With ordinary differential equations, you express the system of equations as dy/dt = f(y(t), t); the rate of change of the system depends on the current state of the system and the current time, but with delay differential equations dy/dt also depends on y(t - τ), where τ is a length of time back into the past. In general these are hard to solve numerically but there is a large class of useful equations with constant delays that are both interesting and tractable.
Integrating delay differential equations allows researchers in our department to model relationships where (say) the number of mosquitos entering a lifecycle phase now depends on the number of people who were bitten several days ago.
dde implements the method of Hairer, Norsett and Wanner (1993) where we use an ODE solver that can accurately interpolate to points within steps that it takes along with a ring buffer to store the history. It only works with non-stiff systems but we have found it to work well on large systems of equations where the DDE support in
deSolve (implemented using
lsoda) stopped working.
dde is now available on CRAN and can be installed with
To get started see the package vignette.