General Solver for Delay Differential Equations.
Function dede is a general solver for delay differential equations, i.e.
equations where the derivative depends on past values of the state variables
or their derivatives.
dede(y, times, func=NULL, parms,
method = c( "lsoda", "lsode", "lsodes", "lsodar", "vode",
"daspk", "bdf", "adams", "impAdams", "radau"), control = NULL, ...)y |
the initial (state) values for the DE system, a vector. If
|
times |
time sequence for which output is wanted; the first
value of |
func |
an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t.
The return value of If method "daspk" is used, then |
parms |
parameters passed to |
method |
the integrator to use, either a string ( |
control |
a list that can supply (1) the size of the history array, as
|
... |
additional arguments passed to the integrator. |
Cubic Hermite interpolation is used by default to obtain an accurate
interpolant at the requested lagged time. For methods adams, impAdams,
a more accurate interpolation method can be triggered by setting
control$interpol = 2.
dede does not deal explicitly with propagated derivative discontinuities,
but relies on the integrator to control the stepsize in the region of a
discontinuity.
dede does not include methods to deal with delays that are smaller than the
stepsize, although in some cases it may be possible to solve such models.
For these reasons, it can only solve rather simple delay differential equations.
When used together with integrator lsodar, or lsode, dde
can simultaneously locate a root, and trigger an event. See last example.
A matrix of class deSolve with up to as many rows as elements in
times and as many
columns as elements in y plus the number of "global" values
returned in the second element of the return from func, plus an
additional column (the first) for the time value. There will be one
row for each element in times unless the integrator returns
with an unrecoverable error. If y has a names attribute, it
will be used to label the columns of the output value.
Karline Soetaert <karline.soetaert@nioz.nl>
## =============================================================================
## A simple delay differential equation
## dy(t) = -y(t-1) ; y(t<0)=1
## =============================================================================
##-----------------------------
## the derivative function
##-----------------------------
derivs <- function(t, y, parms) {
if (t < 1)
dy <- -1
else
dy <- - lagvalue(t - 1)
list(c(dy))
}
##-----------------------------
## initial values and times
##-----------------------------
yinit <- 1
times <- seq(0, 30, 0.1)
##-----------------------------
## solve the model
##-----------------------------
yout <- dede(y = yinit, times = times, func = derivs, parms = NULL)
##-----------------------------
## display, plot results
##-----------------------------
plot(yout, type = "l", lwd = 2, main = "dy/dt = -y(t-1)")
## =============================================================================
## The infectuous disease model of Hairer; two lags.
## example 4 from Shampine and Thompson, 2000
## solving delay differential equations with dde23
## =============================================================================
##-----------------------------
## the derivative function
##-----------------------------
derivs <- function(t,y,parms) {
if (t < 1)
lag1 <- 0.1
else
lag1 <- lagvalue(t - 1,2)
if (t < 10)
lag10 <- 0.1
else
lag10 <- lagvalue(t - 10,2)
dy1 <- -y[1] * lag1 + lag10
dy2 <- y[1] * lag1 - y[2]
dy3 <- y[2] - lag10
list(c(dy1, dy2, dy3))
}
##-----------------------------
## initial values and times
##-----------------------------
yinit <- c(5, 0.1, 1)
times <- seq(0, 40, by = 0.1)
##-----------------------------
## solve the model
##-----------------------------
system.time(
yout <- dede(y = yinit, times = times, func = derivs, parms = NULL)
)
##-----------------------------
## display, plot results
##-----------------------------
matplot(yout[,1], yout[,-1], type = "l", lwd = 2, lty = 1,
main = "Infectuous disease - Hairer")
## =============================================================================
## time lags + EVENTS triggered by a root function
## The two-wheeled suitcase model
## example 8 from Shampine and Thompson, 2000
## solving delay differential equations with dde23
## =============================================================================
##-----------------------------
## the derivative function
##-----------------------------
derivs <- function(t, y, parms) {
if (t < tau)
lag <- 0
else
lag <- lagvalue(t - tau)
dy1 <- y[2]
dy2 <- -sign(y[1]) * gam * cos(y[1]) +
sin(y[1]) - bet * lag[1] + A * sin(omega * t + mu)
list(c(dy1, dy2))
}
## root and event function
root <- function(t,y,parms) ifelse(t>0, return(y), return(1))
event <- function(t,y,parms) return(c(y[1], y[2]*0.931))
gam = 0.248; bet = 1; tau = 0.1; A = 0.75
omega = 1.37; mu = asin(gam/A)
##-----------------------------
## initial values and times
##-----------------------------
yinit <- c(y = 0, dy = 0)
times <- seq(0, 12, len = 1000)
##-----------------------------
## solve the model
##-----------------------------
## Note: use a solver that supports both root finding and events,
## e.g. lsodar, lsode, lsoda, adams, bdf
yout <- dede(y = yinit, times = times, func = derivs, parms = NULL,
method = "lsodar", rootfun = root, events = list(func = event, root = TRUE))
##-----------------------------
## display, plot results
##-----------------------------
plot(yout, which = 1, type = "l", lwd = 2, main = "suitcase model", mfrow = c(1,2))
plot(yout[,2], yout[,3], xlab = "y", ylab = "dy", type = "l", lwd = 2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.