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Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability

Author

Listed:
  • Haiyue Chen

    (Institute of Product and Process Innovation, Leuphana University Lüneburg, Universitätsallee 1, D-21335 Lüneburg, Germany
    These authors contributed equally to this work.)

  • Benedikt Haus

    (Institute of Product and Process Innovation, Leuphana University Lüneburg, Universitätsallee 1, D-21335 Lüneburg, Germany
    These authors contributed equally to this work.)

  • Paolo Mercorelli

    (Institute of Product and Process Innovation, Leuphana University Lüneburg, Universitätsallee 1, D-21335 Lüneburg, Germany
    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. The Lyapunov theory offers the possibility to track desired trajectories, guaranteeing 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.

Suggested Citation

  • Haiyue Chen & Benedikt Haus & Paolo Mercorelli, 2021. "Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability," Mathematics, MDPI, vol. 9(17), pages 1-25, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2076-:d:623475
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    References listed on IDEAS

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    1. Memon, Zaibunnisa & Qureshi, Sania & Memon, Bisharat Rasool, 2021. "Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
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    3. Das, Arghya & Dhar, Abhishek & Goyal, Srashti & Kundu, Anupam & Pandey, Saurav, 2021. "COVID-19: Analytic results for a modified SEIR model and comparison of different intervention strategies," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    4. Zhang, Yong & Yu, Xiangnan & Sun, HongGuang & Tick, Geoffrey R. & Wei, Wei & Jin, Bin, 2020. "Applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of COVID-19," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    5. Postnikov, Eugene B., 2020. "Estimation of COVID-19 dynamics “on a back-of-envelope”: Does the simplest SIR model provide quantitative parameters and predictions?," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
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    2. Alberto Olivares & Ernesto Staffetti, 2021. "Optimal Control Applied to Vaccination and Testing Policies for COVID-19," Mathematics, MDPI, vol. 9(23), pages 1-22, December.

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