A multiscale discrete-to-continuum framework for structured population models

TL;DR

This paper introduces a systematic multiscale framework using asymptotic expansions to derive continuum models from discrete structured population models, improving analytical clarity and boundary condition handling.

Contribution

It develops a novel multiscale approach combining multiple scales and matched asymptotics to accurately upscale discrete population models into continuum PDEs.

Findings

Derived PDEs include nonlinear advection and advection-diffusion equations.

Identified regions where continuum approximation is valid or invalid.

Validated the framework on a lipid-structured model for atherosclerosis.

Abstract

Mathematical models of biological populations commonly use discrete structure classes to capture trait variation among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions can improve analytical tractability and scalability of numerical solutions. Common upscaling approaches based solely on Taylor expansions may, however, introduce ambiguities in truncation order, uniform validity and boundary conditions. To address this, here we introduce a discrete multiscale framework to systematically derive continuum approximations of structured population models. Using the method of multiple scales and matched asymptotic expansions applied to discrete systems, we identify regions of structure space for which a continuum representation is appropriate and derive the corresponding partial differential equations. The leading-order dynamics are given by a nonlinear advection equation in the bulk domain and advection-diffusion processes in small inner layers about the leading wavefronts and stagnation point. We further derive discrete boundary layer descriptions for regions where a continuum representation is fundamentally inappropriate. Finally, we demonstrate the method on a simple lipid-structured model for early atherosclerosis and verify consistency between the discrete and continuum descriptions. The multiscale framework we present can be applied to other heterogeneous systems with discrete structure in order to obtain appropriate upscaled dynamics with asymptotically consistent boundary conditions.