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Well-posedness of Two Mathematical Models for Alzheimer's Disease

Yarmohammadi, Parisa | 2021

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 54445 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Hesaaraki, Mahmoud
  7. Abstract:
  8. In season 1, we introduce a mathematical model of the in vivo progression of Alzheimer’s disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of Aβ -amyloid plaques based on the concentrations of Aβ oligomers, PrPC proteins, and the Aβ-×-PrPC complex, which are hypothesized to be responsible for synaptic tox- icity. We prove the well posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of β-amyloid is constant and also when it is described by a power law. In seson 2, We consider the existence and uniqueness of solutions of an initial-boundary value problem for a coupled system of PDE’s arising in a model for Alzheimer’s disease. Apart from reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are: the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artefacts, but are naturally imposed by modelling considerations. In particular, the use of a probability measure is natural from a modelling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments
  9. Keywords:
  10. Alzheimer ; Stability ; Well Posedness ; Smoluchowski Equations ; Transport and Diffusion Equations ; Prion ; Alzheimer’s Disease Mathematical Models

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