Analysis and Numerical Simulations of a Reaction-Diffusion Model with Fixed Active Bodies

We propose an interaction model for two substances (molecules) of concentrations $f$ and $g$. The substance $f$ is assumed to be involved in the growth process of $g$, and in contrast, the substance $g$ inhibits the production of $f$. Our model is based on a reaction-diffusion system for the interac...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 2021-01, Vol.81 (4), p.1339-1360
Main Authors: Yang, Chang, Tine, Léon Matar
Format: Article
Language:English
Subjects:
Life Sciences
Mathematics
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Summary:We propose an interaction model for two substances (molecules) of concentrations $f$ and $g$. The substance $f$ is assumed to be involved in the growth process of $g$, and in contrast, the substance $g$ inhibits the production of $f$. Our model is based on a reaction-diffusion system for the interaction of two considered substances (molecules). We analyze its well-posedness and perform a numerical scheme to simulate the behavior of the solutions. As an application, we investigate the role of A$\beta$-oligomers in the early stage of Alzheimer's disease propagation, more precisely for the impact of A$\beta$-oligomers in neuron impairment as studied in [M. Andrade-Restrepo et al., Modeling the spatial propagation of A$\beta$ oligomers in Alzheimer's disease, in CEMRACS 2018 - Numerical and Mathematical Modeling for Biological and Medical Applications: Deterministic, Probabilistic and Statistical Descriptions, Marseille, France, 2018, pp. 1--10].
ISSN:0036-1399
1095-712X
DOI:10.1137/20M1343385