Department of Mathematics, Presidency University, Kolkata, 700073, India.
J Biol Phys. 2024 Jun;50(2):149-179. doi: 10.1007/s10867-023-09652-0. Epub 2023 Dec 29.
We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.
我们提出了一个数学模型，旨在探索阿尔茨海默病的进展，特别关注与疾病相关的蛋白质和星形胶质细胞的参与。我们的模型由一个微分方程的耦合系统组成，该系统描绘了淀粉样β斑块、淀粉样β蛋白、tau 蛋白和星形胶质细胞的动态。淀粉样β斑块可以被认为是依赖于斑块大小和时间的原纤维。我们通过对斑块大小进行积分操作将我们的数学模型转换为时间系统。在我们的模型中进行了存在性、唯一性、正定性和有界性的理论分析。我们通过添加两个群体，即朊病毒蛋白和淀粉样β-朊病毒复合物，扩展了我们的数学模型。我们通过定位两个模型的生物可行平衡点及其局部稳定性分析来描述系统动态。所提出模型的特征可以帮助我们深入了解阿尔茨海默病的发展及其复杂的动态。我们通过构建合适的李雅普诺夫函数来研究内部平衡点周围的全局稳定性分析。我们借助广泛的数值说明验证了我们的理论分析。
