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#' 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
#' ## ===================================================== ##
#' ## Illustrate on weekly data ##
#' ## ===================================================== ##
#'
#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
#' ## obtain Rhat when serial distribution has mean of five days
#' res1 <- WP(NT=NT, mu=5/7)
#' res1$Rhat
#' ## obtain Rhat when serial distribution has mean of three days
#' res2 <- WP(NT=NT, mu=3/7)
#' res2$Rhat
#' ## obtain Rhat when serial distribution is unknown
#' ## NOTE: this implementation will take longer to run
#' res3 <- WP(NT=NT)
#' res3$Rhat
#' ## find mean of estimated serial distribution
#' serial <- res3$SD
#' sum(serial$supp * serial$pmf)
#'
#' ## ========================================================= ##
#' ## Compute Rhat using only the first five weeks of data ##
#' ## ========================================================= ##
#'
#' res4 <- WP(NT=NT[1:5], mu=5/7) # serial distribution has mean of five days
#' res4$Rhat
#'
#' @importFrom stats pexp qexp
#'
#' @export
WP <- function(NT, mu=NA, search=list(B=100, shape.max=10, scale.max=10), tol=0.999) {
if (is.na(mu)) {
print("You have assumed that the serial distribution is unknown.")
res <- WP_unknown(NT=NT, B=search$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)))
}
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