#' WP method
#'
#' This function implements an R0 estimation due to White and Pagano (Statistics
#' in Medicine, 2008). The method is based on maximum likelihood estimation in a
#' Poisson transmission model. See details for important implementation notes.
#'
#' This method is based on a Poisson transmission model, and hence may be most
#' most valid at the beginning of an epidemic. In their model, the serial
#' distribution is assumed to be discrete with a finite number of posible
#' values. In this implementation, if \code{mu} is not {NA}, the serial
#' distribution is taken to be a discretized version of a gamma distribution
#' with mean \code{mu}, shape parameter one, and largest possible value based on
#' parameter \code{tol}. When \code{mu} is \code{NA}, the function implements a
#' grid search algorithm to find the maximum likelihood estimator over all
#' possible gamma distributions with unknown mean and variance, restricting
#' these to a prespecified grid (see \code{search} parameter).
#'
#' When the serial distribution is known (i.e., \code{mu} is not \code{NA}),
#' sensitivity testing of \code{mu} is strongly recommended. If the serial
#' distribution is unknown (i.e., \code{mu} is \code{NA}), the likelihood
#' function can be flat near the maximum, resulting in numerical instability of
#' the optimizer. When \code{mu} is \code{NA}, the implementation takes
#' considerably longer to run. Users should be careful about units of time
#' (e.g., are counts observed daily or weekly?) when implementing.
#'
#' The model developed in White and Pagano (2008) is discrete, and hence the
#' serial distribution is finite discrete. In our implementation, the input
#' value \code{mu} is that of a continuous distribution. The algorithm
#' discretizes this input when \code{mu} is not \code{NA}, and hence the mean of
#' the serial distribution returned in the list \code{SD} will differ from
#' \code{mu} somewhat. That is to say, if the user notices that the input
#' \code{mu} and output mean of \code{SD} are different, this is to be expected,
#' and is caused by the discretization.
#'
#' @param NT Vector of case counts.
#' @param mu Mean of the serial distribution (needs to match case counts in time
#'           units; for example, if case counts are weekly and the serial
#'           distribution has a mean of seven days, then \code{mu} should be set
#'           to one). The default value of \code{mu} is set to \code{NA}.
#' @param search List of default values for the grid search algorithm. The list
#'               includes three elements: the first is \code{B}, which is the
#'               length of the grid in one dimension; the second is
#'               \code{scale.max}, which is the largest possible value of the
#'               scale parameter; and the third is \code{shape.max}, which is
#'               the largest possible value of the shape parameter. Defaults to
#'               \code{B=100, scale.max=10, shape.max=10}. For both shape and
#'               scale, the smallest possible value is 1/\code{B}.
#' @param tol Cutoff value for cumulative distribution function of the
#'            pre-discretization gamma serial distribution. Defaults to 0.999
#'            (i.e. in the discretization, the maximum is chosen such that the
#'            original gamma distribution has cumulative probability of no more
#'            than 0.999 at this maximum).
#'
#' @return \code{WP} returns a list containing the following components:
#'         \code{Rhat} is the estimate of R0, and \code{SD} is either the
#'         discretized serial distribution (if \code{mu} is not \code{NA}), or
#'         the estimated discretized serial distribution (if \code{mu} is
#'         \code{NA}). The list also returns the variable \code{check}, which is
#'         equal to the number of non-unique maximum likelihood estimators. The
#'         serial distribution \code{SD} is returned as a list made up of
#'         \code{supp} (the support of the distribution) and \code{pmf} (the
#'         probability mass function).
#'
#' @examples
#' # Weekly data.
#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
#'
#' # Obtain R0 when the serial distribution has a mean of five days.
#' res1 <- WP(NT, mu = 5 / 7)
#' res1$Rhat
#'
#' # Obtain R0 when the serial distribution has a mean of three days.
#' res2 <- WP(NT, mu = 3 / 7)
#' res2$Rhat
#'
#' # Obtain R0 when the serial distribution is unknown.
#' # NOTE: This implementation will take longer to run.
#' res3 <- WP(NT)
#' res3$Rhat
#'
#' # Find the mean of the estimated serial distribution.
#' serial <- res3$SD
#' sum(serial$supp * serial$pmf)
#'
#' @importFrom stats pexp qexp
#'
#' @export
WP <- function(NT, mu = NA,
               search = list(B = 100, shape.max = 10, scale.max = 10),
               tol = 0.999) {
  if (is.na(mu)) {
    print("You have assumed that the serial distribution is unknown.")
    res <- WP_unknown(NT, B = search$B, shape.max = search$shape.max,
                      scale.max = search$scale.max, tol = tol)
    Rhat <- res$Rhat
    p <- res$p
    range.max <- res$range.max
    JJ <- res$JJ
  } else {
    print("You have assumed that the serial distribution is known.")
    range.max <- ceiling(qexp(tol, rate = 1 / mu))
    p <- diff(pexp(0:range.max, 1 / mu))
    p <- p / sum(p)
    res <- WP_known(NT = NT, p = p)
    Rhat <- res
    JJ <- NA
  }

  return(list(Rhat = Rhat,
              check = length(JJ),
              SD = list(supp = 1:range.max, pmf = p)))
}