  We use the McKendrick equation with variable ageing rate and randomly
distributed maturation time to derive a state dependent distributed delay
differential equation. We show that the resulting delay differential equation
preserves non-negativity of initial conditions and we characterise local
stability of equilibria. By specifying the distribution of maturation age, we
recover state dependent discrete, uniform and gamma distributed delay
differential equations. We show how to reduce the uniform case to a system of
state dependent discrete delay equations and the gamma distributed case to a
system of ordinary differential equations. To illustrate the benefits of these
reductions, we convert previously published transit compartment models into
equivalent distributed delay differential equations.
