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Analysis and Numerical Simulations of a Reaction-Diffusion Model with Fixed Active Bodies
SIAM Journal on Applied Mathematics ( IF 1.9 ) Pub Date : 2021-07-01 , DOI: 10.1137/20m1343385 Chang Yang , Léon Matar Tine
SIAM Journal on Applied Mathematics ( IF 1.9 ) Pub Date : 2021-07-01 , DOI: 10.1137/20m1343385 Chang Yang , Léon Matar Tine
SIAM Journal on Applied Mathematics, Volume 81, Issue 4, Page 1339-1360, January 2021.
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].
中文翻译：
具有固定活动体的反应扩散模型的分析与数值模拟
SIAM 应用数学杂志，第 81 卷，第 4 期，第 1339-1360 页，2021 年 1 月。
我们为两种浓度为 $f$ 和 $g$ 的物质（分子）提出了一个相互作用模型。假定物质$f$ 参与了$g$ 的生长过程，相反，物质$g$ 抑制了$f$ 的产生。我们的模型基于两种考虑的物质（分子）相互作用的反应扩散系统。我们分析其适定性并执行数值方案来模拟解决方案的行为。作为一项应用，我们研究了 A$\beta$-寡聚体在阿尔茨海默病传播早期阶段的作用，更准确地说是研究了 A$\beta$-寡聚体对神经元损伤的影响，如 [M. Andrade-Restrepo 等人，模拟 A$\beta$ 寡聚体在阿尔茨海默病中的空间传播，
更新日期：2021-07-18
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].
中文翻译：
具有固定活动体的反应扩散模型的分析与数值模拟
SIAM 应用数学杂志，第 81 卷，第 4 期，第 1339-1360 页，2021 年 1 月。
我们为两种浓度为 $f$ 和 $g$ 的物质（分子）提出了一个相互作用模型。假定物质$f$ 参与了$g$ 的生长过程，相反，物质$g$ 抑制了$f$ 的产生。我们的模型基于两种考虑的物质（分子）相互作用的反应扩散系统。我们分析其适定性并执行数值方案来模拟解决方案的行为。作为一项应用，我们研究了 A$\beta$-寡聚体在阿尔茨海默病传播早期阶段的作用，更准确地说是研究了 A$\beta$-寡聚体对神经元损伤的影响，如 [M. Andrade-Restrepo 等人，模拟 A$\beta$ 寡聚体在阿尔茨海默病中的空间传播，
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