  The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used to
model the accumulation of prionic proteins, which are responsible for many
different neurological disorders. Likely, the most important and studied
misfolded protein in literature is the Amyloid-$\beta$, responsible for the
onset of Alzheimer disease. Starting from medical images we construct a
reduced-order model based on a graph brain connectome. The reaction coefficient
of the proteins is modelled as a stochastic random field, taking into account
all the many different underlying physical processes, which can hardly be
measured. Its probability distribution is inferred by means of the Monte Carlo
Markov Chain method applied to clinical data. The resulting model is
patient-specific and can be employed for predicting the disease's future
development. Forward uncertainty quantification techniques (Monte Carlo and
sparse grid stochastic collocation) are applied with the aim of quantifying the
impact of the variability of the reaction coefficient on the progression of
protein accumulation within the next 20 years.
