The EpiEstim package has been extended to allow users to estimate the time varying reproduction number (R_t) from temporally aggregated incidence data (Nash et al.). This approach reconstructs daily incidence from data supplied at any timescale. This vignette will take you through the different ways aggregated incidence data can be supplied, the additional parameters needed, and an example using aggregated UK COVID-19 data. Please also see the FAQs section.

Input data

Incidence data

Many diseases are not reported on a daily basis. EpiEstim can now use incidence data that has been aggregated in multiple ways, e.g.:

Constant aggregations, such as weekly data
Repeating patterns of aggregations, such as regular reporting 3x per week
Variable aggregations over the course of an outbreak

Daily incidence data is reconstructed from aggregated data using an expectation-maximisation (EM) algorithm. There are three stages of the EM algorithm:

Initialisation. The EM algorithm is initialised with a naive disaggregation of the incidence data. For example, if there were 70 cases over the course of a week, this would be naively split into 10 cases per day.
Expectation. The reproduction number is estimated for each aggregation window, except for the first aggregation window (as there is no past incidence data). This means that the earliest the incidence reconstruction can start is at least the first day of the second aggregation window. Additionally, if the disaggregated incidence in subsequent aggregation windows is too low to estimate the reproduction number, this will mean that the reconstruction will not start until case numbers are sufficiently high.
Maximisation. The reproduction number estimates are then translated into growth rates for each aggregation window (Wallinga & Lipsitch, 2007) and used to reconstruct daily incidence data assuming exponential growth. The daily incidence is adjusted by a constant to ensure that if the daily incidence were to be re-aggregated, it would still sum to the original aggregated totals. The expectation and maximisation steps repeat iteratively until convergence.

The daily incidence that is reconstructed after the final iteration of the EM algorithm is then used to estimate R_t using the same process as the original estimate_R() function, with sliding weekly time windows used as the default.

Aggregation windows

Aggregation windows can be specified using the parameter dt, and can be provided in one of three ways:

A single integer - for constant aggregation windows, e.g. dt = 7L for weekly data
A repeating vector of integers - for repeating aggregation patterns e.g. reporting 3x per week on the same day of the week could be dt = c(2L,2L,3L)
A vector of aggregations matching the length of incidence data supplied

Serial interval distribution

The serial interval can be provided on a daily timescale (as usual), either as the mean and standard deviation (parametric distribution) or the full distribution (non-parametric distribution). See ‘full_EpiEstim_vignette’ for more details.

Estimate R_t from temporally aggregated incidence data

To estimate R_t from temporally aggregated incidence data, we simply use the estimate_R() function with two additional parameters required, dt and dt_out, and some optional parameters, recon_opt, iter, tol, and grid.

`estimate_R()` for aggregated data

estimate_R(incid = aggregated_incidence,
           dt = 7L,
           dt_out = 7L,
           recon_opt = "naive",
           iter = 10L,
           tol = 1e-6,
           grid = list(precision = 0.001, min = -1, max = 1),
           config = config,
           method = method)

As described above, dt can be supplied as a single integer, a vector of repeating integers, or a full vector of integers matching the length of the incidence data.
dt_out is the length of the sliding windows used to estimate R_t from the reconstructed daily incidence data, this is dt = 7L (weekly sliding windows) by default. We recommend that dt_out is at least equal to the length of the longest aggregation window (dt) in the data.
recon_opt can be one of two options: "naive" or "match". This specifies how to handle the initial incidence data that cannot be reconstructed by the EM algorithm (e.g. the incidence data for the aggregation window that precedes the first aggregation window that R can be estimated for). If "naive" is chosen, the naive disaggregation of the incidence data will be kept. If "match" is chosen, the incidence in the preceding aggregation window will be reconstructed by assuming that the growth rate matches that of the first estimation window. This is "naive" by default.

There are three other optional parameters that can be modified, however, we recommend that the default values are used:

iter is the number of iterations of the EM algorithm used to reconstruct the daily incidence data. This is iter = 10L by default.
tol is the tolerance value used for the convergence check. The tolerance is how much the final iteration of the reconstructed daily incidence is allowed to differ from the reconstructed incidence produced in the previous iteration without returning a warning. This is tol = 1e-6 by default.
grid is a list of “precision”, “min”, and “max” values to define a grid of growth rate parameters used inside the EM algorithm. The grid is used to convert reproduction number estimates for each aggregation of incidence data into growth rates, which are then used to reconstruct the daily incidence data assuming exponential growth. The grid will auto-adjust if it is not large enough, so we recommend using the default values.

The SI distibution can be specified as normal using the method and config parameters (see the full_EpiEstim_vignette for more details).

Examples

Estimate R_t from weekly COVID-19 data

This example will take you through a workflow using weekly incidence data for UK COVID-19 cases. (For detailed description of the data see Nash et al.)

incid <- readRDS("./aggregated_data/UK_covid_cases.rds")

Incidence

Let us say we have a vector of weekly incidence data for COVID-19 cases.

incid
#>  [1]      21     241    1503    4714   14294   27408   33124   30126   33288
#> [10]   31944   25581   20435   17320   12195    9259    7102    6948    5596
#> [19]    4376    4258    4287    4704    5507    5964    7408    7500    8244
#> [28]   13823   22060   23924   41711   66725  106925  116472  146015  150832
#> [37]  159310  172001  146202  107282  102508  128194  202680  260353  352685
#> [46]  397790  316549  250624  175157  133220   94886   79373   63185   41334
#> [55]   38741   37560   37213   27387   19128   16934   16182   14287   13422
#> [64]   14743   15882   20430   30671   46961   62642   97320  163612  214681
#> [73]  294984  279099  187392  186748  198318  225124  239123  249063  251378
#> [82]  199736  235620  234391  244202  292109  326688  280646  255388  253735
#> [91]  287826  295546  322535  345941  525261  800717 1034989

We need to specify how the data is aggregated, which in this case, is by constant weekly aggregation windows. We do this by supplying dt with a single integer (7L).

dt <- 7L

We can take an estimate from the literature to specify a parametric SI with a mean of 6.3 days and a standard deviation of 4.2 days (Bi et al 2020).

mean_si <- 6.3
std_si <- 4.2
method <- "parametric_si"
config <- make_config(list(mean_si = mean_si,
                           std_si = std_si))

Estimate R_t

Now that we have our aggregated incidence, our aggregation time window, and SI distribution, we can supply these to the estimate_R() function. We do not need to specify dt_out, iter, tol, or grid because we are going to use the default values.

output <- EpiEstim::estimate_R(incid = incid,
                               dt = dt,
                               recon_opt = "match",
                               method = method,
                               config = config)

The output consists of multiple elements, including the reconstructed daily incidence data:

head(output$I)
#> [1] 3 3 3 3 3 3

And the R_t estimates (in this case, using the default weekly sliding windows):

head(output$R)
#>   t_start t_end  Mean(R)    Std(R) Quantile.0.025(R) Quantile.0.05(R)
#> 1       8    14 5.472053 0.3517569          4.804283         4.906602
#> 2       9    15 4.940857 0.2770256          4.412755         4.494190
#> 3      10    16 4.440229 0.2176621          4.023813         4.088375
#> 4      11    17 4.095384 0.1758868          3.757861         3.810433
#> 5      12    18 3.865754 0.1457340          3.585360         3.629207
#> 6      13    19 3.708135 0.1229331          3.471074         3.508271
#>   Quantile.0.25(R) Median(R) Quantile.0.75(R) Quantile.0.95(R)
#> 1         5.230868  5.464518         5.705025         6.063209
#> 2         4.751292  4.935681         5.124780         5.405182
#> 3         4.291544  4.436673         4.585038         4.804214
#> 4         3.975417  4.092866         4.212605         4.388922
#> 5         3.766486  3.863923         3.963027         4.108548
#> 6         3.624494  3.706777         3.790295         3.912632
#>   Quantile.0.975(R)
#> 1          6.182645
#> 2          5.498376
#> 3          4.876853
#> 4          4.447213
#> 5          4.156555
#> 6          3.952916

In this example, you will notice that R_t estimation does not start until day 8. This is because the daily incidence data cannot be reconstructed, and R_t estimation cannot start, until the first day of the second aggregation window. The start of R_t estimation may also be delayed if incidence is too low, but this was not the case here.

Plot results

As normal, simply plot the full or partial output.

plot(output) # full output

plot(output, "incid") # Reconstructed daily incidence only
plot(output, "R") # Rt estimates only
plot(output, "SI") # SI estimates only

Convergence

Convergence is checked automatically to ensure that the final iteration of the reconstructed daily incidence does not differ from the previous iteration beyond a tolerance of 10⁻⁶ by default. If convergence is not reached, a warning will be returned and the algorithm can be run with more iterations by modifying iter =. The tolerance threshold can also be modified using tol =.

FAQs

Why does R_t estimation start later than the start date of the incidence data supplied?

In order to reconstruct daily incidence data, the method requires that R_t is estimated for each aggregation window in turn, which is translated into a growth rate and used to reconstruct daily incidence assuming exponential growth. As there is no past incidence data beyond the first aggregation window, R_t cannot be estimated and the daily incidence cannot be reconstructed until the first day of the second aggregation window.

Additionally, R_t estimation will not start until case numbers are sufficiently high.

Please also see the FAQ section in the main “EpiEstim Vignette”.

References

Nash RK, Cori A, Nouvellet P. Estimating the epidemic reproduction number from temporally aggregated incidence data: a statistical modelling approach and software tool. medRxiv pre-print.

Bi Q, et al. Epidemiology and transmission of COVID-19 in 391 cases and 1286 of their close contacts in Shenzhen, China: a retrospective cohort study. Lancet. 2020.

Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings of the Royal Society B: Biological Sciences. 2007 Feb 22;274(1609):599–604.

EpiEstim for aggregated incidence data