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#' ID method
#'
#' 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 NT Vector of case counts.
#' @param mu Mean of the serial distribution. This 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{ID} returns a single value, the estimate of R0.
#'
#' @examples
#' # Weekly data:
#' NT <- 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(NT, mu = 5 / 7)
#'
#' # Obtain R0 when the serial distribution has a mean of three days.
#' ID(NT, mu = 3 / 7)
#'
#' @export
ID <- function(NT, mu) {
NT <- as.numeric(NT)
TT <- length(NT)
s <- (1:TT) / mu
y <- log(NT) / s
R0_ID <- exp(sum(y) / TT)
return(R0_ID)
}
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