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

A model of progression of Alzheimer’s disease (AD) incorporating the interactions of A$β$-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.

💡 Research Summary

**
The paper presents a rigorous numerical framework for a biologically motivated model of Alzheimer’s disease (AD) that captures the interactions among amyloid‑β (Aβ) monomers, Aβ oligomers, microglial cells, interleukins, and neuronal stress. The continuous model consists of four coupled parabolic partial differential equations (PDEs) for the spatially distributed species (u₁: oligomers, u₃: monomers, u₄: microglia, u₅: interleukins) and an ordinary differential equation (ODE) for the plaque‑bound oligomer concentration (u₂). A key feature is the chemotactic drift term ∇·(χ(u₄)∇u₁) in the microglial equation, where χ(u₄)=α u₄(ĥm−u₄) vanishes at the recruitment threshold ĥm, modeling volume‑filling effects.

The authors develop an unconditionally stable finite‑volume (FV) discretization on general admissible meshes. Spatial operators (diffusion and chemotaxis) are treated with standard central differences and upwind schemes, while the nonlinear reaction terms are discretized using a semi‑implicit nonstandard finite‑difference (NSFD) approach. This hybrid strategy yields a scheme that preserves the essential qualitative properties of the continuous system: non‑negativity of all concentrations, uniform a‑priori bounds, and the same equilibrium points and stability characteristics as the spatially homogeneous (SH) ODE system. The NSFD treatment allows the reaction step to be implicit in some variables and explicit in others, eliminating the restrictive Courant–Friedrichs–Lewy (CFL) condition typical of explicit methods.

Mathematical analysis proceeds in several stages. First, the authors identify an invariant rectangle R = ∏