Estimating the effective reproduction number for heterogeneous models using incidence data

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.