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

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  08 July 2021

ROMAN M. CHERNIHA Open the ORCID record for ROMAN M. CHERNIHA [Opens in a new window]  and
VASYL V. DAVYDOVYCH
ROMAN M. CHERNIHA
Affiliation:
Institute of Mathematics, NAS of Ukraine, 3 Tereshchenkivs’ka Street, 01004Kyiv, Ukraine emails: r.m.cherniha@gmail.com, davydovych@imath.kiev.ua
VASYL V. DAVYDOVYCH
Affiliation:
Institute of Mathematics, NAS of Ukraine, 3 Tereshchenkivs’ka Street, 01004Kyiv, Ukraine emails: r.m.cherniha@gmail.com, davydovych@imath.kiev.ua
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Abstract

A non-linear 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 non-trivial 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 typical 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.

Keywords

Reaction–diffusion systemcross-diffusionLie symmetryexact solutionmathematical modelling in epidemiology

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 92D30: Epidemiology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 5 , October 2022 , pp. 785 - 802
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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