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

• Type of Material: Article
• Authors: Alex Viguerie, Università di Pavia; Alessandro Veneziani, Emory University; Guillermo Lorenzo, University of Texas Austin; Davide Baroli, Aachen University; Nicole Aretz-Nellesen, Aachen University; Alessia Patton, Università di Pavia; Thomas E. Yankeelov, University of Texas Austin; Alessandro Reali, Università di Pavia; Thomas J. R. Hughes, University of Texas Austin; Ferdinando Auricchio, Università di Pavia
• Language: English
• Date: 2020-08-13
• Publisher: Springer (part of Springer Nature)
• Publication Version: Final Published Version
• Copyright Statement: © The Author(s) 2020.
• License: Creative Commons Attribution 4.0 International
• Final Published Version (URL): http://dx.doi.org/10.1007/s00466-020-01888-0
• Title of Journal or Parent Work: Computational Mechanics
• Volume: 66
• Start Page: 1
• End Page: 22
• Grant/Funding Information: Open access funding provided by Università degli Studi di Pavia within the CRUI-CARE Agreement.
• Keywords: Partial differential equations; Compartmental models; COVID-19; Epidemic
• Research Categories: Computer Science; Health Sciences, Epidemiology; Mathematics; Engineering, Biomedical

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.