Intrinsic and realized generation intervals in infectious-disease transmission

Abstract

The generation interval is the interval between the time when an individual is infected by an infector and the time when this infector was infected. Its distribution underpins estimates of the reproductive number and hence informs public health strategies. Empirical generation-interval distributions are often derived from contact-tracing data. But linking observed generation intervals to the underlying generation interval required for modelling purposes is surprisingly not straightforward, and misspecifications can lead to incorrect estimates of the reproductive number, with the potential to misguide interventions to stop or slow an epidemic. Here, we clarify the theoretical framework for three conceptually different generation-interval distributions: the 'intrinsic' one typically used in mathematical models and the 'forward' and 'backward' ones typically observed from contact-tracing data, looking, respectively, forward or backward in time. We explain how the relationship between these distributions changes as an epidemic progresses and discuss how empirical generation-interval data can be used to correctly inform mathematical models.

Keywords: contact-tracing; epidemiological model; generation interval; reproductive number.

Figure 1.

Illustration of backward and forward generation intervals. (a) Illustration of the example of a primary case (solid circle), infected at time t₀ then infecting three other individuals (open circle), respectively, at times t₁, t₂ and t₃. The generation intervals are defined as G_i = t_i − t₀ for i = 1, 2, 3. (b) Plot of the backward generation intervals (black squares), that is from the infectees' point of view. There is only one backward generation interval per infectee. (c) Plot of the forward generation intervals (black squares) for the primary case. The x-axis represents the infection time of the infector, hence the three forward generation intervals are all defined at time t₀.

Figure 2.

Mean backward generation interval (GI). See main text (§2e) for explanations.

Figure 3.

Mean forward generation interval (GI). See main text (§2e) for explanations.

Figure 4.

Mean generation intervals: theory versus simulations. Numerical validation of forward and backward generation-interval distributions. a(i) The thick line is the mean of the forward generation interval obtained by integrating equation (2.13). The open circles represent the mean of the forward generation intervals from stochastic simulations. The horizontal dashed line depicts the mean intrinsic generation interval. The three squares show the calendar times chosen for the distribution in the second panel. a(ii) Empirical (grey histogram) and theoretical (black line) forward generation-interval distribution at calendar times 5, 40 and 60 days. The solid (resp. dashed) vertical line represents the mean of the theoretical (resp. empirical) distribution. Parts b(i,ii) represent the same quantities as the first and second panels, but for the backward generation interval using equation (2.14). Model parameters: n_E = n_I = 3; mean latency and mean infectious duration both equal 5 days; Monte Carlo iterations = 30; population size = 25 000.

Figure 5.

Erlang SEIR model.

Figure 6.

Comparison between fitting the backward (b) or intrinsic (g) generation interval distribution of an Erlang SEIR model to synthetic data. Model parameters used to generate the data were n_E = n_I = 3; mean latency and mean infectious duration both equal to 5 days; population size at 25 000. (a) Fit to the mean of the backward generation interval distribution. Solid black circles are the simulated backward generation intervals data. The red solid thick curve is the fitted mean backward generation interval b from the Erlang SEIR model to the mean backward generation intervals data. The red dashed thick curve is the fitted mean backward generation interval b when fitting (naively) the intrinsic distribution g from the same Erlang SEIR model to the backward generation intervals data. The thin grey curve is b when using the ‘true’ parameter values that generated the simulated data. (b) Fit to the variance of the backward generation interval distribution. Open circles represent the simulated data. The red thick solid line is the variance of the fitted distribution b when fitting b to the simulated backward generation intervals data. The red thick dashed line represents the variance of the fitted distribution b when (naively) fitting the intrinsic distribution g to the simulated backward generation intervals data. Only the points to the left of the vertical dashed line (at calendar time 50) were used in both fits. An approximate Bayesian computation method with 1000 iterations was performed for both fits. (Online version in colour.)