Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread

Silviu Dumitru Pavăl; Alex Vasilică; Alin Adochiei

doi:10.3934/dcdss.2022094

Article Contents

2023, Volume 16, Issue 1: 33-53. Doi: 10.3934/dcdss.2022094

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Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread

1.
Department of Computer Engineering, "Gheorghe Asachi" Technical University of Iaşi, Strada prof. dr. doc. Dimitrie Mangeron 27, 700050, Iaşi, Romania
2.
Department of Computer Science, University "Al. I. Cuza" of Iaşi, Strada General Berthelot 16, 700483, Iaşi, Romania
3.
Department of Mathematics, University "Al. I. Cuza" of Iaşi, Bd. Carol I 11, 700506, Iaşi, Romania

Received: March 2022

Revised: March 2022

Early access: April 2022

Published: January 2023

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Abstract

The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: $ S_0(x) $, $ E_0(x) $, $ I_0(x) $ $ R_0(x) $, $ D_0(x) $, $ f(t,x) $ and $ w_{_i}(t,x), i = 1,2,3,4,5 $, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $ W^{1,2}_p(Q) $, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.

The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.

Keywords:

Mathematics Subject Classification: Primary: 35Bxx; Secondary: 35K57, 65M06, 92B05.

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Figure 1. The boundary $ \Gamma = \cup_{\ell = 1}^4\Gamma_\ell $ of the rectangle $ \Omega = [0,a]\times [0,b] $

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Figure 2. Initial data

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Figure 3. Epidemic evolution showing the affected areas for the respective populations: S - susceptible, E - exposed, I - infected, R - recovered, D - deceased

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Figure 4. The maximum values evolution for all population categories

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