Dynamically consistent finite volume scheme for a bimonomeric simplified model with inflammation processes for Alzheimer's disease

TL;DR

This paper develops a finite volume scheme for a complex PDE-ODE model of Alzheimer's disease, ensuring mathematical consistency, stability, and convergence, and demonstrates its effectiveness through numerical experiments.

Contribution

It introduces a dynamically consistent finite volume scheme for a bimonomeric Alzheimer's model, with proofs of convergence and stability, and includes numerical validation.

Findings

The scheme converges to a weak solution of the model.

The scheme maintains non-negativity and boundedness of solutions.

Numerical experiments illustrate the model's behavior and scheme's effectiveness.

Abstract

A model of progression of Alzheimer's disease (AD) incorporating the interactions of A$\beta$-monomers, oligomers, microglial cells and interleukins with neurons is considered. The resulting convection-diffusion-reaction system consists of four partial differential equations (PDEs) and one ordinary differential equation (ODE). We develop a finite volume (FV) scheme for this system, together with non-negativity and a priori bounds for the discrete solution, so that we establish the existence of a discrete solution to the FV scheme. It is shown that the scheme converges to an admissible weak solution of the model. The reaction terms of the system are discretized using a semi-implicit strategy that coincides with a nonstandard discretization of the spatially homogeneous (SH) model. This construction enables us to prove that the FV scheme is dynamically consistent with respect to the spatially homogeneous version of the model. Finally, numerical experiments are presented to illustrate the model and to assess the behavior of the FV scheme.