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#'
#' First Derivate of the Weibull-Mafart Model
#'
#' Calculates the first derivative of Weibull-Mafart model at a given time for
#' the model parameters provided and the environmental conditions given.
#'
#' The model is developed from the isothermal Weibull-Mafart model without
#' taking into
#' account in the derivation the time dependence of \eqn{\delta_T} for
#' non-isothermal temperature profiles.
#'
#' This function is compatible with the function
#' \code{\link{predict_inactivation}}.
#'
#' @section Model Equation:
#'
#' \deqn{\frac{dN}{dt} = -N \cdot p \cdot (1/\delta)^p \cdot t^{p-1} }{
#' dN/dt = -N * p * (1/delta)^p * t^(p-1)}
#'
#' \deqn{\delta(T) = \delta_{ref} \cdot 10^{- (T-T_ref)/z} }{
#' delta(T) = delta_ref * 10^(- (T-T_ref)/z )}
#'
#' @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.
#'
#' @return The value of the first derivative of N at time \code{t} as a list.
#'
#' @section Model Parameters:
#' \itemize{
#' \item temp_ref: Reference temperature for the calculation.
#' \item delta_ref: Value of the scale factor at the reference temperature.
#' \item z: z-value.
#' \item p: 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}}
#'
dMafart_model<- function(t, x, parms, temp_profile) {
temp <- temp_profile(t)
with(as.list(c(x, parms)),{
delta <- delta_ref * 10^( -(temp-temp_ref)/z)
dN <- - N * p * (1/delta)^p * t^(p-1) * log(10)
res <- c(dN)
return(list(res))
})
}
# #'
# #' First Derivate of the Mafart Model with Full Derivation
# #'
# #' Calculates the first derivative of Weibull-Mafart model at a given time for
# #' the model parameters provided and the environmental conditions given.
# #'
# #' The model is developed from the isothermal Weibull-Mafart model taking into
# #' account in the derivation the time dependence of \eqn{\delta_T} for
# #' non-isothermal temperature profiles.
# #' For the linearized version, see \code{\link{dMafart_model}}
# #'
# #' This function is compatible with the function
# #' \code{\link{predict_inactivation}}.
# #'
# #' @section Model Equation:
# #'
# #' \deqn{ \frac{dN}{dt} = -N \cdot p \cdot (t/\delta)^{p-1} \cdot
# #' \frac{\mathrm{ln}(10)}{\delta} (1+t \frac{\mathrm{ln}(10)}{z}
# #' \frac{dT}{dt}) }{
# #' dN/dt =-N*p*(t/delta)^(p-1)*log(10)/delta*(1+log(10)*t/z*dtemp)}
# #'
# #' \deqn{\delta(T) = \delta_{ref} \cdot 10^{- (T-T_ref)/z}}{
# #' delta(T) = delta_ref * 10^(- (T-T_ref)/z )}
# #'
# #' @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_ref: Reference temperature for the calculation.
# #' \item delta_ref: Value of the scale factor at the reference temperature.
# #' \item z: z-value.
# #' \item p: 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}}
# #'
# dMafart_model_full<- function(t, x, parms, temp_profile, dtemp_profile) {
#
# temp <- temp_profile(t)
# dtemp <- dtemp_profile(t)
#
# with(as.list(c(x, parms)),{
# delta <- delta_ref * 10^( -(temp-temp_ref)/z)
# dN <- -N * p * (t/delta)^(p-1) * log(10)/delta*abs( 1+log(10)*t/z*dtemp )
# # dN <- -N * p * (t/delta)^(p-1) * log(10)/delta*( 1+log(10)*t/z*abs(dtemp) )
#
# res <- c(dN)
# return(list(res))
# })
#
# }
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