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#' Incidence Decay (ID)
#'
#' This function implements a least squares estimation method of R0 due to
#' Fisman et al. (PloS One, 2013). See details for implementation notes.
#'
#' The method is based on a straightforward incidence decay model. The estimate
#' of R0 is the value which minimizes the sum of squares between observed case
#' counts and cases counts expected under the model.
#'
#' This method is based on an approximation of the SIR model, which is most
#' valid at the beginning of an epidemic. The method assumes that the mean of
#' the serial distribution (sometimes called the serial interval) is known. The
#' final estimate can be quite sensitive to this value, so sensitivity testing
#' is strongly recommended. Users should be careful about units of time (e.g.,
#' are counts observed daily or weekly?) when implementing.
#'
#' @param cases Vector of case counts. The vector must be non-empty and only
#' contain positive integers.
#' @param mu Mean of the serial distribution. This must be a positive number.
#' The value should match the case counts in time units. For example, if case
#' counts are weekly and the serial distribution has a mean of seven days,
#' then `mu` should be set to `1`. If case counts are daily and the serial
#' distribution has a mean of seven days, then `mu` should be set to `7`.
#'
#' @return An estimate of the basic reproduction number (R0).
#'
#' @references
#' [Fisman et al. (PloS One, 2013)](
#' https://doi.org/10.1371/journal.pone.0083622)
#'
#' @seealso [idea()] for a similar method.
#'
#' @export
#'
#' @examples
#' # Weekly data.
#' cases <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
#'
#' # Obtain R0 when the serial distribution has a mean of five days.
#' id(cases, mu = 5 / 7)
#'
#' # Obtain R0 when the serial distribution has a mean of three days.
#' id(cases, mu = 3 / 7)
id <- function(cases, mu) {
exp(sum((log(cases) * mu) / seq_along(cases)) / length(cases))
}
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