#'
#' #' Simulate a partially observed Poisson branching process with discrete generations
#' #'
#' #' For a given reproduction number, this function simulate a Poisson branching and return observed incidence
#' #' (per generation) accounting for under-reporting.
#' #' . This is \code{some code}.
#' #' Bullet list:
#' #'
#' #' \itemize{
#' #'
#' #' \item toto: il est beau
#' #'
#' #' \item tata: il est caca
#' #'
#' #' }
#' #'
#' #'
#' #' @author Pierre Nouvellet (\email{p.nouvellet@@imperial.ac.uk})
#' #'
#' #' @export
#' #'
#' #' @param x this is what x is
#' #'
#' #' @param y this is what y is
#' #'
#' #' @return
#' #' The function returns ....
#' #'
#' #' @examples
#' #'
#' #' x <- simulate_poisson(1, 2)
#' #' x
#' #'
#' simulate_poisson <- function(R.eff,n,rho,t.max) {
#'
#' R <- R.eff$R.effective
#'
#' N <- matrix(1,n,t.max)
#' for (i in 2:t.max){
#' N[,i] <- rpois(n,R*N[,i-1])
#' }
#' n.ongoing <- n - sum(N[,t.max]==0)
#' if (n.ongoing>0) warning(
#' paste0('There are ',n.ongoing,' outbreaks that are not extinct,
#' consider increasing the number of time-steps (especially if R<1).'))
#'
#' z.sim <- rowSums(N)
#' y.obs.full <- rbinom(n,z.sim,rho)
#' f <- which(y.obs.full==0)
#' if (length(f)>0){
#' y.obs <- y.obs.full[-f]
#' }else{
#' y.obs <- y.obs.full
#' }
#' return( list(outbreak.size = z.sim, observed.size.full = y.obs.full,
#' observed.size = y.obs) )
#' }
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