The synergistic interplay between two proteins: a mathematical model for Alzheimer's disease

Abstract:
There is currently a great deal of interest in the scientific community in investigating the effects of the synergistic interplay of Amyloid beta and tau on the onset and progression of Alzheimer’s disease.
Today I will present a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations for the two proteins.  In the model neurons are treated as a continuous medium and structured by their degree of malfunctioning.
Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: 

1.  diffusion and agglomeration of soluble Amyloid beta, 
2.  effects of misfolded tau protein and 
3.  neuron-to- neuron prion-like transmission of the disease.

These processes are modelled 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. I will explain in detail the structure of the model and give a hint of the main results obtained and the techniques used for the purpose.
Eventually I will also show the output of some numerical simulations, of some significance even if performed in an over-simplified 2D geometry.