Alzheimer’s disease (AD) is characterized by the progressive deposition of β-amyloid (Aβ) plaques in the brain, where the Aβ oligomers have been confirmed to produce the critical cytotoxicity during the disease process. In this study, a model is established to describe the effect of Aβ oligomers on the interplay between Aβ and Ca2＋. Mathematical analysis demonstrates the existence and stability of the equilibria and the conditions under which backward bifurcation and saddle–node bifurcation occur are proposed. In addition, the aggregate reproduction number R0 is introduced to characterize the progression of AD. These results may offer valuable insights for studying AD-related medical strategies.

中文翻译：

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阿尔茨海默病模型对寡聚体对 [公式省略] -淀粉样蛋白和 Ca2+ 之间相互作用的病理影响的研究


阿尔茨海默病 （AD） 的特征是 β-淀粉样蛋白 （Aβ） 斑块在大脑中逐渐沉积，其中 Aβ 寡聚体已被证实在疾病过程中产生关键的细胞毒性。在这项研究中，建立了一个模型来描述 Aβ 寡聚体对 Aβ 和 Ca2+ 之间相互作用的影响。数学分析证明了平衡的存在和稳定性，并提出了发生向后分岔和鞍节点分岔的条件。此外，引入了聚集繁殖数 R0 来表征 AD 的进展。这些结果可能为研究 AD 相关的医疗策略提供有价值的见解。