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#' IDEA method
#'
#' This function implements a least squares estimation method of R0 due to Fisman et al. (PloS One, 2013).
#' See details for implementation notes.
#'
#' This method is closely related to that implemented in \code{ID}. The method is based on an 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 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, if case
#' counts are daily and the serial distribution has a mean of seven days, then \code{mu} should be set to seven)
#'
#' @return \code{IDEA} returns a list containing the following components: \code{Rhat} is the estimate of R0 and
#' \code{inputs} is a list of the original input variables \code{NT, mu}.
#'
#' @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
#' IDEA(NT=NT, mu=5/7)
#' ## obtain Rhat when serial distribution has mean of three days
#' IDEA(NT=NT, mu=3/7)
#'
#' ## ========================================================= ##
#' ## Compute Rhat using only the first five weeks of data ##
#' ## ========================================================= ##
#'
#' IDEA(NT=NT[1:5], mu=5/7) # serial distribution has mean of five days
#'
#' @export
IDEA <- function(NT, mu) {
if (length(NT) < 2)
print("Warning: length of NT should be at least two.")
else {
NT <- as.numeric(NT)
TT <- length(NT)
s <- (1:TT) / mu
y1 <- log(NT) / s
y2 <- s^2
y3 <- log(NT)
IDEA1 <- sum(y2) * sum(y1) - sum(s) * sum(y3)
IDEA2 <- TT * sum(y2) - sum(s)^2
IDEA <- exp(IDEA1 / IDEA2)
return(IDEA)
}
}
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