Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2020 Jul;17(168):20200144.
doi: 10.1098/rsif.2020.0144. Epub 2020 Jul 22.

Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: framework and applications to the novel coronavirus (SARS-CoV-2) outbreak

Affiliations

Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: framework and applications to the novel coronavirus (SARS-CoV-2) outbreak

Sang Woo Park et al. J R Soc Interface. 2020 Jul.

Abstract

A novel coronavirus (SARS-CoV-2) emerged as a global threat in December 2019. As the epidemic progresses, disease modellers continue to focus on estimating the basic reproductive number [Formula: see text]-the average number of secondary cases caused by a primary case in an otherwise susceptible population. The modelling approaches and resulting estimates of [Formula: see text] during the beginning of the outbreak vary widely, despite relying on similar data sources. Here, we present a statistical framework for comparing and combining different estimates of [Formula: see text] across a wide range of models by decomposing the basic reproductive number into three key quantities: the exponential growth rate, the mean generation interval and the generation-interval dispersion. We apply our framework to early estimates of [Formula: see text] for the SARS-CoV-2 outbreak, showing that many [Formula: see text] estimates are overly confident. Our results emphasize the importance of propagating uncertainties in all components of [Formula: see text], including the shape of the generation-interval distribution, in efforts to estimate [Formula: see text] at the outset of an epidemic.

Keywords: Bayesian multilevel model; COVID-19; SARS-CoV-2; basic reproductive number; generation interval; novel coronavirus.

PubMed Disclaimer

Conflict of interest statement

We declare no competing interests.

Figures

Figure 1.
Figure 1.
Comparisons of the reported parameter values with our pooled estimates. We inferred point estimates (black), uniform distributions (orange) or confidence/credible intervals (purple) for each parameter from each study, and combined them into pooled estimates using a Bayesian multilevel model (red). Points represent medians calculated from the parameter set (ri,G¯i,κi) for each study i (orange and purple). Error bars represent 95% equi-tailed quantiles calculated from the parameter set (ri,G¯i,κi) for each study i. Red density plots represent distributions of 2000 posterior samples. Open triangle: we assumed κ = 0.5 for study 2, which does not report generation-interval assumptions.
Figure 2.
Figure 2.
Effects of the exponential growth rate r, mean generation interval G¯ and generation-interval dispersion κ on the estimates of the basic reproductive number R0. We compare estimates of R0 under nine scenarios that propagate different parameter uncertainties (a) based on our pooled estimates (μr, μG and μκ) and (b) assuming a fourfold reduction in uncertainty of our pooled estimate of the exponential growth rate (using μ^r=(μr+3×median(μr))/4 instead of μr). Each uncertainty type represents R0 estimates based the posterior distributions of one of three parameters (μr, μG and μκ) while using median estimates of two other parameters. The ‘none’ type represents R0 estimate based on the median estimates of μr, μG and μκ. The ‘all’ type represents R0 estimates based on the joint posterior distributions of μr, μG and μκ (also corresponds to Rpool). Points represent the median estimates. Vertical error bars represent the 95% credible intervals.
Figure 3.
Figure 3.
Sensitivity of the reported R0 estimates with respect to our pooled estimates of the underlying parameters. We calculate substitute estimates by replacing the reported parameter values (growth rate r, mean generation interval G¯ and generation-interval dispersion κ) with our corresponding pooled estimates (μr, μG and μκ) one at a time and recalculating R0. The pooled estimate represents Rpool, which is calculated from the joint posterior distribution of μr, μG and μκ; this corresponds to replacing all reported parameter values with our pooled estimates, which gives identical results across all studies. The reported estimates refer to estimates listed in table 1. Points represent the medians of the reported, base, substitute and pooled estimates. Vertical error bars represent the 95% credible intervals of our base, substitute and pooled estimates (based on 2000 posterior samples). Horizontal dashed lines represent the 95% credible intervals of our pooled estimate.
Figure 4.
Figure 4.
Trace plots of the multilevel model. Each chain is represented by a different colour.
Figure 5.
Figure 5.
Marginal posterior distributions of the multilevel model. Each chain is represented by a different colour.

References

    1. World Health Organization 2020. Pneumonia of unknown cause—China. www.who.int/csr/don/05-january-2020-pneumonia-of-unkown-cause-china/en/ (accessed 30 January 2020).
    1. Andersen KG, Rambaut A, Lipkin WI, Holmes EC, Garry RF. 2020. The proximal origin of SARS-CoV-2. Nat. Med. 26, 450–452. (10.1038/s41591-020-0820-9) - DOI - PMC - PubMed
    1. He X. et al. 2020. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med. 26, 672–675. - PubMed
    1. World Health Organization 2020. Coronavirus disease 2019 (COVID-19) situation report – 112. www.who.int/docs/default-source/coronaviruse/situation-reports/20200511-... (accessed 11 May 2020).
    1. Bedford T, Neher R, Hadfield J, Hodcroft E, Ilcisin M, Müller N. 2020. Genomic analysis of nCoV spread. Situation report 2020-01-23. https://nextstrain.org/narratives/ncov/sit-rep/2020-01-23 (accessed 24 January 2020).

Publication types

Grants and funding