Communications in Mathematical Sciences

Volume 18 (2020)

Number 4

A mathematical model for Alzheimer’s disease: An approach via stochastic homogenization of the Smoluchowski equation

Pages: 1105 – 1134



Bruno Franchi (Dipartimento di Matematica, Università di Bologna, Italy)

Martin Heida (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany)

Silvia Lorenzani (Dipartimento di Matematica, Politecnico di Milano, Italy)


In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski’s coagulation-diffusion equations with nonhomogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer’s disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.


Smoluchowski equation, stochastic homogenization, randomly perforated domains, Alzheimer’s disease

2010 Mathematics Subject Classification

35K55, 35R60, 80A30, 80M40

Received 18 April 2019

Accepted 21 January 2020

Published 28 July 2020