---
title: "White and Pagano: Utilizing the Serial Distribution"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{White and Pagano: Utilizing the Serial Distribution}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, include = FALSE}
library(Rnaught)
```

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`:

```{r}
# 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)
```

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:

```{r}
# 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

# 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
```

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:

```{r, dpi = 192, echo = FALSE}
par(mar = c(4.1, 4.1, 0.5, 0.5))

# Serial distribution.
plot(estimate$supp, estimate$pmf, xlab = "x", ylab = "p(x)",
  col = "black", lty = 1, type = "l"
)

# Serial mean.
abline(v = serial_mean, col = "blue", lty = 2)
# Serial median.
abline(v = serial_med, col = "red", lty = 2)

legend("topright",
  legend = c("Serial distribution", "Serial mean", "Serial median"),
  col = c("black", "blue", "red"),
  lty = c(1, 2, 2), cex = 0.5
)
```
