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We use the McKendrick equation with variable ageing rate and randomlydistributed maturation time to derive a state dependent distributed delaydifferential equation. We show that the resulting delay differential equationpreserves non-negativity of initial conditions and we characterise localstability of equilibria. By specifying the distribution of maturation age, werecover state dependent discrete, uniform and gamma distributed delaydifferential equations. We show how to reduce the uniform case to a system ofstate dependent discrete delay equations and the gamma distributed case to asystem of ordinary differential equations. To illustrate the benefits of thesereductions, we convert previously published transit compartment models intoequivalent distributed delay differential equations.

A Recipe for State Dependent Distributed Delay Differential Equations