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   1 #' Incidence Decay and Exponential Adjustment (IDEA)
   2 #'
   3 #' This function implements a least squares estimation method of R0 due to
   4 #' Fisman et al. (PloS One, 2013). See details for implementation notes.
   5 #'
   6 #' This method is closely related to that implemented in [id()]. The method is
   7 #' based on an incidence decay model. The estimate of R0 is the value which
   8 #' minimizes the sum of squares between observed case counts and case counts
   9 #' expected under the model.
  10 #'
  11 #' This method is based on an approximation of the SIR model, which is most
  12 #' valid at the beginning of an epidemic. The method assumes that the mean of
  13 #' the serial distribution (sometimes called the serial interval) is known. The
  14 #' final estimate can be quite sensitive to this value, so sensitivity testing
  15 #' is strongly recommended. Users should be careful about units of time (e.g.,
  16 #' are counts observed daily or weekly?) when implementing.
  17 #'
  18 #' @param cases Vector of case counts. The vector must be of length at least two
  19 #'   and only contain positive integers.
  20 #' @param mu Mean of the serial distribution. This must be a positive number.
  21 #'   The value should match the case counts in time units. For example, if case
  22 #'   counts are weekly and the serial distribution has a mean of seven days,
  23 #'   then `mu` should be set to `1`. If case counts are daily and the serial
  24 #'   distribution has a mean of seven days, then `mu` should be set to `7`.
  25 #'
  26 #' @return An estimate of the basic reproduction number (R0).
  27 #'
  28 #' @references [Fisman et al. (PloS One, 2013)](
  29 #' https://doi.org/10.1371/journal.pone.0083622)
  30 #'
  31 #' @seealso [id()] for a similar method.
  32 #'
  33 #' @export
  34 #'
  35 #' @examples
  36 #' # Weekly data.
  37 #' cases <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
  38 #'
  39 #' # Obtain R0 when the serial distribution has a mean of five days.
  40 #' idea(cases, mu = 5 / 7)
  41 #'
  42 #' # Obtain R0 when the serial distribution has a mean of three days.
  43 #' idea(cases, mu = 3 / 7)
  44 idea <- function(cases, mu) {
  45   s <- seq_along(cases) / mu
  46
  47   x1 <- sum(s)
  48   x2 <- sum(s^2)
  49   x3 <- log(cases)
  50
  51   y1 <- x2 * sum(x3 / s) - x1 * sum(x3)
  52   y2 <- x2 * length(cases) - x1^2
  53
  54   exp(y1 / y2)
  55 }