Diffusion–reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study

Computational Mechanics, vol. 66, pp. 1131–1152

Abstract

Abstract The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models , in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.

Authors 10

1. Alex Viguerie corresponding

   University of Pavia

   Affiliation as printed

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100 Pavia, PV Italy

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100, Pavia, PV, Italy

2. Alessandro Veneziani

   Emory University

   Affiliation as printed

   Department of Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322 USA

   Department of Mathematics, Emory University, 400 Dowman Drive, Atlanta, GA 30322 USA

   Department of Mathematics, Emory University, 400 Dowman Drive, Atlanta, GA, 30322, USA

3. Guillermo Lorenzo

   The University of Texas at Austin

   Affiliation as printed

   Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, Austin, TX 78712-1229 USA

   Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, Austin, TX, 78712-1229, USA

4. Davide Baroli Aachen Aachen Institute for Advanced Study in Computational Engineering Science (AICES)

   Aachen Institute for Advanced Study in Computational Engineering Science · RWTH Aachen University

   Affiliation as printed

   Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

   Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062, Aachen, Germany

5. Nicole Aretz Aachen Aachen Institute for Advanced Study in Computational Engineering Science (AICES)

   Aachen Institute for Advanced Study in Computational Engineering Science · RWTH Aachen University

   Affiliation as printed

   Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

   Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062, Aachen, Germany

6. Alessia Patton

   University of Pavia

   Affiliation as printed

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100 Pavia, PV Italy

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100, Pavia, PV, Italy

7. Thomas E. Yankeelov

   The University of Texas at Austin

   Affiliation as printed

   Departments of Biomedical Engineering, Diagnostic Medicine, and Oncology, Livestrong Cancer Institutes, The University of Texas at Austin, 107 W. Dean Keeton St., Austin, TX 78712 USA

   Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, Austin, TX 78712-1229 USA

   Departments of Biomedical Engineering, Diagnostic Medicine, and Oncology, Livestrong Cancer Institutes, The University of Texas at Austin, 107 W. Dean Keeton St., Austin, TX, 78712, USA

8. Alessandro Reali

   University of Pavia

   Affiliation as printed

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100 Pavia, PV Italy

   Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100, Pavia, PV, Italy

9. Thomas J.R. Hughes

   The University of Texas at Austin

   Affiliation as printed

   Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, Austin, TX 78712-1229 USA

   Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, Austin, TX, 78712-1229, USA

10. Ferdinando Auricchio

    University of Pavia

    Affiliation as printed

    Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100 Pavia, PV Italy

    Dipartimento di Ingegneria Civile ed Architettura, Università di Pavia, Via Ferrata 3, 27100, Pavia, PV, Italy