Inferring generation-interval distributions from contact-tracing data

Abstract

Generation intervals, defined as the time between when an individual is infected and when that individual infects another person, link two key quantities that describe an epidemic: the initial reproductive number, [Formula: see text], and the initial rate of exponential growth, r. Generation intervals can be measured through contact tracing by identifying who infected whom. We study how realized intervals differ from 'intrinsic' intervals that describe individual-level infectiousness and identify both spatial and temporal effects, including truncating (due to observation time), and the effects of susceptible depletion at various spatial scales. Early in an epidemic, we expect the variation in the realized generation intervals to be mainly driven by truncation and by the population structure near the source of disease spread; we predict that correcting realized intervals for the effect of temporal truncation but not for spatial effects will provide the initial forward generation-interval distribution, which is spatially informed and correctly links r and [Formula: see text]. We develop and test statistical methods for temporal corrections of generation intervals, and confirm our prediction using individual-based simulations on an empirical network.

Keywords: basic reproductive number; contact tracing; generation interval; infectious disease modelling; population structure.

Figure 1.

Comparison of individual- and population-level kernels. (a) An individual-level kernel of an infected individual with a latent period of 11.4 days followed by an infectious period of 5 days; this represents an individual realization of a random process. (b) A population-level kernel of infected individuals with latent and infectious periods exponentially distributed with means of 11.4 days and 5 days, respectively; this represents a population average of a random process. Shaded areas under the curves are equal to individual- and population-level reproductive numbers, both of which are set to 2 in this example. Parameters are chosen to reflect the West African Ebola outbreak [25].

Figure 2.

Temporal variation in the mean backward and aggregated generation interval. A deterministic susceptible–exposed–infectious–recovered (SEIR) model is simulated using Ebola-like parameters [25]: mean latent period 1/σ = 11.4 days, mean infectious period 1/γ = 5 days and the basic reproductive number ${\mathcal{R}}_0=2$. The mean backward and aggregated generation intervals are calculated over the course of an epidemic. The dotted horizontal line represents the mean intrinsic generation interval.

Figure 3.

Spatial effects on realized generation intervals. Theoretical distributions and means are shown in colour (and are the same in each panel, for reference). Simulated distributions and means are shown in black. (a) The intrinsic generation-interval distribution corresponds to all infectious contacts by a focal individual, regardless of whether the contact results in infection. (b) The egocentric generation-interval distribution corresponds to the distribution of all infectious contacts by the focal individual with susceptible individuals, in the case where the focal individual is the only possible infector (simulated on a star network). (c) Realized generation-interval distributions have a shorter mean than egocentric distributions in general, because contacts can be wasted when susceptibles become infected through other routes (simulated on a homogeneous network). All figures were generated using 5000 stochastic simulations on a network with five nodes (one infector and four susceptibles) with Ebola-like parameters [25]: mean latent period 1/σ = 11.4 days and mean infectious period 1/γ = 5 days. Per-pair contact rate λ = 0.25 days⁻¹ is chosen to be sufficiently high so that the differences between generation-interval distributions are clear. Each simulation is run until all individuals are either susceptible or have recovered.

Figure 4.

A summary of spatiotemporal effects on generation intervals. The intrinsic generation-interval distribution describes the expected time distribution of infectious contacts made by a primary case. The realized (forward) generation-interval distribution describes the time between actual infection events—egocentric, local and global depletion of the susceptible pool reduces the mean realized generation intervals. The aggregated generation-interval distribution reflects the contact-tracing process during an ongoing epidemic and is subject to right truncation bias. Our goal is to estimate the initial forward generation-interval distribution from the initial aggregated generation-interval distribution, measured during the exponential growth period.

Figure 5.

Comparison of estimates of reproductive number based on various methods. Using the aggregated generation-interval distributions (based on the first 1000 realized generation intervals) without correcting for right-truncation (labelled as ‘contact tracing’) severely underestimates the reproductive number. Similarly, using the intrinsic generation-interval distribution overestimates the reproductive number because it fails to account for local spatial effects; the egocentric distribution corrects for this only partially. Both population-level and individual-level methods provide estimates of reproductive number that are consistent with the empirical estimates, which we define as the average number of secondary cases generated by the first 75 infected individuals, as well as the estimates based on the initial forward generation intervals, which are calculated by applying the Euler-Lotka equation to the realized generation intervals of all infections caused by the first 75 infected individuals. Boxplots are generated using 100 stochastic simulations of the SEIR model on an empirical network using Ebola-like parameters [25]: mean latent period 1/σ = 11.4 days and mean infectious period 1/γ = 5 days. Per-pair contact rate λ = 0.08 days⁻¹ is chosen to be sufficiently high such that differences are clear.