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

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.

Figures (9)

• Figure 1: The asymptotic structure of the illustrative example is made of one outer region (I) which can be represented with a continuum structured variable, and one boundary layer (II) which is fundamentally discrete.
• Figure 2: Schematic of forward and backward transitions of neighbouring subpopulations with their respective rates, for $n=1,\dots,N-1$.
• Figure 3: Distributions of the discretely structured population, $u_{n}(t)$ for $n=0,1,\dots,N$ found by solving \ref{['eq:dimensional_structured_model', 'eq:linear_functions']} for $N=10,50,100,500$ at times $t=0.1,0.5,2,25$ when $k_f=1,k_b=0.5,\gamma=1.25$ and $u_{n}(0)=N^{-1}$ for all $n$. For $N=500$, we highlight the apparent moving wavefronts with arrows.
• Figure 4: (a) The asymptotic regions we investigate in this paper, via the multiple scales analysis in \ref{['sec:outer regions', 'sec:left_BL', 'sec:right_BL', 'sec:effective BCs', 'sec:zero advection point BL', 'sec:wavefront']}. The outer regions (white, O1–O4) are governed by continuum equations \ref{['eq:secularity_condition_outer']}, while the shaded zones correspond to boundary layers where the dynamics change character. At the domain endpoints, boundary layers B1 and B2 arise; because transitions here are of the same order as the discrete structure, these regions are fundamentally discrete and cannot be described by a continuum approximation. Within the interior, a stagnation point at $s=c(t)$ (\ref{['eq:zero advection point definition']}) generates an inner moving boundary layer (region IN1), while moving fronts at $s=\ell(t)$ (\ref{['eq:wavefront_definition']}) and $s=r(t)$ (\ref{['eq:right_wavefront_definition']}) generate additional inner layers (regions IN2 and IN3). (b) Additional asymptotic regions arise due to the intersections of various boundary layers, which we do not investigate in this paper. At early times, the inner boundary layer at the left wavefront intersects the boundary layer at the left endpoint, forming region B3. Similarly, region B4 arises from the right endpoint-wavefront intersection. As time evolves, the three inner layers intersect to form a transient overlap region (region IN4), which eventually coalesce into a single steady-state inner layer (region IN5).
• Figure 5: Illustrative sketch showing $k^+$ and $k^-$ against $s$. Monotonicity with respect to $n$ imply that $k^+$ and $k^-$ must intersect at a unique interior point in the continuum state space. At this point, the advection term in \ref{['eq:secularity_condition_outer']} is zero and the PDE becomes degenerate.