pandemic to analyze the system behavior. When the outbreak is in an active stage, the model should be not only descriptive, but also suitable for controller design. 1.1. Historical Development of Compartmental Models for Epidemics In the history of mathematical models in epidemiology, the focus has always been on deterministic compartmental models, which can be defined as a sub-categorization of the whole population into different compartments, introducing transfer rates from one category Mathematics 2021, 9, 2076. https://doi.org/10.3390/math9172076 https://www.mdpi.com/journal/mathematics