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).

翻译：再生数在种群动力学中具有基础性作用。对于年龄结构模型，这些量通常被定义为作用于无穷维空间算子的谱半径，因此，若无对模型系数（如年龄特异性传播率的可分离性）的额外假设，其解析计算几乎难以实现，故而需要数值近似方法。本文提出一种基于相关算子伪谱配置的通用数值方法，用于近似有限寿命周期年龄结构模型类别的再生数。据我们所知，这是首个允许在“出生”与“转变”过程中实现完全灵活选择的数值方法，其可行性源于对积分状态等价问题的处理。我们讨论了该方法在具有连续速率的流行病模型以及基于真实数据估计的分段连续速率模型中的应用，通过考虑年龄变量（如实足年龄、感染年龄、疾病年龄）与传播项（如水平传播与垂直传播）的不同诠释，阐明了该方法如何计算不同再生数——包括基础再生数与类型再生数作为特例。