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A numerical method for the stability analysis of linear age-structured models with nonlocal diffusion

Dimitri Breda, Simone De Reggi, Rossana Vermiglio

TL;DR

The paper tackles stability analysis for linear age-structured population models with nonlocal diffusion, where asymptotic stability is governed by the spectrum of the infinitesimal generator and the semigroup lacks spatial regularization. It develops a numerical method that reformulates the problem via integration of the age-state and then discretizes the generator with a spatial spectral projection plus age pseudospectral collocation, reducing the task to finite-dimensional eigenproblems. A rigorous convergence analysis is provided in the separable-coefficient case, showing that discretized eigenvalues converge to the true spectrum, and extensive numerical experiments validate the approach across Dirichlet and Neumann nonlocal diffusion scenarios. The work enables accurate computation of leading eigenvalues that determine stability, offers practical implementation details, and suggests directions for extending the method to more general coefficients and domain geometries with potential applications to R0 computation and bifurcation analysis in age-structured epidemics.

Abstract

We numerically address the stability analysis of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, models with nonlocal diffusion are more challenging since the associated semigroups have no regularizing properties in the spatial variable. Nevertheless, the asymptotic stability of the null equilibrium is determined by the spectrum of the infinitesimal generator associated to the semigroup. We propose a numerical method to approximate the leading part of this spectrum by first reformulating the problem via integration of the age-state and then by discretizing the generator combining a spectral projection in space with a pseudospectral collocation in age. A rigorous convergence analysis proving spectral accuracy is provided in the case of separable model coefficients. Results are confirmed experimentally and numerical tests are presented also for the more general instance.

A numerical method for the stability analysis of linear age-structured models with nonlocal diffusion

TL;DR

The paper tackles stability analysis for linear age-structured population models with nonlocal diffusion, where asymptotic stability is governed by the spectrum of the infinitesimal generator and the semigroup lacks spatial regularization. It develops a numerical method that reformulates the problem via integration of the age-state and then discretizes the generator with a spatial spectral projection plus age pseudospectral collocation, reducing the task to finite-dimensional eigenproblems. A rigorous convergence analysis is provided in the separable-coefficient case, showing that discretized eigenvalues converge to the true spectrum, and extensive numerical experiments validate the approach across Dirichlet and Neumann nonlocal diffusion scenarios. The work enables accurate computation of leading eigenvalues that determine stability, offers practical implementation details, and suggests directions for extending the method to more general coefficients and domain geometries with potential applications to R0 computation and bifurcation analysis in age-structured epidemics.

Abstract

We numerically address the stability analysis of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, models with nonlocal diffusion are more challenging since the associated semigroups have no regularizing properties in the spatial variable. Nevertheless, the asymptotic stability of the null equilibrium is determined by the spectrum of the infinitesimal generator associated to the semigroup. We propose a numerical method to approximate the leading part of this spectrum by first reformulating the problem via integration of the age-state and then by discretizing the generator combining a spectral projection in space with a pseudospectral collocation in age. A rigorous convergence analysis proving spectral accuracy is provided in the case of separable model coefficients. Results are confirmed experimentally and numerical tests are presented also for the more general instance.
Paper Structure (16 sections, 7 theorems, 80 equations, 6 figures)

This paper contains 16 sections, 7 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Age-structured models with Dirichlet nonlocal diffusion
  3. Abstract setting
  4. Rates independent of the space variable
  5. The numerical approach
  6. Convergence analysis
  7. Convergence in space
  8. Convergence in age
  9. Final convergence result
  10. Implementation
  11. Numerical results
  12. Example 1
  13. Example 2
  14. Example 3
  15. Nonlocal diffusion of Neumann type
  16. ...and 1 more sections

Key Result

Proposition 3.1

$\mathcal{J}^x$ has only real eigenvalues smaller than $1$, say $\{\theta_j\}_{j=0}^{\infty}$ ordered as $1>\theta_0>\theta_1\ge \theta_2\ge\cdots$ and they accumulate at $0$. Moreover, $\theta_0$ is the unique eigenvalue associated to a positive eigenfunction $g_0\in L^2(\Omega,\mathbb{R})$, it is

Figures (6)

  • Figure 7.1: \ref{['ex1par']}: computed spectra in the complex plane with $l=1$ and $d=1$ for $\mathcal{K}^a$ (left), $J^x$ (center) and $\mathcal{B}$ (right) with $N=M=100$.
  • Figure 7.3: \ref{['ex1par']}: computed spectra of $\mathcal{B}$ in the complex plane with $l=1$, $d=1$ and $R=1.5$ (left), $R\approx 0.7277628676660066$ (center) and $R=0.1$ (right) with $N=M=100$.
  • Figure 7.4: \ref{['ex1par']}: level curves for $\lambda_0$ as a function of $(d,l)$ and stability boundary (thick line) with $R=0.7$ and $N=M=30$.
  • Figure 7.5: \ref{['ex2par']}: spectrum of $\mathcal{B}$ in the complex plane computed with $N=M=50$ (left) and convergence diagram for the first two rightmost eigenvalues $\lambda_0$ and $\lambda_1$ (right) with $\beta_0=8$ and $d=l=1$. The reference values $\lambda_0\approx-0.248872934970194$ and $\lambda_1\approx-0.739612643296491$ are obtained with $N=M=100$.
  • Figure 7.8: \ref{['ex2par']}: level curves for $\lambda_0$ as a function of $(d,l)$ and stability boundary (thick line) with $\beta_0=10$ and $N=M=30$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Proposition 3.1
  • Theorem 3.2
  • Theorem 5.1
  • proof
  • Corollary 5.2
  • proof
  • Proposition 5.3
  • proof
  • Theorem 5.4
  • proof
  • ...and 2 more