Well-posedness for a diffusion-reaction compartmental model simulating the spread of COVID-19

TL;DR

This paper proves the existence and uniqueness of solutions for a nonlinear diffusion-reaction SEIR model describing COVID-19 spread, using mathematical analysis and time discretization techniques.

Contribution

It provides the first rigorous mathematical analysis of a nonlinear diffusion-reaction SEIR model with nonlinear diffusions for COVID-19.

Findings

Existence of solutions is established under general conditions.

Uniqueness of solutions is proved for constant diffusion coefficients and regular data.

Uniform bounds and regularity results are obtained for the solutions.

Abstract

This paper is concerned with the well-posedness of a diffusion-reaction system for a Susceptible-Exposed-Infected-Recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio-temporal spread of the COVID-19 pandemic and is a variation of the one recently introduced, discussed and tested in [A. Viguerie et al, Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study, Comput. Mech. 66 (2020) 1131-1152]. Here, we deal with the mathematical analysis of the resulting Cauchy-Neumann problem: the existence of solutions is proved in a rather general setting and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.