Delayed state for delay differential equations

delay(state, T) evaluates an ODE state at the past time t - T, turning an ordinary differential equation model into a delay differential equation (DDE). The semantics match the delay() function of Monolix.

Usage

delay(state, T)

Arguments

state: An ODE state (compartment) defined in the model whose past value is required.
T: The delay duration. May be a constant, a parameter, a covariate, or any model expression. The value returned is the state at time t - T.

Value

Inside an rxode2 model, the value of state at the past time t - T. Before the start of integration the constant initial-condition history is used.

Details

Delayed states are interpolated from the solver's dense output using the same 8th-order Dormand-Prince interpolant as the integrator, so a delayed value is obtained to the full accuracy of the solution.

Because this requires dense output, delay models are solved on a dense path. When a model uses delay(), the default solving method becomes the dense AutoSwitch composite "dop853+ros4" and dense output is enabled automatically; method = "dop853" also works. The composite switches between dop853 and ros4 per segment in dense mode, so a delay model that is non-stiff in one region and stiff in another is solved efficiently in a single pass. Stiff delay models can also be solved with method = "ros4" directly, whose dense Rosenbrock output is likewise recorded and interpolated for delay(). The integrator step size is automatically capped to the smallest delay so that short delays remain accurate. Methods that cannot record dense history raise an error.

The dense-output and delay-history machinery is adapted from the dde package by Rich FitzJohn and Wes Hinsley (Imperial College of Science, Technology and Medicine), following the approach of Hairer, Norsett and Wanner.

Author

Matthew L. Fidler

Examples

# \donttest{
# Classic linear delay differential equation y'(t) = -y(t - 1)
dde <- rxode2({
  y(0) <- 1
  d/dt(y) <- -delay(y, 1)
})
#>  
#>  

s <- rxSolve(dde, et(seq(0, 5, by = 0.1)))
#>  
#>  

# Delayed (Hutchinson) logistic growth
hutch <- rxode2({
  r <- 0.5
  K <- 10
  tau <- 1
  N(0) <- 2
  d/dt(N) <- r * N * (1 - delay(N, tau) / K)
})
#>  
#>  

s2 <- rxSolve(hutch, et(seq(0, 40, by = 0.5)))
#>  
#>  
# }