Are epidemic growth rates more informative than reproduction numbers?

Abstract

statistics, often derived from simplified models of epidemic spread, inform public health policy in real time. The instantaneous reproduction number, $R_t$ , is predominant among these statistics, measuring the average ability of an infection to multiply. However, $R_t$ encodes no temporal information and is sensitive to modelling assumptions. Consequently, some have proposed the epidemic growth rate, $r_t$ , that is, the rate of change of the log-transformed case incidence, as a more temporally meaningful and model-agnostic policy guide. We examine this assertion, identifying if and when estimates of $r_t$ are more informative than those of $R_t$ . We assess their relative strengths both for learning about pathogen transmission mechanisms and for guiding public health interventions in real time.

Keywords: COVID‐19; epidemic modelling; growth rate; infectious disease; reproduction number; situational awareness.

FIGURE 1

Instantaneous reproduction numbers and growth rates. We simulate a seasonally varying epidemic with incidence $I_t$, according to the renewal model with true transmissibility $R_t$ and serial interval distribution estimated for Ebola virus from Van Kerkhove et al. (2015). In panels A and B, we estimate the instantaneous reproduction number ${\widehat{R}}_t$ (with 95% credible intervals) using EpiFilter (see Parag, 2021) and provide one‐step‐ahead predictions ${\widehat{I}}_t$ using ${\widehat{R}}_t$. In panels C and D we derive three growth rate estimates, ${\widehat{r}}_t$ using: ${\widehat{R}}_t$ (via the Wallinga and Lipsitch, method), a smoothed version of the incidence curve $S_t$ (via SG filters) and the total infectiousness of the epidemic ${\mathrm{\Lambda }}_t$ by treating it as a type of SG filter. The latter two estimates have to be right and left shifted respectively by $\frac{1}{2}\tau$ due to the effects of filtering, with τ ≈ 8 days as the mean generation time or serial interval. We show that a left‐shifted version of the total infectiousness ${\mathrm{\Lambda }}_{t+\tau }$ effectively approximates a smoothed incidence curve.

FIGURE 2

Misspecified estimates of reproduction numbers and growth rates. We repeat the simulation of Figure 1 but our estimates now assume a misspecified Ebola virus generation time distribution. This distribution has a mean that is 33% smaller than the one used to generate the epidemic data (which is from Van Kerkhove et al. 2015). Panel A provides estimates of instantaneous reproduction numbers, ${\widehat{R}}_t$, under the true and misspecified distributions (with 95% credible intervals) using EpiFilter (Parag, 2021). Panel B presents corresponding growth rate estimates (and 95% credible intervals), ${\widehat{r}}_t$, which are derived from the various ${\widehat{R}}_t$ in A (Wallinga & Lipsitch, 2007).