Paper List

期刊: ArXiv Preprint
发布日期: 2026-03-16
Mathematical BiologyApplied Mathematics

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

Mathematical Institute, University of Oxford, OX2 6GG Oxford, UK | Ludwig Institute for Cancer Research, University of Oxford, OX3 7DQ Oxford, UK

Eleonora Agostinelli, Keith L. Chambers, Helen M. Byrne, Mohit P. Dalwadi
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IN SHORT: This paper addresses the core challenge of systematically deriving uniformly valid continuum approximations from discrete structured population models, overcoming ambiguities in truncation order and boundary conditions inherent in traditional Taylor expansion methods.

核心创新

  • Methodology Introduces a discrete multiscale framework combining the method of multiple scales with matched asymptotic expansions to systematically derive continuum approximations, identifying regions where continuum representation is appropriate versus fundamentally discrete.
  • Methodology Provides asymptotically consistent boundary conditions through discrete boundary layer analysis, resolving the ambiguity in boundary condition selection that plagues traditional Taylor expansion approaches.
  • Methodology Demonstrates the framework on a lipid-structured model for early atherosclerosis, showing consistency between discrete and continuum descriptions and validating the method's practical applicability.

主要结论

  • The method identifies distinct asymptotic regions: outer regions (e.g., O1-O4) describable by continuum PDEs (nonlinear advection equations) and inner boundary layers (e.g., IN1-IN5, B1-B4) that remain fundamentally discrete and require separate analysis.
  • For the paradigm problem (Eq. 1), the framework yields a composite solution (Eq. 16) asymptotically consistent with the exact discrete steady state (Eq. 10), unlike the truncated PDE solution (Eq. 9) which predicts an incorrect decay rate (a/(εb) vs. log((2b+a)/(2b-a))/ε).
  • The framework successfully derives a continuum approximation for a lipid-structured atherosclerosis model, verifying consistency and demonstrating transferability to biological systems with discrete internal states (e.g., lipid accumulation in macrophages).
研究空白： Current discrete-to-continuum upscaling methods based solely on Taylor expansions suffer from ambiguities in truncation order, lack uniform validity across the domain (especially near boundaries and internal layers), and provide unclear guidance on appropriate boundary conditions for the resulting PDEs.

摘要: 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.