Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations

Dimitri Breda; Toshikazu Kuniya; Jordi Ripoll; Rossana Vermiglio

doi:10.1007/s10915-020-01339-1

Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations

J Sci Comput. 2020;85(2):40. doi: 10.1007/s10915-020-01339-1. Epub 2020 Oct 31.

Authors

Dimitri Breda¹, Toshikazu Kuniya², Jordi Ripoll³, Rossana Vermiglio¹

Affiliations

¹ CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle scienze 206, 33100 Udine, Italy.
² Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501 Japan.
³ Department of Computer Science, Applied Mathematics and Statistics, University of Girona, Campus Montilivi, 17003 Girona, Spain.

Abstract

We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.

Keywords: Basic reproduction number; Next-generation operator; Pseudospectral collocation; Spectral approximation; Spectral radius; Stability analysis of equilibria; Structured population dynamics.