Reproduction numbers for epidemiological models with realistic age structure (RAS models)

Abstract The dynamics of childhood diseases are strongly influenced by children’s contact patterns, particularly those shaped by school term schedules and classroom structures. Class compositions often remain stable for multiple school years, with new cohorts entering annually. To capture these dynamics, researchers have developed realistic age-structured (RAS) epidemiological models. Despite their widespread use, RAS models lack a formal definition of the basic reproduction number ( $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> ) and a standardized method for its computation. These models may incorporate seasonal forcing aligned with school terms, as well as discontinuities resulting from annual grade transitions. In this work, we first demonstrate that recently developed $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> theory can accommodate the hybrid features of RAS models. We define the basic reproduction number as the spectral radius of a specifically constructed operator and prove its role as a threshold parameter for the stability of the disease-free solution. Furthermore, we present a practical numerical method for calculating $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . To illustrate the implications of age structuring, we analyze a two-age-group SIR model and show that distinct modeling choices can produce significantly different reproduction numbers, demonstrating how different modeling approaches can yield significantly varied reproduction numbers. Finally, we apply our methodology to a fitted RAS model of measles transmission in the UK, highlighting its utility and relevance for public health planning.