Calculation of the effective reproduction number

The basic reproduction number [Formula: see text] is a key epidemiological quantity, representing the expected number of new cases obtained from receiving a pathogen from one infected individual in totally susceptible population. Here is studied the effective reproduction number [Formula: see text] as its generalized version, by considering that in the population might be present other infected persons, besides the one introduced, which are absent in the definition of [Formula: see text]. To calculate the theoretical value of the effective reproduction number, we use an integral formula based on the age of infection. The approach is applied to the classical SIR model and the two-strain SIIS model. It is obtained that such theoretical values deviate from the classical values which assume constant susceptible population, if the susceptible compartment size changes substantially during the typical period of infectiousness. Numerical simulations of the stochastic version of the models give values that match very well to those from the integral formula. The findings suggest that in scenarios where the population has nonnegligible number of infected individuals and the change of the compartment sizes is considerable, one has to be cautious in using the classical results. The numerical simulations also show that the shape of the generation interval distribution depends on whether the susceptible population is changing. Thus, the estimates of the generation interval distribution should be considered carefully as well.