  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.
