On the convergence of the pseudospectral approximation of reproduction numbers for age-structured models

TL;DR

The paper addresses the numerical computation of reproduction numbers in age-structured models with finite age span by introducing an operator-based reformulation on the space of absolutely continuous functions and applying Chebyshev pseudospectral discretization to the associated Next Generation operator $\mathcal{H}$. It proves norm convergence of the discretized operator $\widehat{\mathcal{H}}_N$ to the continuous operator $\mathcal{H}$ and establishes eigenvalue/eigenfunction convergence rates that depend on the regularity of model coefficients, supported by explicit invertibility results for the discretized transition operator. The authors provide detailed implementation guidance and validate the approach on epidemiological models, including SIR- and HBV-like systems, demonstrating spectral or high-order convergence under smooth coefficients and robust performance with piecewise data. Overall, the framework offers a rigorous, flexible tool for threshold analysis in age-structured dynamics and supports data-informed decision-making by reliably computing $R$, $R_0$, or $T$ in complex settings.

Abstract

We rigorously investigate the convergence of a new numerical method, recently proposed by the authors, to approximate the reproduction numbers of a large class of age-structured population models with finite age span. The method consists in reformulating the problem on a space of absolutely continuous functions via an integral mapping. For any chosen splitting into birth and transition processes, we first define an operator that maps a generation to the next one (corresponding to the Next Generation Operator in the case of R0). Then, we approximate the infinite-dimensional operator with a matrix using pseudospectral discretization. In this paper, we prove that the spectral radius of the resulting matrix converges to the true reproduction number, and the (interpolation of the) corresponding eigenvector converges to the associated eigenfunction, with convergence order that depends on the regularity of the model coefficients. Results are confirmed experimentally and applications to epidemiology are discussed.

Figures (7)

• Figure 6.1: \ref{['Example 1']} with $q(a)=e^{-2a}$ (analytic, left) and $q(a)=e^{-(x-0.5)^{-2}}(x-0.5)^{-2}\chi_{[0.5, a^\dagger]}(a)$ ($C^\infty$, right), with $a^\dagger = \gamma =1$. Log-log plot for increasing $N$ of the approximation error on $R$ (white dots for the $N+1$ Chebyshev extrema and grey dots for the $N$ Chebyshev zeros extended with the left endpoint) and the error $|(\mathcal{H}-\widehat{\mathcal{H}}_N)\psi_{|_{a=0}}|_{\mathbb{C}^d}+\|((\mathcal{H}-\widehat{\mathcal{H}}_N)\psi)'\|_\infty$ for the discretization at the Chebyshev zeros extended with the left endpoint (black dots). Infinite order of convergence is observed in both panels, in agreement with \ref{['corollbound']}.
• Figure 6.2: \ref{['Example 1']} with $q(a)=(0.5-a)^2|0.5-a|$ ($W^{3, \infty}$) and $a^\dagger = \gamma =1$. Left panel: log-log plot for increasing $N$ of the approximation error on $R$ (white dots for the $N+1$ Chebyshev extrema and grey dots for the $N$ Chebyshev zeros extended with the left endpoint) computed as $\rho(\mathcal{H}_N)$, and the error $|(\mathcal{H}-\widehat{\mathcal{H}}_N)\psi_{|_{a=0}}|_{\mathbb{C}^d}+\|((\mathcal{H}-\widehat{\mathcal{H}}_N)\psi)'\|_\infty$ for the discretization at the Chebyshev zeros extended with the left endpoint (black dots). Convergence of order $4$ and order $3$ is observed for $|R-R_N|$ and $|(\mathcal{H}-\widehat{\mathcal{H}}_N)\psi_{|_{a=0}}|_{\mathbb{C}^d}+\|((\mathcal{H}-\widehat{\mathcal{H}}_N)\psi)'\|_\infty$, respectively. Right panel: log-log plot for increasing $N$ of the approximation error on $R$ (white dots for the $N+1$ Chebyshev extrema and grey dots for the $N$ Chebyshev zeros extended with the left endpoint) computed as $\rho(\mathcal{M}^{-1}_N\mathcal{B}_N)$, and the error $\|((\mathcal{M}-\widehat{\mathcal{M}}^{-1}_N)\mathcal{B}\phi)'\|_\infty$ for the discretization at the Chebyshev zeros extended with the left endpoint. Convergence of order $4$ is observed for all the errors, in agreement with \ref{['remOrder']}. The dashed lines have slope $-4$, while the solid line has slope $-3$.
• Figure 6.3: \ref{['Example 1']} with $q(a)=e^{-2a}$. Log-log plot for increasing $N$ of the approximation error on $R$ (white dots for the $N+1$ Chebyshev extrema and grey dots for the $N$ Chebyshev zeros extended with the left endpoint) varying $a^\dagger$ and $\gamma$.
• Figure 6.4: \ref{['Example 1']} with $q(a)=e^{-2a}$, $a^\dagger =30$ and $\gamma =100$. Log-log plot for increasing $N$ of the piecewise approximation error on $R$ (white dots for the $N+1$ Chebyshev extrema, grey and black dots for the $N$ Chebyshev zeros extended with the left endpoint with $N$ even and $N$ odd, respectively). Note the different behavior of the error for odd and even $N$ for the Chebyshev zeros.
• Figure 6.5: \ref{['Example 2']}. Log-log plot of the approximation error for increasing $N$ on $R=1$ (white dots for the $N+1$ Chebyshev extrema and grey dots for the $N$ Chebyshev zeros extended with the left endpoint), for $k=2$ (left) and $k=\pi$ (right). The slope of the dashed line is $-6.5$.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• proof