On the Numerical Computation of \({R_0}\) in Periodic Environments

Abstract. We propose a novel approach to approximate the basic reproduction number [Formula: see text] as the spectral radius of the Next-Generation Operator in time-periodic population models by characterizing the latter via evolution semigroups. Once birth/infection and transition operators are identified, we discretize them via either Fourier or Chebyshev collocation methods. Then [Formula: see text] is obtained by solving a generalized matrix eigenvalue problem. The order of convergence of the approximating reproduction numbers to the true one is shown to depend on the regularity of the model coefficients, and spectral accuracy is proved. We validate the theoretical results by discussing applications to epidemiology, viz. a large-size multigroup epidemic model with periodic contact rates, and a vector-borne disease model with seasonal vector recruitment. We illustrate how the method facilitates implementation compared to existing approaches and how it can be easily adapted to also compute type-reproduction numbers. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.com/SIMONE.DEREGGI/R0_periodicODEs and in the supplementary materials ( R0_periodicODEs-main.zip [14.9KB]). [Formula: see text]