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

2021 ◽  
pp. 1-18
Author(s):  
ROMAN M. CHERNIHA ◽  
VASYL V. DAVYDOVYCH
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Interesting Case ◽  
Lie Symmetry ◽  
Point Of View ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

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.

ABRAHAMS–TSUNETO REACTION–DIFFUSION SYSTEM IN SUPERCONDUCTIVITY AND SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (10) ◽  
pp. 1417-1423 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Reaction Diffusion ◽  
Analytic Solutions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Ginzburg Landau ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Exact Analytic Solutions ◽  
Self Consistent

The reaction–diffusion systems represent various problems in the real world. For the Abrahams–Tsuneto reaction–diffusion system arising in superconductivity, we perform computerized symbolic computation and find its new exact analytic solutions, which are solitonic. We see the possibility that by way of the shock waves, the self–consistent superconducting interaction drives the Ginzburg–Landau order parameter, which might be observable.

Multiparameter bifurcation for some particular reaction–diffusion systems

1989 ◽  
Vol 112 (1-2) ◽  
pp. 135-143 ◽  
Author(s):  
J. Esquinas ◽  
J. López-Gómez
Keyword(s):  
Linear Operator ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Operator Polynomial ◽  
Reaction Diffusion Systems ◽  
Abstract Equation ◽  
Diffusion Systems ◽  
Multiparameter Bifurcation

SynopsisIn some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

EPIDEMIC REACTION-DIFFUSION SYSTEM WITH CROSS-DIFFUSION: MODELING AND NUMERICAL SOLUTION

1995 ◽  
Vol 03 (03) ◽  
pp. 733-746 ◽  
Author(s):  
V. CAPASSO ◽  
A. DI LIDDO ◽  
F. NOTARNICOLA ◽  
L. RUSSO
Keyword(s):  
Population Dynamics ◽  
Numerical Analysis ◽  
Numerical Solution ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Diffusion Modeling ◽  
Qualitative Behaviour ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

Reaction-diffusion systems with cross-diffusion are analyzed here for modeling the population dynamics of epidemic systems. In this paper specific attention is devoted to the numerical analysis and simulation of such systems to show that, far from possible pathologies, the qualitative behaviour of the systems may well interpret the dynamics of real systems.

A geometrical approach to wave-type solutions of excitable reaction-diffusion systems

1991 ◽  
Vol 433 (1887) ◽  
pp. 151-164 ◽  
Keyword(s):  
Numerical Solution ◽  
Wave Front ◽  
Eikonal Equation ◽  
Reaction Diffusion ◽  
Stationary Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Wave Boundary

We formulate the eikonal equation approximation for travelling waves in excitable reaction-diffusion systems, which have been proposed as models for a large number of biomedical situations. This formulation is particularly suited, in a natural way, to numerical solution by finite difference methods. We show how this solution is independent of the parametric variable used for expressing the eikonal equation, and how a reduction of dimensionality implies a major saving over the time taken to solve the original reaction-diffusion system. Neumann boundary conditions on reactants in the original system translate into a geometric constraint on the wave boundary itself. We show how this leads to geometrically stable stationary wave boundaries in appropriately shaped non-convex domains. This analytical prediction is verified by numerical solution of the eikonal equation on a domain which supports geometrically stable stationary wave boundary configurations. We show how the concepts of geometrical stability and wave-front stability relate to a problem where a bi-stable reaction-diffusion system has a stable stationary wave-front configuration.

scholarly journals Coloured Noise from Stochastic Inflows in Reaction–Diffusion Systems

2020 ◽  
Vol 82 (4) ◽  
Author(s):  
Michael F. Adamer ◽  
Heather A. Harrington ◽  
Eamonn A. Gaffney ◽  
Thomas E. Woolley
Keyword(s):  
Steady State ◽  
Power Spectrum ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Coloured Noise ◽  
Reaction Systems ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Parameter Values

Abstract In this paper, we present a framework for investigating coloured noise in reaction–diffusion systems. We start by considering a deterministic reaction–diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction–diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction–diffusion system.

Unique continuation for a reaction-diffusion system with cross diffusion

2019 ◽  
Vol 27 (4) ◽  
pp. 511-525 ◽  
Author(s):  
Bin Wu ◽  
Ying Gao ◽  
Zewen Wang ◽  
Qun Chen
Keyword(s):  
Unique Continuation ◽  
Reaction Diffusion ◽  
Cauchy Data ◽  
Carleman Estimate ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Lateral Boundary ◽  
Strongly Coupled ◽  
Cross Diffusion ◽  
Coupled Reaction

Abstract This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

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