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Equivalences between age structured models and state dependent distributed delay differential equations

Tyler Cassidy Morgan Craig Antony R. Humphries

*Corresponding author: Tyler Cassidy tyler.cassidy@mail.mcgill.ca

MBE2019,5,5419doi:10.3934/mbe.2019270

We use the McKendrick equation with variable ageing rate and randomly distributed mat-uration 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 depen-dent 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.

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