Estimating effective reproduction number using generation time versus serial interval, with application to covid-19 in the Greater Toronto Area, Canada

Abstract

Background: The effective reproduction number R _e (t) is a critical measure of epidemic potential. R _e (t) can be calculated in near real time using an incidence time series and the generation time distribution: the time between infection events in an infector-infectee pair. In calculating R _e (t), the generation time distribution is often approximated by the serial interval distribution: the time between symptom onset in an infector-infectee pair. However, while generation time must be positive by definition, serial interval can be negative if transmission can occur before symptoms, such as in covid-19, rendering such an approximation improper in some contexts.

Methods: We developed a method to infer the generation time distribution from parametric definitions of the serial interval and incubation period distributions. We then compared estimates of R _e (t) for covid-19 in the Greater Toronto Area of Canada using: negative-permitting versus non-negative serial interval distributions, versus the inferred generation time distribution.

Results: We estimated the generation time of covid-19 to be Gamma-distributed with mean 3.99 and standard deviation 2.96 days. Relative to the generation time distribution, non-negative serial interval distribution caused overestimation of R _e (t) due to larger mean, while negative-permitting serial interval distribution caused underestimation of R _e (t) due to larger variance.

Implications: Approximation of the generation time distribution of covid-19 with non-negative or negative-permitting serial interval distributions when calculating R _e (t) may result in over or underestimation of transmission potential, respectively.

Keywords: COVID-19; Generation time; Incubation period; Reproduction number; Serial interval.

Fig. 1

Random variables involved in the serial interval Notation — i: infector index; i + 1: infectee index; f_i: time of infection; y_i: time of symptom onset; G(τ): generation time distribution; H(τ): incubation time distribution; S(τ): serial interval distribution.

Fig. 2

Recovered generation time distribution $\widehat{G}(\tau |{\theta }^{\ast })$ based on MLE approximation of the serial interval distribution S(τ) by $\widehat{S}(\tau |{\theta }^{\ast })$ and the incubation period distribution H(τ).

Fig. 3

R_e(t) of covid-19 in GTA using serial interval versus generation timeNotation — S(τ): serial interval; G(τ): generation time; [NP]: negative-permitting; [NN]: non-negative. See Figure A2 for zoom-in of later dates.

Fig. A.1

Illustration of reported serial interval and generation time distributions used for calculating R_e(t) in covid-19.

Fig. A.2

R_e(t) of covid-19 in GTA using serial interval versus generation time (zoom of March 30 to May 4) Notation — S(τ): serial interval; G(τ): generation time; [NP]: negative-permitting; [NN]: non-negative.