  We propose a mathematical model for the onset and progression of Alzheimer's
disease based on transport and diffusion equations. We treat brain neurons as a
continuous medium and structure them by their degree of malfunctioning. Three
different mechanisms are assumed to be relevant for the temporal evolution of
the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii)
effects of phosphorylated tau protein and iii) neuron-to-neuron prion-like
transmission of the disease. We model these processes by a system of
Smoluchowski equations for the Amyloid beta concentration, an evolution
equation for the dynamics of tau protein and a kinetic-type transport equation
for the distribution function of the degree of malfunctioning of neurons. The
latter equation contains an integral term describing the random onset of the
disease as a jump process localized in particularly sensitive areas of the
brain. We are particularly interested in investigating the effects of the
synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's
disease. The output of our numerical simulations, although in 2D with an
over-simplified geometry, is in good qualitative agreement with clinical
findings concerning both the disease distribution in the brain, which varies
from early to advanced stages, and the effects of tau on the dynamics of the
disease.
