Approximating reproduction numbers: a general numerical approach for age-structured models

Reproduction numbers play a fundamental role in population dynamics. For age-structured models, these quantities are typically defined as spectral radius of operators acting on infinite dimensional spaces. As a result, their analytical computation is hardly achievable without additional assumptions on the model coefficients (e.g., separability of age-specific transmission rates) and numerical approximations are needed. In this paper we introduce a general numerical approach, based on pseudospectral collocation of the relevant operators, for approximating the reproduction numbers of a class of age-structured models with finite life span. To our knowledge, this is the first numerical method that allows complete flexibility in the choice of the ``birth'' and ``transition'' processes, which is made possible by working with an equivalent problem for the integrated state. We discuss applications to epidemic models with continuous rates, as well as models with piecewise continuous rates estimated from real data, illustrating how the method can compute different reproduction numbers-including the basic and the type reproduction number as special cases-by considering different interpretations of the age variable (e.g., chronological age, infection age, disease age) and the transmission terms (e.g., horizontal and vertical transmission).