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   1 #' WP method
   2 #'
   3 #' This function implements an R0 estimation due to White and Pagano (Statistics
   4 #' in Medicine, 2008). The method is based on maximum likelihood estimation in a
   5 #' Poisson transmission model. See details for important implementation notes.
   6 #'
   7 #' This method is based on a Poisson transmission model, and hence may be most
   8 #' most valid at the beginning of an epidemic. In their model, the serial
   9 #' distribution is assumed to be discrete with a finite number of posible
  10 #' values. In this implementation, if \code{mu} is not {NA}, the serial
  11 #' distribution is taken to be a discretized version of a gamma distribution
  12 #' with mean \code{mu}, shape parameter one, and largest possible value based on
  13 #' parameter \code{tol}. When \code{mu} is \code{NA}, the function implements a
  14 #' grid search algorithm to find the maximum likelihood estimator over all
  15 #' possible gamma distributions with unknown mean and variance, restricting
  16 #' these to a prespecified grid (see \code{search} parameter).
  17 #'
  18 #' When the serial distribution is known (i.e., \code{mu} is not \code{NA}),
  19 #' sensitivity testing of \code{mu} is strongly recommended. If the serial
  20 #' distribution is unknown (i.e., \code{mu} is \code{NA}), the likelihood
  21 #' function can be flat near the maximum, resulting in numerical instability of
  22 #' the optimizer. When \code{mu} is \code{NA}, the implementation takes
  23 #' considerably longer to run. Users should be careful about units of time
  24 #' (e.g., are counts observed daily or weekly?) when implementing.
  25 #'
  26 #' The model developed in White and Pagano (2008) is discrete, and hence the
  27 #' serial distribution is finite discrete. In our implementation, the input
  28 #' value \code{mu} is that of a continuous distribution. The algorithm
  29 #' discretizes this input when \code{mu} is not \code{NA}, and hence the mean of
  30 #' the serial distribution returned in the list \code{SD} will differ from
  31 #' \code{mu} somewhat. That is to say, if the user notices that the input
  32 #' \code{mu} and output mean of \code{SD} are different, this is to be expected,
  33 #' and is caused by the discretization.
  34 #'
  35 #' @param NT Vector of case counts.
  36 #' @param mu Mean of the serial distribution (needs to match case counts in time
  37 #'           units; for example, if case counts are weekly and the serial
  38 #'           distribution has a mean of seven days, then \code{mu} should be set
  39 #'           to one). The default value of \code{mu} is set to \code{NA}.
  40 #' @param search List of default values for the grid search algorithm. The list
  41 #'               includes three elements: the first is \code{B}, which is the
  42 #'               length of the grid in one dimension; the second is
  43 #'               \code{scale.max}, which is the largest possible value of the
  44 #'               scale parameter; and the third is \code{shape.max}, which is
  45 #'               the largest possible value of the shape parameter. Defaults to
  46 #'               \code{B = 100, scale.max = 10, shape.max = 10}. For both shape
  47 #'               and scale, the smallest possible value is 1/\code{B}.
  48 #' @param tol Cutoff value for cumulative distribution function of the
  49 #'            pre-discretization gamma serial distribution. Defaults to 0.999
  50 #'            (i.e. in the discretization, the maximum is chosen such that the
  51 #'            original gamma distribution has cumulative probability of no more
  52 #'            than 0.999 at this maximum).
  53 #'
  54 #' @return \code{WP} returns a list containing the following components:
  55 #'         \code{Rhat} is the estimate of R0, and \code{SD} is either the
  56 #'         discretized serial distribution (if \code{mu} is not \code{NA}), or
  57 #'         the estimated discretized serial distribution (if \code{mu} is
  58 #'         \code{NA}). The list also returns the variable \code{check}, which is
  59 #'         equal to the number of non-unique maximum likelihood estimators. The
  60 #'         serial distribution \code{SD} is returned as a list made up of
  61 #'         \code{supp} (the support of the distribution) and \code{pmf} (the
  62 #'         probability mass function).
  63 #'
  64 #' @examples
  65 #' # Weekly data.
  66 #' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
  67 #'
  68 #' # Obtain R0 when the serial distribution has a mean of five days.
  69 #' res1 <- WP(NT, mu = 5 / 7)
  70 #' res1$Rhat
  71 #'
  72 #' # Obtain R0 when the serial distribution has a mean of three days.
  73 #' res2 <- WP(NT, mu = 3 / 7)
  74 #' res2$Rhat
  75 #'
  76 #' # Obtain R0 when the serial distribution is unknown.
  77 #' # NOTE: This implementation will take longer to run.
  78 #' res3 <- WP(NT)
  79 #' res3$Rhat
  80 #'
  81 #' # Find the mean of the estimated serial distribution.
  82 #' serial <- res3$SD
  83 #' sum(serial$supp * serial$pmf)
  84 #'
  85 #' @importFrom stats pexp qexp
  86 #'
  87 #' @export
  88 WP <- function(NT, mu = NA,
  89                search = list(B = 100, shape.max = 10, scale.max = 10),
  90                tol = 0.999) {
  91   if (is.na(mu)) {
  92     print("You have assumed that the serial distribution is unknown.")
  93     res <- WP_unknown(NT, B = search$B, shape.max = search$shape.max,
  94                       scale.max = search$scale.max, tol = tol)
  95     Rhat <- res$Rhat
  96     p <- res$p
  97     range.max <- res$range.max
  98     JJ <- res$JJ
  99   } else {
 100     print("You have assumed that the serial distribution is known.")
 101     range.max <- ceiling(qexp(tol, rate = 1 / mu))
 102     p <- diff(pexp(0:range.max, 1 / mu))
 103     p <- p / sum(p)
 104     res <- WP_known(NT = NT, p = p)
 105     Rhat <- res
 106     JJ <- NA
 107   }
 108
 109   return(list(Rhat = Rhat,
 110               check = length(JJ),
 111               SD = list(supp = 1:range.max, pmf = p)))
 112 }