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+#' WP method background function WP_known
+#'
+#' This is a background/internal function called by \code{WP}. It computes the maximum
+#' likelihood estimator of R0 assuming that the serial distribution is known and finite discrete.
+#'
+#' @param NT Vector of case counts.
+#' @param p Discretized version of the serial distribution.
+#'
+#' @return The function returns the maximum likelihood estimator of R0.
+#'
+#' @keywords internal
+WP_known <- function(NT, p) {
+    k <- length(p)
+    TT <- length(NT) - 1
+	mu_t <- rep(0, TT)
+
+    for (i in 1:TT) {
+        Nt <- NT[i:max(1, i-k+1)]
+        mu_t[i]	<- sum(p[1:min(k, i)] * Nt)
+    }
+
+    Rhat <- sum(NT[-1]) / sum(mu_t)
+	return(Rhat)
+}
+
+#' WP method background function WP_unknown
+#'
+#' This is a background/internal function called by \code{WP}. It computes the maximum likelihood estimator
+#' of R0 assuming that the serial distribution is unknown but comes from a discretized gamma distribution.
+#' The function then implements a simple grid search algorithm to obtain the maximum likelihood estimator
+#' of R0 as well as the gamma parameters.
+#'
+#' @param NT Vector of case counts.
+#' @param B Length of grid for shape and scale (grid search parameter).
+#' @param shape.max Maximum shape value (grid \code{search} parameter).
+#' @param scale.max Maximum scale value (grid \code{search} parameter).
+#' @param tol cutoff value for cumulative distribution function of the serial distribution (defaults to 0.999).
+#'
+#' @return The function returns \code{Rhat}, the maximum likelihood estimator of R0, as well as the maximum
+#'         likelihood estimator of the discretized serial distribution given by \code{p} (the probability mass
+#'         function) and \code{range.max} (the distribution has support on the integers one to \code{range.max}).
+#'         The function also returns \code{resLL} (all values of the log-likelihood) at \code{shape} (grid for
+#'         shape parameter) and at \code{scale} (grid for scale parameter), as well as \code{resR0} (the full
+#'         vector of maximum likelihood estimators), \code{JJ} (the locations for the likelihood for these), and
+#'         \code{J0} (the location for the maximum likelihood estimator \code{Rhat}). If \code{JJ} and \code{J0}
+#'         are not the same, this means that the maximum likelihood estimator is not unique.
+#'
+#' @importFrom stats pgamma qgamma
+#'
+#' @keywords internal
+WP_unknown <- function(NT, B=100, shape.max=10, scale.max=10, tol=0.999) {
+	shape <- seq(0, shape.max, length.out=B+1)
+	scale <- seq(0, scale.max, length.out=B+1)
+	shape <- shape[-1]
+	scale <- scale[-1]
+
+	resLL <- matrix(0,B,B)
+	resR0 <- matrix(0,B,B)
+
+    for (i in 1:B) {
+        for (j in 1:B) {
+            range.max <- ceiling(qgamma(tol, shape=shape[i], scale=scale[j]))
+            p <- diff(pgamma(0:range.max, shape=shape[i], scale=scale[j]))
+            p <- p / sum(p)
+            mle <- WP_known(NT, p)
+            resLL[i,j] <- computeLL(p, NT, mle)
+            resR0[i,j] <- mle
+        }
+    }
+	
+    J0 <- which.max(resLL)
+    R0hat <- resR0[J0]
+    JJ <- which(resLL == resLL[J0], arr.ind=TRUE)
+    range.max <- ceiling(qgamma(tol, shape=shape[JJ[1]], scale=scale[JJ[2]]))
+    p <- diff(pgamma(0:range.max, shape=shape[JJ[1]], scale=scale[JJ[2]]))
+    p <- p / sum(p)
+	
+    return(list(Rhat=R0hat, J0=J0, ll=resLL, Rs=resR0, scale=scale, shape=shape, JJ=JJ, p=p, range.max=range.max))
+}
+
+#' WP method background function computeLL
+#'
+#' This is a background/internal function called by \code{WP}.  It computes the log-likelihood.
+#'
+#' @param NT Vector of case counts.
+#' @param p Discretized version of the serial distribution.
+#' @param R0 Basic reproductive ratio.
+#'
+#' @return This function returns the log-likelihood at the input variables and parameters.
+#'
+#' @keywords internal
+computeLL <- function(p, NT, R0) {
+    k <- length(p)
+    TT <- length(NT) - 1
+    mu_t <- rep(0, TT)
+
+    for (i in 1:TT) {
+        Nt <- NT[i:max(1, i-k+1)]
+        mu_t[i]	<- sum(p[1:min(k, i)] * Nt)
+    }
+
+    mu_t <- R0 * mu_t
+    LL <- sum(NT[-1] * log(mu_t)) - sum(mu_t)
+
+	return(LL)
+}