Figures
Abstract
The time-varying reproduction number (Rt) is an important measure of epidemic transmissibility that directly informs policy decisions and the optimisation of control measures. EpiEstim is a widely used opensource software tool that uses case incidence and the serial interval (SI, time between symptoms in a case and their infector) to estimate Rt in real-time. The incidence and the SI distribution must be provided at the same temporal resolution, which can limit the applicability of EpiEstim and other similar methods, e.g. for contexts where the time window of incidence reporting is longer than the mean SI. In the EpiEstim R package, we implement an expectation-maximisation algorithm to reconstruct daily incidence from temporally aggregated data, from which Rt can then be estimated. We assess the validity of our method using an extensive simulation study and apply it to COVID-19 and influenza data. For all datasets, the influence of intra-weekly variability in reported data was mitigated by using aggregated weekly data. Rt estimated on weekly sliding windows using incidence reconstructed from weekly data was strongly correlated with estimates from the original daily data. The simulation study revealed that Rt was well estimated in all scenarios and regardless of the temporal aggregation of the data. In the presence of weekend effects, Rt estimates from reconstructed data were more successful at recovering the true value of Rt than those obtained from reported daily data. These results show that this novel method allows Rt to be successfully recovered from aggregated data using a simple approach with very few data requirements. Additionally, by removing administrative noise when daily incidence data are reconstructed, the accuracy of Rt estimates can be improved.
Author summary
EpiEstim is a tool used to estimate the time-varying reproduction number, Rt, using only daily incidence data and the serial interval (SI) distribution—the estimated time between symptom onset in a case and their infector. Rt indicates whether case numbers are rising (Rt > 1) or falling (Rt < 1), with implications for disease control programmes. Frequently, incidence data are not reported daily, and when the SI of a disease is shorter than the temporal aggregation of incidence data, tools such as EpiEstim cannot be applied. Here, a novel method allows Rt to be estimated directly from reconstructed daily incidence data, using only aggregated incidence and the SI distribution. We validate our approach using influenza and COVID-19 data, alongside simulated incidence, to explore numerous epidemic scenarios. Trends in Rt estimated from reported daily incidence data can be recovered from daily incidence reconstructed from aggregated data. When administrative noise (e.g., weekend effects) is added to simulated incidence, Rt estimates from reconstructed data are more accurate, mitigating the impact of prominent ‘noise’ in daily reported incidence. Now implemented in EpiEstim, the method typically takes just a few seconds to run, making the tool applicable to a wider range of epidemic contexts.
Citation: Nash RK, Bhatt S, Cori A, Nouvellet P (2023) Estimating the epidemic reproduction number from temporally aggregated incidence data: A statistical modelling approach and software tool. PLoS Comput Biol 19(8): e1011439. https://doi.org/10.1371/journal.pcbi.1011439
Editor: Eric HY Lau, The University of Hong Kong, CHINA
Received: December 14, 2022; Accepted: August 18, 2023; Published: August 28, 2023
Copyright: © 2023 Nash et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: We provide a GitHub repository containing the data and code to reproduce the analyses (https://github.com/rebeccanash/EM_EpiEstim_Nash2023) and used Zenodo to assign a DOI to the repository (10.5281/zenodo.8091058). The code to apply the method developed is available in the open-source R package EpiEstim: https://github.com/mrc-ide/EpiEstim.
Funding: RKN acknowledges funding from the Medical Research Council (MRC) Doctoral Training Partnership (grant reference MR/N014103/1). AC acknowledges the Academy of Medical Sciences Springboard, funded by the Academy of Medical Sciences, Wellcome Trust, the Department for Business, Energy and Industrial Strategy, the British Heart Foundation, and Diabetes UK (reference SBF005\1044). SB acknowledges support from the Novo Nordisk Foundation via The Novo Nordisk Young Investigator Award (NNF20OC0059309), The Eric and Wendy Schmidt Fund For Strategic Innovation via the Schmidt Polymath Award (G-22-63345), and the Danish National Research Foundation via a chair position. AC and SB acknowledge funding from the National Institute for Health and Care Research (NIHR) Health Protection Research Unit in Modelling and Health Economics, a partnership between the UK Health Security Agency, Imperial College London and LSHTM (grant code NIHR200908), and acknowledge funding from the MRC Centre for Global Infectious Disease Analysis (reference MR/R015600/1), jointly funded by the UK Medical Research Council (MRC) and the UK Foreign, Commonwealth & Development Office (FCDO), under the MRC/FCDO Concordat agreement, and is also part of the EDCTP2 programme supported by the European Union. Disclaimer: The views expressed are those of the author(s) and not necessarily those of the NIHR, UK Health Security Agency or the Department of Health and Social Care. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: I have read the journal’s policy and the authors of this manuscript have the following competing interests: AC has received payment from Pfizer for teaching on a course for mathematical modelling of infectious disease transmission and vaccination.
Introduction
As infectious disease outbreaks become more common, it is increasingly important to rapidly characterise the threat of emerging and re-emerging pathogens [1]. Transmissibility, i.e. a pathogen’s ability to spread through a population, can be quantified using the time-varying reproduction number, Rt, defined as the average number of infections that are caused by a primary case at time t of an outbreak. Rt signals whether an outbreak is growing (Rt > 1) or declining (Rt < 1), and whether current interventions are sufficient to control the spread of the disease.
One of the most popular tools for real-time Rt estimation, the R package EpiEstim, relies on observing the incidence data and supplying an estimated serial interval (SI) distribution—the time between symptom onset in a case and their infector. EpiEstim requires that the SI distribution and incidence data are supplied using the same time units. This can be problematic when daily incidence data is not reported, which is common for many diseases, such as influenza, Zika virus disease, and most notifiable diseases in countries such as the UK and the US [2–5]. Additionally, several studies intentionally aggregate data to reduce the impact of daily reporting variability; administrative noise, such as “weekend effects”, are characterised by a drop in reported cases over weekends, due to reduced care seeking and longer delays in reporting, followed by a peak on Mondays [6,7]. A commonly used workaround is to aggregate the SI distribution to match the frequency of incidence reporting [8,9], however this is not possible if the SI is shorter than the aggregation of data. For example, influenza-like illness is typically reported on a weekly basis, but influenza has an estimated mean SI of 2–4 days [10,11]. Similarly, reporting of COVID-19, which has an estimated SI of 3–7 days, has typically moved from daily to weekly [12,13]. Therefore, enabling estimation of Rt from temporally aggregated data is critical to ensure methods such as EpiEstim are widely applicable [14].
In this study, we combine an expectation-maximisation (EM) algorithm with the renewal equation approach implemented in EpiEstim to reconstruct daily incidence from aggregated data and estimate Rt. We assess the performance of the method using influenza and COVID-19 data, in addition to an extensive simulation study.
Methods
EpiEstim
EpiEstim uses the renewal equation (Eq 1), a form of branching process model [15]. In this formulation, the incidence of new symptomatic cases at time t (It) is approximated by a Poisson process, where It-s is the past incidence, and gs is the probability mass function of the serial interval.
(1)
With EpiEstim, Rt can be assumed to remain constant within user defined time windows, which smooth out estimates.
Extending EpiEstim for coarsely aggregated data
We extended EpiEstim to estimate Rt from aggregated incidence data, where each aggregation window (w) is >1 day, whilst still conditioning on an assumed serial interval distribution (gs). We use an EM algorithm to iteratively reconstruct daily incidence from aggregated data, and in turn estimate Rt. We present the method with weekly data in mind, but the method and software can be applied to any temporal aggregation (Fig 1 and S1 Appendix pp 23–24).
Schematic of the EM algorithm approach used to reconstruct daily incidence (I) from temporally aggregated incidence data (in this case weekly, A). The algorithm is initialised with a naive disaggregation of the weekly incidence (assuming constant daily incidence throughout the aggregation window, left panel). The resulting daily incidence is then used to estimate the reproduction number for each aggregation window, in this case for each week, R* (expectation step, central panel). R* is converted into a growth rate (see Eq 7), which is in turn used to reconstruct daily incidence data, whilst ensuring that if I were to be reaggregated it would still sum to the original weekly totals (maximisation step, right panel). The process cycles between the expectation and maximisation steps until convergence.
We define:
- I = {It}t=1,…,T the vector of unobserved daily incidence,
- A = {Aw}w=1,…,W the vector of observed aggregated incidence, so that for each aggregation window w
the vector of reproduction numbers corresponding to each incidence aggregation window. The * indicates that this is only used in the EM algorithm to reconstruct the daily incidence I, and is distinct from the final estimated R.
We use the following indexes:
- t for days (t = 1, …, T),
- w for aggregation windows (w = 1, …, W),
- i for iterations of the EM algorithm (i = 0, …, 10).
In the following, the bold notation signifies vectors. Our overall goal is to maximise the marginal likelihood function
(2)
This marginal likelihood seeks to compute daily incidence while marginalising over the conditional probability distribution of the reproduction number. Loosely, the statistical goal is to produce a series of reproduction numbers that can reconstruct daily incidence while still being consistent with the observed aggregated incidence. We use an expectation maximisation scheme to approximate this marginal likelihood at low computational cost. The algorithm involves three steps: initialisation, expectation, and maximisation.
Initialisation
The algorithm is initialised (step i = 0) by disaggregating the aggregated incidence by piecewise constant functions (constant daily incidences across each aggregation window). That is, for aggregation window w covering days t = tw−1 + 1, …, tw:
(3)
Note that this allows non-integer incidence counts.
For iteration i ≥ 1 of the algorithm, we iterate over two steps: expectation and maximisation.
Expectation
First, we compute the expectation
(4)
This expectation computes the average reproduction number over each data aggregation window, given the original observed aggregated incidence and the previous (ith) iteration of the estimate of daily incidence. To compute this, we use the renewal approach from EpiEstim where the posterior distribution of reproduction numbers is found analytically as , with
~ Gamma(shape = αw, scale = βw) and where
and
, where a and b are the shape and scale of the Gamma prior distribution for Rw.[15] The expected value for the reproduction number is therefore calculated as:
(5)
Maximisation
The maximisation step consists of recovering the most likely daily incidence from the expected R* i.e. maximising ℙ(I, R*│A), or maximising ℙ(I, R*) ∝ ∏t≥1 ℙ(It│I0, …, It−1, R*), subject to the constraint that daily incidence sums to the aggregated incidence i.e. .
In our renewal equation context, It│I0, …, It−1, R follows a Poisson distribution with mean Rw ∑s It−s gs (where w is such that tw−1 < t ≤ tw), and therefore has mode which we approximate as:
(6)
Wallinga and Lipsitch [16] demonstrate, conditional on the generation time distribution, analytical correspondence between reproduction number R and growth rate r through the link function:
(7)
We therefore assume local exponential growth so that Eq 6 is equivalent to:
(8)
where
is the exponential growth rate over aggregation window w, obtained from
using the link function in Eq 7.
kw is calculated to ensure the sum of daily incidence values adds up to the observed weekly totals:
(9)
We then use the estimate of I from Eq 8 in the maximisation step and iterate, thus completing the algorithm (Eq 5 for the expectation step and Eq 8 for the maximisation step). At this point, I can be used to estimate the full posterior distribution of R over any time window using EpiEstim (hereafter, this final R is referred to as Rt).
Given the rapid computational time and convergence (see S1 Appendix pp 10 and 22), the default number of iterations was set to 10 in the R package. However, a convergence check ensures that the final iteration of the reconstructed daily incidence does not differ from the previous iteration beyond a tolerance of 10−6, and the number of iterations can be modified by the user.
Case studies
We chose datasets where incidence data was available daily, and then artificially aggregated them to weekly counts. Rt was estimated from daily incidence that was reconstructed from weekly aggregated data using our new approach, and compared to Rt estimates obtained from the reported daily incidence using the original EpiEstim R package. All Rt estimates were made using both daily and weekly sliding time windows, and we refer to those estimates as daily Rt estimates and weekly Rt estimates respectively.
We considered three characteristics: 1) mean Rt estimates, 2) uncertainty in the Rt estimates, and 3) the classification of Rt as increasing, uncertain or declining (S1 Appendix pp 8–9). To compare the performance of this approach to the original method, we assessed the correlations between each of the three characteristics when using the reported and reconstructed incidence. For the mean Rt estimates and uncertainty in Rt estimates, we assessed the linear relationships using the Pearson correlation coefficient (where values closer to +1 are indicative of a strong positive correlation).
The gamma distributed priors for R* and Rt were set to a mean and standard deviation of 5 (shape = 1, scale = 5), which is the default prior parameterisation used in EpiEstim. The rationale behind this choice is that it ensures that one will not conclude R < 1 unless the data strongly supports that. The user can set the prior themselves.
Influenza
We obtained a five-week subset of a dataset (11th December 2009—14th January 2010) on US active component military personnel (employed by the military as their full-time occupation) that made an outpatient visit to a permanent military treatment facility describing a respiratory-related illness. This daily incidence by date of presentation at a clinic was originally obtained by Riley et al. from the Armed Forces Health Surveillance Center and were digitally extracted for use here.[17] We used a mean SI of 3.6 days and SD of 1.6 days.[10]
COVID-19
Incidence of UK COVID-19 cases and deaths were taken from the UK government website [18]. For COVID-19 cases, we obtained ninety-seven weeks of data (21st February 2020 to 30th December 2021) for incidence by date of specimen, which is the date that a sample was taken from an individual which later tested positive. For COVID-19 deaths, we used ninety-six weeks of data (2nd March 2020 to 2nd January 2022) for incidence by date of death within twenty-eight days of a positive test. We assumed a mean SI of 6.3 days and SD of 4.2 days [12].
In the S1 Appendix, we also apply the EM algorithm to weekly incidence data for Zika virus disease to assess the performance of the method on a non-respiratory pathogen.
Simulation study
We considered scenarios where Rt either remained constant or varied over time, with a stepwise or gradual change. For each scenario, one hundred seventy-day epidemic trajectories were simulated using a Poisson branching process as implemented in the R package projections [19]. Daily datasets were aggregated weekly and used to estimate Rt using the proposed method; these values were compared to Rt estimates obtained from simulated daily data using the original EpiEstim R package. We explored the impact of weekend effects on Rt estimates, the ability to supply alternative temporal aggregations of data e.g., three-day, ten-day, or two-weekly aggregations, the ability to detect mid-aggregation variations in transmissibility, and finally, the number of iterations required to reach convergence when reconstructing daily incidence data. The full simulation study description and details can be found in S1 Appendix.
Results
Hereafter, we refer to reported and reconstructed incidence data, these are the reported daily incidence and the daily incidence that has been reconstructed from weekly aggregated data, respectively.
Influenza
The reconstructed incidence of influenza was much smoother than the reported incidence, which showed clear weekend effects and lower reported cases on two public holidays, both occurring on Fridays (Fig 2A and S1 Appendix p 8). Considering weekly sliding Rt first, there was a high correlation in both the mean Rt estimates derived from each dataset (R2 = 0.91, Fig 2C and S1 Appendix p 2) and their associated uncertainty (R2 = 0.93, Fig 2D). The overall agreement in the classification of Rt reached 81.8% (see methods and S1 Appendix p 9).
A) The reported (grey) and reconstructed (green) daily incidence of influenza by date of presentation at a military clinic. B) Squared error of the daily (orange) and weekly sliding (pink) Rt estimates that were made from reconstructed daily data compared to those obtained from the reported daily data. Rt estimation starts on the first day of the second aggregation window (day 8—18th December 2009) and is plotted on the last day of the time window used for estimation (i.e., starting on day 9 (19th December) for daily estimates and day 14 (24th December) for weekly estimates). Note: the x-axis is shared with the incidence plot above. C & E) Correlation between the weekly sliding (C) and daily (E) mean Rt estimates using reconstructed data (y-axis) and reported daily data (x-axis). Vertical and horizontal lines depict the 95% credible intervals (95% CrIs) and dotted lines show the threshold of Rt = 1. D & F) Correlation between the uncertainty in the weekly sliding (D) and daily (F) Rt estimates, defined as the width of the 95% credible intervals, using the reconstructed (y-axis) and reported (x-axis) daily data. The colour of the points in panels C-F correspond to the epidemic phase, i.e. the early (19th– 30th December for daily estimates, or 24th– 30th December for weekly sliding estimates), middle (31st December– 6th January) or late (7th– 14th January) phase of the data, shown by the strip in panel A. Solid lines show the linear model fit with 95% confidence intervals (grey shading). Dashed lines represent the x = y line.
In contrast, mean daily Rt estimates differed markedly depending on whether the reported or reconstructed data were used, with an R2 of 0.13 and much higher mean Rt and uncertainty in estimates obtained from reported data (Fig 2E and 2F). Higher mean Rt estimates coincided with large peaks in the reported daily incidence (typically on Mondays), as daily Rt estimates were not smoothed and therefore more affected by intra-weekly variability (S1 Appendix p 2). The overall agreement in the classification of daily Rt estimates was much lower, with only 44.4% agreement (S1 Appendix p 9).
In this case study, the greatest differences in Rt estimates tended to correspond to time periods when the reported and reconstructed incidence data were most dissimilar (Fig 2B and S1 Appendix p 3). There was no apparent pattern in the estimates with regard to the outbreak phase, i.e. early, mid or late-phase, but this is likely due to this dataset being a snapshot of incidence taken from within an established epidemic (Fig 2).
COVID-19 cases
The reconstructed incidence of COVID-19 smoothed out intra-weekly variability, caused by factors such as weekend effects (Fig 3A and S1 Appendix pp 7–8). Weekly sliding Rt estimates obtained from reconstructed and reported incidence were similar, both in their means (R2 = 0.98) and their level of uncertainty (R2 = 0.99, Fig 3C and 3D and S1 Appendix p 4). Mean daily Rt estimates were less well correlated (R2 = 0.67), although the difference is less marked than in the influenza case study (Fig 3E), and the uncertainty in the estimates was similar across both approaches (R2 = 0.97, Fig 3F). Most of the discrepant Rt estimates and higher levels of uncertainty coincide with the early phase of the outbreak when incidence was lower (Fig 3E and 3F). Outside of periods of low incidence, the largest differences in Rt estimates tended to correspond to time periods with greater disparities between the reported and reconstructed incidence data (Fig 3B and S1 Appendix p 5). The overall agreement in the classification of Rt estimates was higher than for influenza, with 74.4% and 94.9% agreement for daily and weekly sliding Rt estimates respectively (S1 Appendix p 9).
A) The reported (grey) and reconstructed (green) daily incidence of COVID-19 by date of specimen. B) Squared error of the daily (orange) and weekly sliding (pink) Rt estimates made from reconstructed data compared to those obtained from the reported daily data. Rt estimation starts on the first day of the second aggregation window (day 8—28th February 2020) and is plotted on the last day of the time window used for estimation (i.e., starting on day 9 (29th February) for daily estimates and day 14 (5th March) for weekly estimates). Note: the x-axis is shared with the incidence plot above and the y-axis has been limited to 0.5 for clarity. C & E) Correlation between the weekly sliding (C) and daily (E) mean Rt estimates using reconstructed (y-axis) and reported (x-axis) daily data, starting on day 30 due to low incidence. Vertical and horizontal lines depict the 95% credible intervals (95% CrIs) and dotted lines show the threshold of Rt = 1. D & F) Correlation between the uncertainty in the weekly sliding (D) and daily (F) Rt estimates, defined as the width of the 95% credible intervals, using the reconstructed (y-axis) and reported (x-axis) daily data. The colour of the points in panels C-F correspond to the epidemic phase, i.e. the early (21st March– 12th October 2020), middle (13th October 2020—22nd May 2021) or late (23rd May– 30th December 2021) phase of the data, shown by the strip in panel A. Solid lines show the linear model fit with 95% confidence intervals (grey shading). Dashed lines represent the x = y line.
COVID-19 deaths
The reported incidence of COVID-19 deaths was much less influenced by day-to-day variation. The reconstructed daily incidence was more similar to the observed daily data than in the previous case studies (Fig 4A). Both weekly and daily Rt estimates obtained from weekly data were highly consistent with those obtained from daily observations (R2 = 0.98 and R2 = 0.80 respectively, Fig 4C and 4E). The overall agreement in Rt classifications for daily estimates was the highest of all case studies at 85.8%, and 93.3% for weekly Rt estimates (S1 Appendix p 9). Discrepancies between the two mostly coincide with periods of particularly low incidence of deaths (Fig 4B and S1 Appendix p 7). The overall lower incidence of COVID-19 deaths compared to COVID-19 cases means there is greater uncertainty in Rt estimates in this case study (Fig 4D and 4F and S1 Appendix p 6). However, there was minimal difference in the uncertainty of estimates obtained from daily and weekly data (Fig 4D and 4F).
A) The reported (grey) and reconstructed (green) daily incidence of COVID-19 by date of death within 28 days of a positive test. B) Squared error of the daily (orange) and weekly sliding (pink) Rt estimates that were made from reconstructed data compared to those obtained from the reported daily data. Rt estimation starts on the first day of the second aggregation window (day 8—9th March 2020) and is plotted on the last day of the time window used for estimation (i.e., starting on day 9 (10th March) for daily estimates and day 14 (15th March) for weekly estimates). Note: the x-axis is shared with the incidence plot above and the y-axis has been limited to 0.5 for clarity. C & E) Correlation between the weekly sliding (C) and daily (E) mean Rt estimates using reconstructed (y-axis) and reported daily data (x-axis), starting on day 30 due to low incidence. Vertical and horizontal lines depict the 95% credible intervals (95% CrIs) and dotted lines show the threshold of Rt = 1. D & F) Correlation between the uncertainty in the weekly sliding (D) and daily (F) Rt estimates, defined as the width of the 95% credible intervals, using the reconstructed (y-axis) and reported daily (x-axis) data. The colour of the points in panels C-F correspond to the epidemic phase, i.e. the early (31st March–20th October 2020), middle (21st October 2020—28th May 2021) or late (29th May 2021—2nd January 2022) phase of the data, shown by the strip in panel A. Solid lines show the linear model fit with 95% confidence intervals (grey shading). Dashed lines represent the x = y line.
In all case-studies, incidence reconstructions converged within 10 iterations of the EM algorithm. The overall process of Rt estimation from weekly aggregated data took three seconds or less to run on MacOS (2 GHz Quad-Core Intel Core i5) 16GB RAM (S1 Appendix p 10); the influenza scenario, with over 57,000 cases, took two seconds to run, whilst the COVID-19 cases and deaths scenarios, with an overall incidence over 149,000 and 13 million cases respectively, took three seconds to run.
Simulation study
The method performed well across all scenarios, successfully estimating Rt from the aggregated simulated data, but unable to recover mid-aggregation window variations in transmissibility (S1 Appendix pp 10–24). Convergence of the EM algorithm was quick, with negligible differences in the reconstructed incidence beyond 5 iterations (S1 Appendix pp 22–23).
When introducing weekend effects into simulated data, Rt estimates from reconstructed incidence were more successful at recovering the true value of Rt than when using reported incidence (S1 Appendix pp 20–21). The method can also be successfully applied to other temporal aggregations of data, e.g. three-, ten- or fourteen-day windows (S1 Appendix pp 23–25).
Discussion
Estimates of the time-varying reproduction number (Rt) have frequently been used to inform and guide policymaking during outbreaks, and a commonly used approach to estimate Rt is EpiEstim, which relies on daily incidence data. However, maintaining daily incidence databases requires substantial time and investment in resources, which is not always feasible, particularly for less acute or routinely reported diseases. Therefore, in practice, many diseases are not reported on a daily basis, including influenza and other notifiable diseases in the UK and US [2–5]. As the COVID-19 pandemic persists, daily reporting is also becoming less common [20]. Coarsely aggregated data can be challenging to deal with in the context of Rt estimation methods, restricting their applications in certain contexts. In this study, we develop a statistical framework and tool that allows Rt estimation from aggregated incidence without introducing bias. Using influenza and COVID-19 data, alongside a simulation study, we demonstrate how a simple expectation-maximisation algorithm approach can rapidly reconstruct daily incidence data and accurately estimate Rt.
In all case studies, direct comparisons between weekly sliding Rt estimates show that very similar estimates can be made from the reported daily incidence and the reconstructed daily incidence from weekly aggregated data. However, daily Rt estimates are more influenced by noise, such as intra-weekly variability, leading to greater disparities in estimates between datasets. There are clear weekend effects exhibited in the influenza and COVID-19 case data (S1 Appendix p 8), leading to peaks and troughs in the reported incidence and the resulting daily Rt estimates (Figs 2 and 3, S1 Appendix pp 2 and 4). Using reconstructed incidence considerably smoothed the daily Rt estimates, removing the impact of weekend-effects. The overall agreement in the classification of Rt as increasing, uncertain, or declining between estimates made from each dataset rose substantially when some of the variability in the reported data was smoothed by estimating Rt using weekly sliding windows (S1 Appendix pp 8–9).
Despite both being affected by weekly periodicity in reporting, concordance of Rt estimates obtained from COVID-19 case data is considerably better than for influenza, perhaps due to the greater quantity of data, with a very strong positive correlation between daily and weekly Rt estimates (Fig 3). This is reflected in the high overall agreement in the classification of Rt estimates obtained from the reported and reconstructed datasets. It is important to note that outlying and much larger Rt estimates obtained from both datasets coincide with the early phase of the epidemic, when incidence was lower and the prior for Rt (μ = 5, σ = 5) had more weight on estimates.
During the early stages of epidemics, despite there being far fewer deaths than cases, death data can sometimes be considered more reliable [21,22]. For example, case reporting is affected by surveillance system quality and the robustness of testing practices, which can vary considerably over the course of an epidemic, especially early on. COVID-19 incidence by date of death is much less influenced by administrative noise in the data (S1 Appendix p 8), and the reconstructed incidence is most similar to the reported daily incidence of any case study. Therefore, the greatest differences in Rt estimates from death data coincide with periods of low incidence (S1 Appendix p 7) when uncertainty increases. Weekly sliding Rt estimates are equally as correlated as those from COVID-19 case data, but daily Rt estimates are the most strongly correlated of any dataset (Fig 4). Additionally, there is very high overall agreement in the classification of daily and weekly Rt (S1 Appendix p 9). This provides further support that differences between daily Rt estimates for influenza and COVID-19 cases is likely due to the reconstructed incidence smoothing out weekly periodicity in reporting.
To investigate further, weekend effects were artificially introduced to data in the simulation study (S1 Appendix p 20). We have shown that, when using reported incidence, Rt estimates are all strongly influenced by weekend effects (regardless of the smoothing time-window). Reconstructing daily incidence from weekly data completely removes the effect of noise from resulting Rt values, greatly improving the accuracy of estimates. This demonstrates that it may be beneficial to artificially aggregate daily data, as has been done in previous studies [6,7]. However, we did assume quite an extreme level of administrative noise, so in instances where the pattern is less prominent, it may have less of an impact on estimates. Furthermore, this smoothing effect could disguise genuine variations in transmissibility that occur mid-aggregation window, for instance, increased/decreased transmission over weekends (S1 Appendix section 3g). Disentangling important temporal trends in Rt from noise in the data can be difficult, and if aggregated data is used it will be at the cost of reduced temporal resolution in Rt estimates.
This can be seen when the method is applied to data aggregated over longer timescales, such as ten- to fourteen-days (S1 Appendix pp 23–25). This approach requires two layers of smoothing: 1) the incidence is smoothed over each aggregation window during the reconstruction process and 2) Rt estimates are smoothed by the sliding window chosen by the user. If a change in Rt occurs at the end of an aggregation window (i.e. on the last day), such as a sudden decrease in Rt due to a strict lockdown, that change is detected with a lag, corresponding to the length of the sliding window used for Rt estimation (S1 Appendix p 24). However, if the event occurs mid-aggregation window, then in addition to the usual lag caused by the sliding window, estimates will be affected by the smoothing of the incidence within the aggregation window during reconstruction (S1 Appendix p 25). The change in Rt will seem more gradual over the period that data are aggregated over and will appear to start earlier (corresponding to the first day of the aggregation window). It is important for users to be aware of this, particularly when using longer aggregations of data.
Another consideration is that the reconstructed incidence can have discontinuities in the borders between aggregation windows (S1 Appendix pp 11–12). This occurs because in reconstructing daily incidence we impose that, if it were to be re-aggregated, it would match the original data. Methods that simply fit smoothing splines to weekly data, inferring daily case counts from the daily difference in cumulative counts, are not affected by this [23,24]. To circumvent this problem, we recommend that sliding windows used to estimate Rt are at least equal to or longer than the length of aggregation windows to reduce the impact of discontinuities on estimates (S1 Appendix pp 23–25).
Alternative approaches include simple smoothing splines or LOESS to reconstruct daily incidence from aggregated data (see S1 Appendix section 4), and modelling frameworks implemented in the Epidemia and EpiNow2 R packages [6,21,25]. Daily infections are modelled as a latent process, back-calculated from observed data on cases or deaths, depending on an appropriate infection to observation distribution. In addition, Epidemia integrates further information, such as the infection ascertainment rate (for cases) or the infection fatality rate (for deaths) [21]. This facilitates a ‘nowcasting’ approach, allowing users to estimate Rt directly from the unobserved infections, but they typically require more data (e.g. incidence of deaths and cases), more assumptions (e.g. delay distributions and ascertainment rates), and are much more computationally intensive, which can be a barrier to the adoption of such methods by users [14].
Here, Rt estimates are based on a single daily incidence reconstruction, meaning Rt can be estimated very rapidly from aggregated data, which is particularly desirable during real-time outbreak analysis [14]. A potential downside is that uncertainty in Rt estimates could be underestimated. However, the simulation study showed that the 95% credible interval of estimates encompassed the correct value of Rt the majority of the time, and we found no substantial indication that this approach detrimentally affected our characterisation of the uncertainty.
Given that this method is directly derived from EpiEstim, it relies on similar assumptions and caveats [15,26]. As time of infection is more difficult to observe than symptom onset, the SI is typically used as an approximation of the generation time in the renewal equation, which may introduce bias [27]. The SI, the level of undetected cases, and the reporting rate are assumed to remain constant, which is often not the case in practice. Factors such as changes in population immunity, and the introduction of interventions, can alter the SI throughout an epidemic [28]. Whilst changing case definitions, new testing practices, and increased healthcare-seeking behaviour, can all affect case ascertainment.[15] Parameters chosen by users can also influence estimation accuracy, for instance, the time window length for temporal smoothing and the prior for Rt [26]. Finally, EpiEstim’s assumption of a Poisson likelihood may be a limitation in instances when data is substantially overdispersed [29,30].
To make the method simple to implement for current and future users of EpiEstim, this extension has been fully integrated with the ‘estimate_R()’ function in the original R package on GitHub [31]. Just one additional parameter is required—the number of days data are aggregated over (with some other optional parameters). The reconstructed daily incidence is also generated as an output, so it is possible to use it in other analysis pipelines involving alternative R estimation methods, which may perform better than EpiEstim in certain contexts, e.g. in retrospective analysis (S1 Appendix pp 29–30) [30], or in the presence of delays in reporting [25]. More details regarding the applications of this method can be found in the package vignette and associated examples [31].
Conclusion
We extended the widely used Rt estimation approach proposed by Cori et al., [15] and implemented in the R package EpiEstim, to incorporate a new feature which allows Rt to be easily estimated from any temporal aggregation of incidence data. We have demonstrated that the method performs well using both simulated and real-world data, recovering or even improving upon the estimates that would have been made from reported daily data. This extension is easy to use and computationally efficient, which will enable epidemiologists and other public health professionals to apply EpiEstim to a wider range of diseases and epidemic contexts.
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