  The effective reproduction number, R(t), is a central point in the study of
infectious diseases. It establishes in an explicit way the extent of an
epidemic spread process in a population. The current estimation methods for the
time evolution of R(t), using incidence data, rely on the generation interval
distribution, g(\tau), which is usually obtained from empirical data or already
known distributions from the literature. However, there are systems, especially
highly heterogeneous ones, in which there is a lack of data and an adequate
methodology to obtain g(\tau). In this work, we use mathematical models to
bridge this gap. We present a general methodology for obtaining an explicit
expression of the reproduction numbers and the generation interval
distributions provided by an arbitrary compartmental model. Additionally, we
present the appropriate expressions to evaluate those reproduction numbers
using incidence data. To highlight the relevance of such methodology, we apply
it to the spread of Covid-19 in municipalities of the state of Rio de janeiro,
Brazil. Using two meta-population models, we estimate the reproduction numbers
and the contributions of each municipality in the generation of cases in all
others. Our results point out the importance of mathematical modelling to
provide epidemiological meaning of the available data.
