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White and Pagano: Utilizing the Serial Distribution

The serial distribution of an infectious disease is the distribution of the time from when an infectious individual – the infector – becomes symptomatic, to when another individual who is infected by the infector becomes symptomatic. The serial interval refers to a range of likely values from this distribution, although it is typically reported as the mean.

In the White and Pagano method, wp(), the serial distribution is assumed to be a discretized, finite version of a gamma distribution. Setting the parameter serial to TRUE causes this discretized distribution to be returned in addition to the estimate of R0. Furthermore, the method can be used whether or not the serial interval (specified as the parameter mu) is known. When mu is specified, it is taken to be the mean of a continuous gamma distribution (i.e., before the discretization). As such, the mean computed from the returned serial distribution may differ slightly from mu:

# Case counts.
cases <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)

estimate <- wp(cases, mu = 3.333, serial = TRUE)

# `supp` is the support of the distribution, and `pmf` is its probability mass
# function.
sum(estimate$supp * estimate$pmf)
#> [1] 3.840047

When mu is unspecified (left to its default value of NA), the method performs a maximum likelihood estimation over all (discretized) gamma distributions via a grid search, whose range of parameters are specified via grid_length, max_shape and max_scale (see ?wp for more details). It is useful to return the estimated serial distribution in this case, as it can provide estimates of the serial interval when it is unknown:

# The grid search parameters specified below are the default values.
estimate <- wp(cases, serial = TRUE,
  grid_length = 100, max_shape = 10, max_scale = 10
)

serial_mean <- sum(estimate$supp * estimate$pmf)
serial_mean
#> [1] 3.564191

# Compute the (discrete) median for an alternative estimate of the serial
# interval.
cdf <- cumsum(estimate$pmf)
serial_med <- estimate$supp[which(cdf >= 0.5 & estimate$pmf - cdf + 1 >= 0.5)]
serial_med
#> [1] 2

Below is a graph of the above results, containing the serial distribution as well as its mean and median, which could be used as estimates of the serial interval:

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