On the relationship between serial interval, infectiousness profile and generation time

Abstract

The timing of transmission plays a key role in the dynamics and controllability of an epidemic. However, observing generation times-the time interval between the infection of an infector and an infectee in a transmission pair-requires data on infection times, which are generally unknown. The timing of symptom onset is more easily observed; generation times are therefore often estimated based on serial intervals-the time interval between symptom onset of an infector and an infectee. This estimation follows one of two approaches: (i) approximating the generation time distribution by the serial interval distribution or (ii) deriving the generation time distribution from the serial interval and incubation period-the time interval between infection and symptom onset in a single individual-distributions. These two approaches make different-and not always explicitly stated-assumptions about the relationship between infectiousness and symptoms, resulting in different generation time distributions with the same mean but unequal variances. Here, we clarify the assumptions that each approach makes and show that neither set of assumptions is plausible for most pathogens. However, the variances of the generation time distribution derived under each assumption can reasonably be considered as upper (approximation with serial interval) and lower (derivation from serial interval) bounds. Thus, we suggest a pragmatic solution is to use both approaches and treat these as edge cases in downstream analysis. We discuss the impact of the variance of the generation time distribution on the controllability of an epidemic through strategies based on contact tracing, and we show that underestimating this variance is likely to overestimate controllability.

Keywords: SARS-CoV-2; contact tracing; epidemiology; generation time; infectiousness; modelling.

Figure 1.

A schematic of how the assumptions about infectiousness and symptoms affect the relationship between the serial interval and generation time distributions. (a) Definitions of: serial interval S_ij, time from symptom onset of infector i to symptom onset of infectee j; generation time G_ij, time from infection of i to infection of j; incubation time I_i, time from infection of i to symptom onset of i; and P_ij, time from symptom onset of i to infection of j. (b) Illustration of how infectiousness relates to the point of infection and onset of symptoms under the two different assumptions. Under assumption 1 (P_ij and I_i independent), the infectiousness is fixed with reference to symptom onset. Under assumption 2 (G_ij and I_i independent), the infectiousness is fixed with reference to the point of infection. (c) The relationship between the generation time distribution, the infectiousness profile and the serial interval distribution under assumptions 1 and 2.

Figure 2.

A schematic showing the impact of functional form and variance on the timing of onward transmission. The plots show cumulative generation time distributions, i.e. the proportion of transmission occurring within x days of infection. All distributions have a mean of 5 days. The illustrated variances correspond to standard deviations of 5.0, 4.1, 3 and 1 days. Note that lognormal (a) and gamma (b) distributions have support on (0, ∞), implying an infinite infectious period, which is not correct. However, in practice, this is an acceptable approximation when the probability density in the tail of the distribution is very low.