Realized generation times: contraction and impact of infectious period, reproduction number and population size

Andrea Torneri; Amin Azmon; Christel Faes; Eben Kenah; Gianpaolo Scalia Tomba; Jacco Wallinga; Niel Hens

In infectious disease epidemiology, mathematical models are increasingly being used to 2 study the transmission dynamics of infectious agents in a population and thereby 3 providing fundamental tools for developing control policies. An optimal control strategy 4 is based on an appropriate prediction model that in turn requires reliable estimates of 5 the key epidemic parameters. 6 Most research has focused on the 'basic reproduction number', R 0 , which is defined 7 as the expected number of secondary cases resulting from introducing a typical infected 8 person into an entirely susceptible population [2] . The inference of its value in the 9 ascending phase of an epidemic is based either explicitly or implicitly on assumptions 10 about the generation interval distribution [3] . 11 The generation interval, or generation time, is defined to be the time interval 12 between the infection time of an infectee and the infection time of its infector [4] . 13 Generation times are lengths of time intervals and thus there is not a unequivocal 14 procedure to define their dependence on a precise calendar time t. To account for the 15 evolution over time a choice has to be made weather considering generations from the 16 infectee or infector point of view. In the former case the time coordinate refers to the 17 time that has evolved since the infector of an infected person was infected. This is 18 called 'backward', or 'period', generation interval. In the latter case, known as 'forward', 19 or 'cohort', generation interval the average time required to infect another individual is 20 recorded [5, 6] . Considered in the forward scheme, the generation interval distributions 21 is commonly used to estimate infectious disease parameters such as the basic 22 reproduction number [6] [7] [8] [9] . 23 More ambiguity arises in the estimation of the mean generation time because actual 24 data often concern the onset of clinical symptoms rather than the time of infection. symptoms for an infectee appear prior to that of its infector [11] . Note that the serial 30 interval is only defined for symptomatic individuals; an issue that we will not discuss 31 here. 32 Statistical development led to approaches for the estimation of the generation time 33 distribution [7, 12, 13] or jointly of the basic reproduction number and the generation 34 time distribution [14] [15] [16] . The usefulness of the aforementioned approaches has been 35 demonstrated in the analysis of epidemic data during e.g. SARS outbreaks and the 36 pandemic influenza A(H1N1)V2009 outbreak [17] [18] [19] . Most of these estimation methods 37 assume the generation or serial time distribution to remain constant during the 38 epidemic. However, several authors described a non-constant evolution over time for 39 both backward and forward generation interval [5, 6, 8, 20] . In the former case as the 40 epidemic evolves the generation time increases while in the latter case the generation 41 time contracts reaching a minimum approximately at the peak of the outbreak [6, 8] . 42 We will refer to this phenomenon in the forward scheme as 'contraction' to stress the 43 particular shape that the mean generation time assumes over time. The non-constant 44 evolution of the generation interval has stimulated a search for different approaches to 45 estimate the reproduction number that avoid assuming a constant generation intervals 46 distribution through time [8, 9, 13] . Kenah et al. (2008) proposed an hazard-b