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The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used tomodel the accumulation of prionic proteins, which are responsible for manydifferent neurological disorders. Likely, the most important and studiedmisfolded protein in literature is the Amyloid-$\beta$, responsible for theonset of Alzheimer disease. Starting from medical images we construct areduced-order model based on a graph brain connectome. The reaction coefficientof the proteins is modelled as a stochastic random field, taking into accountall the many different underlying physical processes, which can hardly bemeasured. Its probability distribution is inferred by means of the Monte CarloMarkov Chain method applied to clinical data. The resulting model ispatient-specific and can be employed for predicting the disease's futuredevelopment. Forward uncertainty quantification techniques (Monte Carlo andsparse grid stochastic collocation) are applied with the aim of quantifying theimpact of the variability of the reaction coefficient on the progression ofprotein accumulation within the next 20 years.

Uncertainty Quantification for Fisher-Kolmogorov Equation on Graphs with Application to Patient-Specific Alzheimer Disease