Practical considerations for measuring the effective reproductive number, Rt - PubMed (original) (raw)

. 2020 Dec 10;16(12):e1008409.

doi: 10.1371/journal.pcbi.1008409. eCollection 2020 Dec.

Lauren McGough 1, Edward B Baskerville 1, Sam Abbott 2, Keya Joshi 3, Christine Tedijanto 3, Rebecca Kahn 3, Rene Niehus 3, James A Hay 3, Pablo M De Salazar 3, Joel Hellewell 2, Sophie Meakin 2, James D Munday 2, Nikos I Bosse 2, Katharine Sherrat 2, Robin N Thompson 2 4, Laura F White 5, Jana S Huisman 6 7, Jérémie Scire 7 8, Sebastian Bonhoeffer 6, Tanja Stadler 7 8, Jacco Wallinga 9 10, Sebastian Funk 2, Marc Lipsitch 3, Sarah Cobey 1

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Practical considerations for measuring the effective reproductive number, Rt

Katelyn M Gostic et al. PLoS Comput Biol. 2020.

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Abstract

Estimation of the effective reproductive number Rt is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using Rt to assess the effectiveness of interventions and to inform policy. However, estimation of Rt from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of Rt, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting Rt estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in Rt estimation.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1

Fig 1. Instantaneous reproductive number as estimated by the method of Cori et al. vs. cohort reproductive number estimated by Wallinga and Teunis.

For each definition of R t, arrows show the times at which infectors (upwards) and their infectees (downwards) appear in the data. Curves show the generation interval distribution (A, B), or serial interval distribution (C). (A) The instantaneous reproductive number quantifies the number of new infections incident at a single point in time (_t_i, blue arrow), relative to the number of infections in the previous generation (green arrows) and their current infectiousness (green curve). The methods of Cori et al. and of Bettencourt and Ribeiro estimate the case reproductive number. This figure illustrates the Cori method. (B-C) The case reproductive number is defined as the average number of new infections that an individual who becomes infected on day _t_i (green arrows in B) or symptomatic on day _t_s (yellow arrows in C) will eventually go on to cause (blue downward arrows in B and C). The first definition applies when estimating the case reproductive number using inferred times of infection, and the second applies when using data on times of symptom onset. The method of Wallinga and Teunis estimates the case reproductive number.

Fig 2

Fig 2. Accuracy of R t estimation methods given ideal, synthetic data.

Solid and dashed black lines show the instantaneous and case reproductive numbers, respectively, calculated from synthetic data. Colored lines show estimates and confidence or credible intervals. To mimic an epidemic progressing in real time, the time series of infections or symptom onset events up to t = 150 was input into each estimation method (inset). Terminating the time series while R t is falling or rising produces similar results as in S1 Fig. (A) By assuming an SIR model (rather than SEIR, the source of the synthetic data), the method of Bettencourt and Ribeiro systematically underestimates R t when the true value is substantially higher than 1. The method is also biased as transmission shifts. (B) The Cori method accurately measures the instantaneous reproductive number. (C) The Wallinga and Teunis method estimates the cohort reproductive number, which incorporates future changes in transmission. Thus, the method produces R t estimates that lead the instantaneous effective reproductive number and becomes unreliable for real-time estimation at the end of the observed time series without adjustment for right truncation [4,29]. In panels A and B, the colored lines show the posterior mean and the shaded region the 95% credible interval. In panel C, the colored line shows the maximum likelihood estimate and the shaded region the 95% confidence interval.

Fig 3

Fig 3

Biases from misspecification of the generation interval mean (A) or variance (B). Demonstrated using the method of Cori et al. GI, generation interval.

Fig 4

Fig 4. R t is a measure of transmission at time t.

Observations after time t must be adjusted. ICU, intensive care unit.

Fig 5

Fig 5. Pitfalls of simple methods to adjust for delays to observation when estimating R t.

Infections back calculated from (A) observed cases or (B) observed deaths either by shifting the observed curve back in time by the mean observation delay (shift), by subtracting a random sample from the delay distribution from each individual time of observation (convolve), or by deconvolution (deconvolve). Only the deconvolved time series is adjusted for right truncation. Deconvolution most accurately recovers peaks or valleys in the true infection curve. Shifting is less accurate, and convolution is least accurate. Errors from back-calculation increase with the variance of the delay distribution (B vs. A). (C) Posterior mean and credible interval of R t estimates from the Cori et al. method. Inaccuracies in the inferred incidence curves affect R t estimates, especially when R t is changing (here, R t was estimated using shifted values from panels A and B). Finally, we note that shifting the observed curves back in time without adjustment for right truncation leads to a gap between the last date in the inferred time series of infection and the last date in the observed data, as shown by the dashed lines and horizontal arrows in panels A–C.

Fig 6

Fig 6. Accuracy of R t estimates given smoothing window width and location of t within the smoothing window.

Estimates were obtained using synthetic data drawn from the S_→_E transition of a stochastic SEIR model (inset) as an input to the method of Cori et al. Colored estimates show the posterior mean and 95% credible interval. Black line shows the exact instantaneous R t calculated from synthetic data.

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References

    1. Pan A, Liu L, Wang C, Guo H, Hao X, Wang Q et al. Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan. JAMA. 2020;323(19):1915 10.1001/jama.2020.6130 - DOI - PMC - PubMed
    1. Sciré J, Nadeau SA, Vaughan TG, Gavin B, Fuchs S, Sommer J et al. Reproductive number of the COVID-19 epidemic in Switzerland with a focus on the Cantons of Basel-Stadt and Basel-Landschaft. Swiss Med Wkly. 2020;150(19–20):w20271 10.4414/smw.2020.20271 - DOI - PubMed
    1. Kucharski AJ, Russell TW, Diamond C, Liu Y, Edmunds J, Funk S et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study. Lancet Infect Dis. 2020. 10.1016/S1473-3099(20)30144-4 - DOI - PMC - PubMed
    1. Cauchemez S, Boëlle PY, Donnelly CA, Ferguson NM, Guy T, Leung GM et al. Real-time Estimates in Early Detection of SARS. Emerg Infect Dis. 2006;12(1):110–3. 10.3201/eid1201.050593 - DOI - PMC - PubMed
    1. Flaxman S, Mishra S, Gandy A, et al. Estimating the number of infections and the impact of nonpharmaceutical interventions on COVID-19 in 11 European countries. 2020. Available from: 10.25561/77731. - DOI

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