  Even in large systems, the effect of noise arising from when populations are
initially small can persist to be measurable on the macroscale. A deterministic
approximation to a stochastic model will fail to capture this effect, but it
can be accurately approximated by including an additional random time-shift to
the initial conditions. We present a efficient numerical method to compute this
time-shift distribution for a large class of stochastic models. The method
relies on differentiation of certain functional equations, which we show can be
effectively automated by deriving rules for different types of model rates that
arise commonly when mass-action mixing is assumed. Explicit computation of the
time-shift distribution can be used to build a practical tool for the efficient
generation of macroscopic trajectories of stochastic population models, without
the need for costly stochastic simulations. Full code is provided to implement
this and we demonstrate our method on an epidemic model and a model of
within-host viral dynamics.
