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

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.

A comprehensive spatial-temporal infection model

“…It is the objective of this paper to create such a methodology to model the spread of an epidemic in general, with application to COVID-19, in particular. Our work parallels related previous continuum models, such as ( Noble, 1974 ) or particularly the very recent work of Viguerie et al (2020) , but differs in a number of aspects. For example, the recent work of Viguerie et al (2020) uses a compartmental modeling approach using continuum mechanics, that allows one to obtain results including diffusion and reaction similar to the work here.…”

“…Our work parallels related previous continuum models, such as ( Noble, 1974 ) or particularly the very recent work of Viguerie et al (2020) , but differs in a number of aspects. For example, the recent work of Viguerie et al (2020) uses a compartmental modeling approach using continuum mechanics, that allows one to obtain results including diffusion and reaction similar to the work here. An important difference is that the view of contagion in the form of chemical reaction processes allows non-trivial additional insights, including the ab initio representation of the kinetics of the infection, as well as the effect of mobility through advection.…”

“…of non-linear diffusion, in which the diffusion coefficients are functions of the individual densities, e.g. see ( Viguerie et al, 2020 ). In crowded environments of high density, diffusion will certainly be affected by the total density, , although it will likely be independent of the individual density fractions.…”

“…(1) are expressed in terms of an extended species vector, with reactions (9) , (10) extended appropriately, possibly involving a product of a reaction matrix with the species vector. A related multicomponent description, although based on a different continuum model using applied mechanics methodologies, was presented in Viguerie et al (2020) . For simplicity, this generalization will not be further considered here.…”

“…The idea is to simulate how infection can spread in a “country” consisting of four hypothetical quadrants, with different densities, hence different values of . A similar simulation of the spreading of infection in the Lombard region of Italy using a different continuum model was provided in Viguerie et al (2020) . In our work, we have assumed that the two diagonally opposite quadrants (e.g.…”