White and Pagano: Utilizing the Serial Distribution â¢ Rnaught

+ + + +

The serial distribution of an infectious disease is the distribution +of the time from when an infectious individual â the infector â 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, 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: