A new framework and software to estimate time-varying reproduction numbers during epidemics

Abstract

The quantification of transmissibility during epidemics is essential to designing and adjusting public health responses. Transmissibility can be measured by the reproduction number R, the average number of secondary cases caused by an infected individual. Several methods have been proposed to estimate R over the course of an epidemic; however, they are usually difficult to implement for people without a strong background in statistical modeling. Here, we present a ready-to-use tool for estimating R from incidence time series, which is implemented in popular software including Microsoft Excel (Microsoft Corporation, Redmond, Washington). This tool produces novel, statistically robust analytical estimates of R and incorporates uncertainty in the distribution of the serial interval (the time between the onset of symptoms in a primary case and the onset of symptoms in secondary cases). We applied the method to 5 historical outbreaks; the resulting estimates of R are consistent with those presented in the literature. This tool should help epidemiologists quantify temporal changes in the transmission intensity of future epidemics by using surveillance data.

Keywords: SARS; incidence; influenza; measles; reproduction number; smallpox; software.

Figure 1.

The first row shows daily epidemic curves (from left to right) for measles in Hagelloch, Germany, October 1861–January 1862; pandemic influenza in Baltimore, Maryland, September–November 1918; smallpox in Kosovo, February–April 1972; severe acute respiratory syndrome (SARS) in Hong Kong, February–June 2003; and pandemic influenza in a school in Pennsylvania, April–May 2009. The second row shows daily estimates of the instantaneous reproduction numbers R over sliding weekly windows; the black lines show the posterior medians and the grey zones show the 95% credible intervals; the horizontal dashed lines indicate the threshold value R = 1. The third row shows daily estimates of the case reproduction numbers R^c over sliding weekly windows; the black dots show the mean estimates, and the bars show the 95% confidence intervals; the horizontal dashed lines indicate the threshold value R^c = 1. The fourth row shows the serial interval distributions used for estimation of R and R^c.

Figure 2.

Estimated reproduction number for pandemic influenza in Baltimore, Maryland, September–November 1918. A) Daily epidemic curve; B) daily estimates of the reproduction number R over sliding weekly windows (the black line shows the posterior medians and the grey zones show the 95% credible intervals; the horizontal dashed line indicates the threshold value R = 1); C) histogram of the mean serial intervals explored; and D) histogram of the standard deviations of the serial interval explored.

Figure 3.

Estimated reproduction number for pandemic influenza in Baltimore, Maryland, September–November 1918, with several time windows. A) Daily epidemic curve; B) serial interval distribution; C–F) daily estimates of the reproduction numbers R over 1-day windows; C), over sliding weekly windows; D), over sliding 2-week windows; E) and over sliding 4-week windows; F) black lines show the posterior medians, and grey zones show the 95% credible intervals; the horizontal dashed lines indicate the threshold value R = 1.