On the basic reproduction number in continuously structured populations

In the framework of population dynamics, the basic reproduction number (Formula presented.) is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments, it is calculated as the spectral radius of the so-called next-generation operator. In continuously structured populations defined in a Banach lattice X with concentrated states at birth, one cannot define the next-generation operator in X. In the present paper, we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which (Formula presented.) can be computed as the spectral radius of the next-generation operator. We apply these results to some examples: the (classical) size-dependent model, a size-structured cell population model, a size-structured model with diffusion in structure space (under some particular assumptions), and a (physiological) age-structured model with diffusion in structure space.