Realized generation times: contraction and impact of infectious period, reproduction number and population size
Authors: Torneri Andrea , Azmon Amin , Faes Christel , Kenah Eben , Tomba Gianpaolo Scalia , Wallinga Jacco , Hens Niel 
Published: March 2019

One of the key characteristics of the transmission dynamics of infectious diseases is thegeneration timewhich refers to the time interval between the infection of a secondary case and the infection of its infector. The generation time distribution together with the reproduction number determines the rate at which an infection spreads in a population. When defining the generation time distribution at a calendar timettwo definitions are plausible according whether we regardtas the infection time of the infector or the infection time of the infectee. The resulting measurements are respectively calledforward generation timeandbackward generation time. It has been observed that the mean forward generation time contracts around the peak of an epidemic. This contraction effect has previously been attributed to either competition among potential infectors or depletion of susceptibles in the population. The first explanation requires many infectives for contraction to occur whereas the latter explanation suggests that contraction occurs even when there are few infectives. With a simulation study we show that both competition and depletion cause the mean forward generation time to contract. Our results also reveal that the distribution of the infectious period and the reproduction number have a strong effect on the size and timing of the contraction, as well as on the mean value of the generation time in both forward and backward scheme.

Author summary

Infectious diseases remain one of the greatest threats to human health and commerce, and the analysis of epidemic data is one of the most important applications of statistics in public health. Thus, having reliable estimates of fundamental infectious diseases parameters is critical for public health decision-makers in order to take appropriate actions for the global prevention and management of outbreaks and other health emergencies. A key example is given by the prediction models of the reproduction numbers: these rely on the generation time distribution that is usually estimated from contact tracing data collected at a precise calendar time. The forward scheme is used in such a prediction model and the knowledge of its evolution over time is crucial to correctly estimate the parameters of interest. It is therefore important to characterize the causes that lead to the contraction of the mean forward generation time during the course of an outbreak.

In this paper, we firstly identify the impact of the epidemiological quantities as reproduction number, infectious period and population size on the mean forward and backward generation time. Moreover, we analyze the phenomena of competition among infectives and depletion of susceptible individuals highlighting their effects on the contraction of the mean forward generation time. The upshot of this investigation is that the variance of the infectious period distribution and the reproduction number have a strong impact on the generation times affecting both the mean value and the evolution over time. Furthermore, competition and depletion can both cause contraction even for small values of the reproduction number suggesting that, in epidemic models where the generation time is considered time-inhomogeneous, estimators accounting for both depletion and competing risks are to be preferred in the inference of the generation interval distributions.