The EpiSewer R package provides a Bayesian generative model to
estimate effective reproduction numbers and other epidemiological
parameters from concentration measurements at a wastewater treatment
plant (or other sampling site) over time. This allows to track the
transmission dynamics of a pathogen in the associated catchment
population. The EpiSewer model is tailored to the specifics of
wastewater concentration measurements, offers comprehensive uncertainty
quantification via MCMC sampling in stan, and provides easily
configurable modeling components.
Measurements
⭐ Non-daily / missing measurements
⭐ Multiple measurements (replicates) per sample
⭐ Multi-day composite samples
⭐ Accurate dPCR noise model
⭐ Limit of detection (LOD) model
Sampling
⭐ Integrated outlier detection
⭐ Sample batch effects (e.g. weekday or age-of-sample effects)
Sewer
⭐ Flow normalization
⭐ Sewer residence time distributions
Shedding
⭐ Incubation period and shedding load distributions
⭐ Individual-level shedding load variation
Infections
⭐ Stochastic infection model with overdispersion
⭐ Flexible
⭐ Transmission indicators:
Forecast
⭐ Probabilistic forecasts of
The development version of EpiSewer can be installed from GitHub as
shown below. Please note that the package is still in early development
and may be subject to breaking changes.
remotes::install_github("adrian-lison/EpiSewer", dependencies = TRUE)EpiSewer also requires CmdStan to be installed on your system. This
can be done using the install_cmdstan() function from cmdstanr. If
you experience any problems installing CmdStan, see the cmdstanr
vignette for help.
cmdstanr::check_cmdstan_toolchain()
cmdstanr::install_cmdstan(cores = 2) # use more cores to speed upThe stan models used by EpiSewer need to be compiled for your device.
This is only necessary once - after installing or updating the package -
and can be done using the sewer_compile() function.
EpiSewer::sewer_compile()If the models are not successfully compiled, please ensure that
cmdstan is properly set up and try updating it to a newer version
using cmdstanr::install_cmdstan(). If the problem persists, please run
EpiSewer::sewer_compile(verbose = TRUE) and post the output in a new
issue on GitHub, along with your cmdstanr::cmdstan_version().
This is a quick introduction to using the EpiSewer package. To learn
more about modeling with EpiSewer, see the model
specification
and detailed
example
vignettes on the package
website.
EpiSewer comes with data.table for data manipulation and ggplot2
for plotting.
library(EpiSewer)
library(data.table)
library(ggplot2)The below example of wastewater data is from Zurich, Switzerland. Data are provided by EAWAG (Swiss Federal Institute of Aquatic Science and Technology) to the public domain (CC BY 4.0 license).
data_zurich <- SARS_CoV_2_Zurich
names(data_zurich)
#> [1] "measurements" "flows" "cases" "units"measurements:EpiSewerrequires a time series of concentration measurements. Measurements should ideally be in gc/mL (gc = gene copies).flows: Data about the daily wastewater volume flowing through the sampling site (should be in mL/day). Flow data is highly recommended, but not strictly necessary (see documentation offlows_assume()).cases: Data about confirmed cases in the catchment area. This is optional but can help informing some of our prior assumptions.
The measurements data contains daily viral concentration measurements
(in gc/mL) for the SARS-CoV-2 N1 gene at the wastewater treatment plant
in Zurich. Some days have missing measurements, but this is no problem:
EpiSewer naturally accounts for missing values during estimation.
data_zurich$measurements
#> date concentration
#> 1: 2022-01-01 NA
#> 2: 2022-01-02 NA
#> 3: 2022-01-03 455.7580
#> 4: 2022-01-04 747.8792
#> 5: 2022-01-05 573.7020
#> ---
#> 116: 2022-04-26 182.3664
#> 117: 2022-04-27 379.2421
#> 118: 2022-04-28 439.7427
#> 119: 2022-04-29 394.0033
#> 120: 2022-04-30 293.1242To show the handling of missing data more clearly, we make our data artificially sparse by keeping only measurements that were made on Mondays and Thursdays. This means we will only have two measurements per week.
measurements_sparse <- data_zurich$measurements[,weekday := weekdays(data_zurich$measurements$date)][weekday %in% c("Monday","Thursday"),]
head(measurements_sparse, 10)
#> date concentration weekday
#> 1: 2022-01-03 455.7580 Monday
#> 2: 2022-01-06 330.7298 Thursday
#> 3: 2022-01-10 387.6885 Monday
#> 4: 2022-01-13 791.1111 Thursday
#> 5: 2022-01-17 551.7701 Monday
#> 6: 2022-01-20 643.9910 Thursday
#> 7: 2022-01-24 741.9150 Monday
#> 8: 2022-01-27 770.1810 Thursday
#> 9: 2022-01-31 627.1725 Monday
#> 10: 2022-02-03 561.2913 ThursdayThe flows data tracks the daily flow (in mL/day) at the treatment
plant in Zurich. The flow data will be used to normalize the
concentration measurements. This helps to account for environmental
factors such as rainfall. It is important that the flow uses the same
volume unit as the concentration (mL here in both cases).
data_zurich$flows
#> date flow
#> 1: 2022-01-01 3.41163e+11
#> 2: 2022-01-02 3.41163e+11
#> 3: 2022-01-03 1.58972e+11
#> 4: 2022-01-04 1.60849e+11
#> 5: 2022-01-05 3.72301e+11
#> ---
#> 116: 2022-04-26 3.30659e+11
#> 117: 2022-04-27 2.06046e+11
#> 118: 2022-04-28 1.58373e+11
#> 119: 2022-04-29 1.52156e+11
#> 120: 2022-04-30 2.08685e+11❗ Note: It’s not a problem if you only have flow data on dates with
measurements. However, for each measured date, there must also be a flow
value. If flow information is missing for measured dates, please make
sure to impute it using a suitable method before passing it to
EpiSewer (or drop those dates).
Finally, if we have case data available, we can also pass this to
EpiSewer. This is then used to calibrate the model so that the
estimated number of infections approximately matches the observed number
of cases.
data_zurich$cases
#> date cases
#> 1: 2022-01-01 NA
#> 2: 2022-01-02 NA
#> 3: 2022-01-03 1519.5313
#> 4: 2022-01-04 1727.3722
#> 5: 2022-01-05 1638.5321
#> ---
#> 116: 2022-04-26 211.9338
#> 117: 2022-04-27 214.9822
#> 118: 2022-04-28 206.2498
#> 119: 2022-04-29 151.9567
#> 120: 2022-04-30 134.8545If you don’t have case data: don’t worry! In general, the assumed
shedding load per case does not have a large effect on the effective
reproduction number, and EpiSewer will use a robust default. You just
have to keep in mind that the absolute number of infections estimated by
EpiSewer should not be interpreted as true incidence or
prevalence.
We combine the measurements data and flow data using the helper
function sewer_data(). We here use our artificially sparse
measurements (only Mondays and Thursdays). For calibration purposes, we
also supply the case data, as explained above.
ww_data <- sewer_data(
measurements = measurements_sparse,
flows = data_zurich$flows,
cases = data_zurich$cases # cases are optional
)In order to estimate the effective reproduction number from wastewater concentration measurements, we have to make a number of assumptions.
- Generation time distribution: Distribution of the time between a primary infection and its resulting secondary infections
- Shedding load distribution: Distribution of the load shed by an average individual over time. We also specify that our distribution is relative to the day of symptom onset.
- Incubation period distribution: Because our shedding load distribution is in days since symptom onset, we also need to make an assumption about the time between infection and symptom onset.
The generation time, shedding load and incubation period distribution
are all disease-specific and are typically obtained from literature.
EpiSewer requires these distributions to be discretized, and offers
specific functions to obtain discretized versions of popular continuous
probability distributions.
ww_assumptions <- sewer_assumptions(
generation_dist = get_discrete_gamma_shifted(gamma_mean = 3, gamma_sd = 2.4),
shedding_dist = get_discrete_gamma(gamma_shape = 0.929639, gamma_scale = 7.241397),
shedding_reference = "symptom_onset", # shedding load distribution is relative to symptom onset
incubation_dist = get_discrete_gamma(gamma_shape = 8.5, gamma_scale = 0.4)
)💡 Sometimes, shedding load distributions are instead specified in days
since infection. In that case, you can use
shedding_reference = "infection" and don’t need to supply an
incubation period distribution.
Now that we have the data and necessary assumptions, we can use
EpiSewer to estimate the effective reproduction number. We here use
the default model and settings provided by EpiSewer. With the helper
function set_fit_opts() we specify our sampling approach: we apply
Hamiltonian MCMC sampling via stan, using 4 chains with 500 warmup and
500 sampling iterations each.
Stan regularly provides updates about the progress of the sampler. The overall runtime will depend on your hardware resources, the size of the data, the complexity of the model used, and how well the model actually fits the data. On a MacBook Pro (2 GHz Quad-Core Intel Core i5) the example below takes about 5 minutes to run.
options(mc.cores = 4) # allow stan to use 4 cores, i.e. one for each chain
ww_result <- EpiSewer(
data = ww_data,
assumptions = ww_assumptions,
fit_opts = set_fit_opts(sampler = sampler_stan_mcmc(
iter_warmup = 500, iter_sampling = 500, chains = 4, seed = 42
))
)💡 We are here using a shorthand version of the EpiSewer() function
which uses all available default settings. To find out how the
EpiSewer model can be customized, see the model
specification
and detailed
example
vignettes.
Great, the sampling has completed! Now we can inspect the results of our model. Plots are a convenient way to get a quick overview.
It is good practice to first assess how well the model actually fitted to the data. For this, we plot the observed concentration measurements against the ones predicted by the model. The inner and outer ribbons show the 50% and 95% credible intervals. The black dots show measurements that we have actually observed in our artificially sparse data set (only Mondays and Thursdays). The grey crosses show all the other measurements that were not available to the model. As we can see, the model still achieved a decent fit on these missing days.
plot_concentration(ww_result, measurements = data_zurich$measurements)
By default, the function plot_concentration() shows absolute
concentrations, which strongly depend on the daily wastewater flow.
Alternatively, we can plot the flow-normalized concentration by setting
normalized = TRUE. This will show the hypothetical concentration if
the flow was at its median value. For this option, we need to provide
the flow data as well (see below).
plot_concentration(ww_result, measurements = data_zurich$measurements, flows = data_zurich$flows, normalized = TRUE)
Since the model fit looked decent at first glance, we now inspect our main parameter of interest, the effective reproduction number over time. Again, the ribbons show the 50% and 95% credible intervals. We see that even with only two measurements per week, we get a fairly clear signal for R being above or below the critical threshold of 1.
plot_R(ww_result)
Note that the estimates for R go back further into the past than our observations. This is due to the delay from infection to shedding, i.e. concentration measurements observed today are mostly a signal of infections in the past. This is also why the R estimates close to the present are strongly uncertain. The measurements observed until the present provide only a delayed signal about transmission dynamics, so the most recent R estimates are informed by only little data.
We can also display a growth report which shows the probability that
infections have been growing for a prolonged period of time. By default,
EpiSewer selects the most recent date for which reliable estimates are
possible as a reference.
plot_growth_report(ww_result)
As the
model was fitted at the end of the seasonal wave in Zurich, there is
high posterior support that infections are currently not growing for a
prolonged period of time.
You can also display a growth report for other dates. Let’s for example have a look at the report for mid-February 2022. Here we can see that there is strong posterior support that infections have been growing for at least a week. At the same time, it seems rather unlikely that they have already been growing for 3 weeks or longer.
plot_growth_report(ww_result, date = "2022-02-19")
💡 EpiSewer also provides further indicators of transmission dynamics,
see e.g. plot_growth_rate() and plot_doubling_time(). These can
however be more volatile and sometimes difficult to interpret.
We can also inspect other internal parameters of the model. This may give us a better understanding of the model fit and may help to diagnose certain problems.
First, we plot the estimated load over time. This is the expected total load that arrived at the sampling side on a given day. We can see this is roughly on the order 1e14 to 3e14.
plot_load(ww_result)
Next, we plot
the estimated number of infections over time. The time series follows a
very similar trend as the load. Compared to the load, the peak in
infections seems to be a bit more spiked, and also a bit earlier. This
is because infected individuals only begin shedding after their
infection and then shed over a longer period of time (as defined by the
incubation period and shedding load distribution). This makes the load a
slightly delayed and blurred signal of the infections.
plot_infections(ww_result)
Note that because we supplied case data, the estimated infections rougly match the case numbers (plotted in blue below). Nevertheless, the infections plot shown here should only be used for diagnostic purposes, for example to ensure that the trend looks sensible and that the scale of infections is not unrealistic (not millions of infections or 0.01 infections per day). Please do not interpret the estimated number of infections in terms of true absolute incidence or prevalence.
plot_infections(ww_result) + geom_step(data = data_zurich$cases, aes(x = date, y = cases), color = "#000080")
We can also compare the prior and posterior distribution for certain parameters of the model to see how much they were informed by the data. For example, we can inspect the coefficient of variation (CV) of the observation noise:
plot_prior_posterior(ww_result, "measurement_noise_cv")
We can further inspect our results object. It has three attributes:
names(ww_result)
#> [1] "job" "stan_model" "checksums" "summary" "fitted"
#> [6] "diagnostics" "runtime"The job attribute stores all information about the job that was
defined via EpiSewer. It contains the data used, alongside
meta-information, and the settings for the sampler. By calling
run(ww_result$job), the job can be run again.
names(ww_result$job)
#> [1] "job_name" "jobarray_size" "data" "model"
#> [5] "init" "fit_opts" "results_opts" "priors_text"
#> [9] "metainfo" "overwrite"In particular, we can print a concise summary of the modeling details
(the EpiSewer defaults in this case):
ww_result$job$model
#> measurements
#> |- concentrations_observe
#> |- noise_estimate
#> |- LOD_none
#>
#> sampling
#> |- outliers_none
#> |- sample_effects_none
#>
#> sewage
#> |- flows_observe
#> |- residence_dist_assume
#>
#> shedding
#> |- incubation_dist_assume
#> |- shedding_dist_assume (shedding_reference = symptom_onset)
#> |- load_per_case_calibrate
#> |- load_variation_estimate
#>
#> infections
#> |- generation_dist_assume
#> |- R_estimate_splines
#> |- seeding_estimate_rw
#> |- infection_noise_estimate (overdispersion = TRUE)➡️ Check out the model
specification
and detailed
example
vignettes to better understand the modeling details and customize the
EpiSewer model.
The summary attribute stores summarized results for important
parameters from the model.
names(ww_result$summary)
#> [1] "samples" "R"
#> [3] "expected_infections" "infections"
#> [5] "growth_rate" "doubling_time"
#> [7] "days_growing" "expected_load"
#> [9] "expected_concentration" "concentration"
#> [11] "normalized_concentration"For example, we can access the exact estimates for the reproduction number.
head(ww_result$summary$R, 5)
#> date mean median lower_0.95 lower_0.5 upper_0.5 upper_0.95
#> 1: 2021-12-03 1.036807 1.035745 0.6965128 0.9325218 1.135077 1.380163
#> 2: 2021-12-04 1.037644 1.036200 0.7072554 0.9352365 1.131915 1.375141
#> 3: 2021-12-05 1.038698 1.035635 0.7225687 0.9409083 1.130870 1.360814
#> 4: 2021-12-06 1.039966 1.037090 0.7374740 0.9479347 1.126985 1.354385
#> 5: 2021-12-07 1.041443 1.038955 0.7485678 0.9517000 1.126372 1.344230
#> type seeding
#> 1: estimate TRUE
#> 2: estimate TRUE
#> 3: estimate TRUE
#> 4: estimate TRUE
#> 5: estimate TRUEThe fitted attribute provides access to all details of the fitted stan
model (see
cmdstanr for
details). For example, we can use this to show sampler diagnostics for
each chain:
ww_result$fitted$diagnostic_summary()
#> $num_divergent
#> [1] 0 0 0 0
#>
#> $num_max_treedepth
#> [1] 0 0 0 0
#>
#> $ebfmi
#> [1] 0.9634045 0.9527539 1.0770535 1.0579167Finally, the checksums attribute gives us several checksums that
uniquely identify the job which was run. These can be used to check
whether two job results used the same or different models, input data,
fitting options, or inits. If all checksums are identical (and the seed
is not NULL), then the results should also be identical.
ww_result$checksums
#> $model
#> [1] "9ab6b8f9db0836e27cf9e9ac6ab1ea02"
#>
#> $input
#> [1] "c5d9acd8e8877d4ffbb24019996c07f9"
#>
#> $fit_opts
#> [1] "5309bbbc3cd1cc109eac60d2fc82de45"
#>
#> $results_opts
#> [1] "e92f83d0ca5d22b3bb5849d62c5412ee"
#>
#> $init
#> [1] "4329118480e9cb86b22d61c739d8c77e"To cite EpiSewer in a publication, please use:
@misc{lisonAdrianEpiSewerPackage,
title = {{EpiSewer: Estimate Epidemiological Parameters from Wastewater Measurements}},
author = {Lison, Adrian},
year = {2024},
doi = {10.5281/zenodo.10569101},
howpublished = {Zenodo}
}
If you use the dPCR-specific model of EpiSewer (see
noise_estimate_dPCR),
please also cite:
@article{lisonImprovingInferenceWastewaterbased2024,
title = {Improving Inference in Wastewater-Based Epidemiology by Modelling the Statistical Features of Digital {{PCR}}},
author = {Lison, Adrian and Julian, Timothy and Stadler, Tanja},
year = {2024},
journal = {bioRxiv preprint},
doi = {10.1101/2024.10.14.618307}
}