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#'
#' First Derivate of the Weibull-Peleg Model
#'
#' Calculates the first derivative of Weibull-Peleg model at a given time for
#' the model parameters provided and the environmental conditions given.
#'
#' The model is developed from the isothermal Weibull model without
#' taking into
#' account in the derivation the time dependence of \eqn{b} for
#' non-isothermal temperature profiles.
#'
#' This function is compatible with the function
#' \code{\link{predict_inactivation}}.
#'
#' @section Model Equation:
#' \deqn{\frac{d(\mathrm{log}_{10}(S))}{dt}=-b(T) \cdot n \cdot
#' (- log10(S)/b(T))^{(n-1)/n)} }{
#' d(log10(S))/dt = -b * n *( - log10(S)/b )^( (n-1)/n)}
#'
#' \deqn{b(T) = \mathrm{ln}(1 + exp( k_b*(T - T_{crit}) ))}{
#' b(T) = ln( 1 + exp( k_b*(T - T_crit) ) )}
#'
#'
#' @param t numeric vector indicating the time of the experiment.
#' @param x list with the value of logS at t.
#' @param parms parameters for the secondary model. No explicit check of their validity
#' is performed (see section \bold{Model Parameters}).
#' @param temp_profile a function that provides the temperature at a given time.
#'
#' @return The value of the first derivative of logS at time \code{t} as a list.
#'
#' @section Model Parameters:
#' \itemize{
#' \item temp_crit: Temperature below which there is inactivation.
#' \item k_b: slope of the b ~ temp line for temperatures above the
#' critical one.
#' \item n: shape factor of the Weibull distribution.
#' }
#'
#' @section Note:
#' For logS=0, dlogS = 0 unless n=1. Hence, a small shift needs to be introduced
#' to logS.
#'
#' @seealso \code{\link{predict_inactivation}}
#'
dPeleg_model <- function(t, x, parms, temp_profile) {
temp <- temp_profile(t)
with(as.list(c(x, parms)),{
b <- log( 1 + exp( k_b*(temp - temp_crit) ) )
dlogS <- -b * n * ( - logS/b ) ^( (n-1)/n)
res <- c(dlogS)
return(list(res))
})
}
# #'
# #' First Derivate of the Peleg Model with Full Derivation
# #'
# #' Calculates the first derivative of Weibull-Peleg model at a given time for
# #' the model parameters provided and the environmental conditions given.
# #'
# #' The model is developed from the isothermal Weibull model
# #' taking into
# #' account in the derivation the time dependence of \eqn{b} for
# #' non-isothermal temperature profiles.
# #' For the version with full derivation, see \code{\link{dPeleg_model}}.
# #'
# #' This function is compatible with the function
# #' \code{\link{predict_inactivation}}.
# #'
# #' @section Model Equation:
# #'
# #' \deqn{\frac{dN}{dt}= -N \cdot t^n \cdot (b(T) \frac{n}{t} +
# #' \frac{db}{dT}\frac{dT}{dt})}{
# #' dN/dt = -N * t^n * (b(T)*n/t + db/dT*dT/dt)}
# #'
# #' \deqn{b(T) = \mathrm{ln}(1 + exp( k_b*(T - T_{crit}) ))}{
# #' b(T) = ln( 1 + exp( k_b*(T - T_crit) ) )}
# #'
# #' @param t numeric vector indicating the time of the experiment.
# #' @param x list with the value of N at t.
# #' @param parms parameters for the secondary model. No explicit check of their validity
# #' is performed (see section \bold{Model Parameters}).
# #' @param temp_profile a function that provides the temperature at a given time.
# #' @param dtemp_profile a function that provides the first derivative of the
# #' temperature at a given time.
# #'
# #' @return The value of the first derivative of N at time \code{t} as a list.
# #'
# #' @section Model Parameters:
# #' \itemize{
# #' \item temp_crit: Temperature below which there is inactivation.
# #' \item k_b: slope of the b ~ temp line for temperatures above the
# #' critical one.
# #' \item n: shape factor of the Weibull distribution.
# #' }
# #'
# #' @section Note:
# #' For t=0, dN = 0 unless n=1. Hence, a small shift needs to be introduced
# #' to t.
# #'
# #' @seealso \code{\link{predict_inactivation}}
# #'
# dPeleg_model_full<- function(t, x, parms, temp_profile, dtemp_profile) {
#
# temp <- temp_profile(t)
# dtemp <- dtemp_profile(t)
#
# with(as.list(c(x, parms)),{
#
# A <- exp(k_b*(temp - temp_crit))
# b <- log( 1 + A)
# db <- k_b * A / (1+A)
#
# dN <- -N * t^n * abs(b*n/t + db*dtemp)
# # dN <- -N * t^n * (b*n/t + db*abs(dtemp))
#
# res <- c(dN)
# return(list(res))
# })
#
# }
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