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

TL;DR

This paper proves that a pseudospectral numerical method for approximating reproduction numbers in age-structured models converges, with convergence rate depending on model regularity, and confirms results through experiments and epidemiological applications.

Contribution

The authors rigorously establish convergence of a new pseudospectral method for age-structured population models, linking spectral radius approximation to true reproduction numbers.

Findings

Spectral radius of the discretized operator converges to the true reproduction number.

Eigenvector approximations converge to the true eigenfunction.

Convergence order depends on the regularity of model coefficients.

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.