Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-30T12:04:13.122Z Has data issue: false hasContentIssue false
  • English
  • Français

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, 01004 Kyiv, 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, 01004 Kyiv, Ukraine emails: r.m.cherniha@gmail.com, davydovych@imath.kiev.ua
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 5 , October 2022 , pp. 785 - 802
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abdelkader, M. A. (1982) Travelling wave solutions for a generalized Fisher equation. J. Math. Anal. Appl. 85, 287–290.CrossRefGoogle Scholar
Ablowitz, M. & Zeppetella, A. (1979) Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840.CrossRefGoogle Scholar
Akhatov, I. S., Gazizov, R. K. & Ibragimov, N.H. (1991) Nonlocal symmetries. Heuristic approach. J. Sov. Math. 55, 1401–1450.Google Scholar
Anderson, R. M. & May, R. M. (1982) Directly transmitted infectious diseases: control by vaccination. Science 215, 1053–1060.CrossRefGoogle ScholarPubMed
Available online: https://www.worldometers.info/coronavirusf (accessed on 31 December 2020).Google Scholar
Bluman, G. W., Cheviakov, A. F. & Anco, S. C. (2010) Applications of Symmetry Methods to Partial Differential Equations. New York. Springer.CrossRefGoogle Scholar
Brauer, F. & Castillo-Chavez, C. (2012) Mathematical Models in Population Biology and Epidemiology. Springer, New York.CrossRefGoogle Scholar
Brovchenko, I. et al. (2020) Development of a mathematical model for the spread of COVID-19 epidemic in Ukraine. Svitohlyad 82, 2–14 (in Ukrainian).Google Scholar
Cherniha, R. & Davydovych, V. (2017) Nonlinear Reaction-Diffusion Systems — Conditional Symmetry, Exact Solutions and Their Applications in Biology. Lecture Notes in Math., Vol. 2196. Springer, Cham.CrossRefGoogle Scholar
Cherniha, R. & Davydovych, V. (2020) A mathematical model for the COVID-19 outbreak. ArXiv, arXiv:2004.01487v2.CrossRefGoogle Scholar
Cherniha, R. & Davydovych, V. (2020) A mathematical model for the COVID-19 outbreak and its applications. Symmetry 12, 12 pp.CrossRefGoogle Scholar
Cherniha, R., Davydovych, V. & Muzyka, L. (2017) Lie symmetries of the Shigesada–Kawasaki–Teramoto system. Comm. Nonlinear Sci. Numer. Simulat. 45, 81–92.CrossRefGoogle Scholar
Cherniha, R. & King, J. R. (2006) Lie symmetries and conservation laws of nonlinear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math. 71, 391–408.CrossRefGoogle Scholar
Cherniha, R., Serov, M. & Pliukhin, O. (2018) Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications. Chapman and Hall/CRC, New York.Google Scholar
Cherniha, R. M. & Wilhelmsson, H. (1996) Symmetry and exact solution of heat-mass transfer equations in thermonuclear plasma. Ukr. Math. J. 48, 1434–1449.CrossRefGoogle Scholar
Diekmann, O. & Heesterbeek, J. (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley, Chichester.Google Scholar
Dietz, K. (1976) The Incidence of Infectious Diseases Under the Influence of Seasonal Fluctuations. Lecture Notes in Biomathematics, Vol. 11. Springer, Berlin, pp. 1–15.CrossRefGoogle Scholar
Efimov, D. & Ushirobira, U. (2020) On interval prediction of COVID-19 development based on a SEIR epidemic model. Research report. Inria Lille Nord Europe–Laboratoire CRIStAL–Universite de: Lille, France.CrossRefGoogle Scholar
Fushchych, W. I., Cherniha, R. & Chopyk, V. (1996) On unique symmetry of two nonlinear generalizations of the SchrÖdinger equation. J. Nonlinear Math. Phys. 3, 296–301.CrossRefGoogle Scholar
Gilding, B. H. & Kersner, R. (2004) Travelling Waves in Nonlinear Diffusion-Convection Reaction. Birkhauser Verlag, Basel.CrossRefGoogle Scholar
Ibragimov, N. H., Torrisi, M. & Valenti, A. (1991) Preliminary group classification of equations $v_{tt} = f(x, v_x) v_{xx} + g(x, v_x)$ . J. Math. Phys. 32, 2988–2995.CrossRefGoogle Scholar
Jordan, P., & Puri, A. (2002) Qualitative results for solutions of the steady Fisher-KPP equation. Applied Mathematics Letters 15, 239–250.CrossRefGoogle Scholar
Keeling, M. J. & Rohani, P. (2008) Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton.CrossRefGoogle Scholar
Kermack, W.O. & McKendrick, A. G. (1927) A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A 115, 700–721.Google Scholar
Luo, X. et al. (2020) Analysis of potential risk of COVID-19 infections in China based on a pairwise epidemic model. Preprints, doi: 10.20944/preprints202002.0398.v1.Google Scholar
Mammeri, Y. (2020) A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France. Comput. Math. Biophys. 8, 102–113.CrossRefGoogle Scholar
Murray, J. D. (1989) Mathematical Biology. Springer, Berlin.CrossRefGoogle Scholar
Murray, J. D. (2003) Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin.CrossRefGoogle Scholar
Nikitin, A. G. (2005) Group classification of systems of nonlinear reaction-diffusion equations. Ukr. Math. Bull. 2, 153–204.Google Scholar
Ovsiannikov, L. V. (1980) The Group Analysis of Differential Equations. Academic Press, New York.Google Scholar
Peng, L. et al. (2020) Epidemic analysis of COVID-19 in China by dynamical modeling. ArXiv, arXiv:2002.06563.CrossRefGoogle Scholar
Polyanin, A. D. & Zaitsev, V. F. (2018) Handbook of Ordinary Differential Equations for Scientists and Engineers. CRC Press Company, Boca Raton.CrossRefGoogle Scholar
Qiang, L., Wang, B. G., & Wang, Z. C. (2020) A reaction-diffusion epidemic model with incubation period in almost periodic environments. Eur. J. Appl. Math. (Published online 25 September 2020)CrossRefGoogle Scholar
Roda, W. C., Varugheseb, M. B., Han, D. & Li, M. Y. (2020) Why is it difficult to accurately predict the COVID-19 epidemic? Infectious Disease Modelling 5, 271–281.CrossRefGoogle ScholarPubMed
Serov, M. I., Karpaliuk, T. O., Pliukhin, O. G. & Rassokha, I. V. (2015) Systems of reaction-convection-diffusion equations invariant under Galilean algebras. J. Math. Anal. Appl. 422, 185–211.CrossRefGoogle Scholar
Shao, N. et al. (2020) Dynamic models for coronavirus disease 2019 and data analysis. Math. Meth. Appl. Sci. 43, 4943–4949.CrossRefGoogle ScholarPubMed
Stewart, J. M., Broadbridge, P. & Goard, J. M. (2002) Symmetry analysis and numerical modelling of invasion by malignant tumour tissue. Nonlinear Dynamics 28, 175–193.CrossRefGoogle Scholar
Tian, J. et al. (2020) Modeling analysis of COVID-19 based on morbidity data in Anhui, China. MBE 17, 2842–2852.CrossRefGoogle Scholar
Torrisi, M., Tracina, R. & Valenti, A. (1996) A group analysis approach for a non linear differential system arising in diffusion phenomena. J. Math. Phys. 37, 4758–4767.CrossRefGoogle Scholar
Viguerie, A. et al. (2021) Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion. Appl. Math. Lett. 111, 106617, 9 pp.Google Scholar