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Estimate a severity measure that varies over time

Source: R/cfr_time_varying.R
cfr_time_varying.Rd

Calculates how the severity of a disease changes over time while optionally correcting for reporting delays using an epidemiological delay distribution of the time between symptom onset and outcome (e.g. onset-to-death for the fatality risk).

Usage

cfr_time_varying(
  data,
  delay_density = NULL,
  burn_in = 7,
  smoothing_window = NULL
)

Arguments

data

A <data.frame> containing the outbreak data. A daily time series with dates or some other absolute indicator of time (e.g. epiday or epiweek) and the numbers of new cases and new deaths at each time point. Note that the required columns are "date" (for the date), "cases" (for the number of reported cases), and "deaths" (for the number of reported deaths) on each day of the outbreak.

Note that the <data.frame> is required to have an unbroken sequence of dates with no missing dates in between. The "date" column must be of class Date (see as.Date()).

Note also that the total number of cases must be greater than the total number of reported deaths.

delay_density

An optional argument that controls whether delay correction is applied in the severity estimation. May be NULL, for no delay correction, or a function that returns the density function of a distribution to evaluate density at user-specified values, e.g. function(x) stats::dgamma(x = x, shape = 5, scale = 1).

burn_in

A single integer-like value for the number of time-points (typically days) to disregard at the start of the time-series, if a burn-in period is desired.

Defaults to 7, which is a sensible default value that disregards the first week of cases and deaths, assuming daily data.

To consider all case data including the start of the time-series, set this argument to 0.

smoothing_window

An odd number determining the smoothing window size to use when smoothing the case and death time-series, using a rolling median procedure (as the k argument to stats::runmed()) before calculating the time-varying severity.

The default behaviour is to apply no smoothing. The minimum value of this argument is 1.

Value

A <data.frame> with the date, maximum likelihood estimate and 95% confidence interval of the daily severity estimates, named "severity_estimate", "severity_low", and "severity_high", with one row for each day in the original data.frame.

Details: Adjusting for delays between two time series

This function estimates the number of cases which have a known outcome over time, following Nishiura et al. (2009). The function calculates a quantity \(k_t\) for each day within the input data, which represents the number of cases estimated to have a known outcome, on day \(t\). \(k_t\) is calculated in the following way: $$k_t = \sum_{j = 0}^t c_t f_{j - t}$$

We then assume that the severity measure, for example CFR, of interest is binomially distributed, in the following way:

$$d_t \sim {\sf Binomial}(k_t, \theta_t)$$

We use maximum likelihood estimation to determine the value of \(\theta_t\) for each \(t\), where \(\theta\) represents the severity measure of interest.

The epidemiological delay distribution passed to delay_density is used to obtain a probability mass function parameterised by time; i.e. \(f(t)\) which gives the probability of the binary outcome of a case (survival or death) being known by time \(t\). The delay distribution is parameterised with disease-specific parameters before it is supplied here.

Note that the function arguments burn_in and smoothing_window are not explicitly used in this calculation. burn_in controls how many estimates at the beginning of the outbreak are replaced with NAs — the calculation above is not applied to the first burn_in data points. The calculation is applied to the smoothed data, if a smoothing_window is specified.

References

Nishiura, H., Klinkenberg, D., Roberts, M., & Heesterbeek, J. A. P. (2009). Early Epidemiological Assessment of the Virulence of Emerging Infectious Diseases: A Case Study of an Influenza Pandemic. PLOS ONE, 4(8), e6852. doi:10.1371/journal.pone.0006852

Examples

# get data pre-loaded with the package
data("covid_data")
df_covid_uk <- covid_data[covid_data$country == "United Kingdom" &
covid_data$date <= as.Date("2020-09-01"), ]

# estimate time varying severity without correcting for delays
cfr_time_varying <- cfr_time_varying(
  data = df_covid_uk,
  burn_in = 7L
)
# View
tail(cfr_time_varying)
#>           date severity_estimate severity_low severity_high
#> 238 2020-08-27       0.017322835  0.010887155    0.02611008
#> 239 2020-08-28       0.011527378  0.006602850    0.01865240
#> 240 2020-08-29       0.007542426  0.003903177    0.01313800
#> 241 2020-08-30       0.007352941  0.003804975    0.01280885
#> 242 2020-08-31       0.006504065  0.002812052    0.01277518
#> 243 2020-09-01       0.009393681  0.004698353    0.01674560

# estimate time varying severity while correcting for delays
# obtain onset-to-death delay distribution parameters from Linton et al. 2020
# J. Clinical Medicine: <https://doi.org/10.3390/jcm9020538>
# view only the first values
cfr_time_varying <- cfr_time_varying(
  data = df_covid_uk,
  delay_density = function(x) dlnorm(x, meanlog = 2.577, sdlog = 0.440),
  burn_in = 7L
)
tail(cfr_time_varying)
#>           date severity_estimate severity_low severity_high
#> 238 2020-08-27       0.021317829  0.013406829    0.03209837
#> 239 2020-08-28       0.015281757  0.008759411    0.02469826
#> 240 2020-08-29       0.011310085  0.005857363    0.01967302
#> 241 2020-08-30       0.011204482  0.005802549    0.01949011
#> 242 2020-08-31       0.007407407  0.003203259    0.01454312
#> 243 2020-09-01       0.010110294  0.005057522    0.01801795