A reaction-diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modeling

TL;DR

This paper uses Lie symmetry methods to analyze a reaction-diffusion model with cross-diffusion for COVID-19, deriving exact solutions that qualitatively match real pandemic data in Ukraine.

Contribution

It provides a complete Lie symmetry classification for the COVID-19 reaction-diffusion system with cross-diffusion, revealing unique symmetries and exact solutions applicable to pandemic modeling.

Findings

Derived exact solutions match COVID-19 spread data in Ukraine

Identified nontrivial Lie symmetries unique to this system

Demonstrated applicability of symmetry methods in pandemic modeling

Abstract

A nonlinear reaction-diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly-specified parameters admits highly nontrivial Lie symmetry operators, which do not occur for all known reaction-diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess all necessary properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.