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Mathematical Biosciences and Engineering, 2019, 16(5): 5419-5450. doi: 10.3934/mbe.2019270. Research article Special Issues

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

Tyler Cassidy, Morgan Craig, Antony R. Humphries

1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, H3A 0B9, Canada
2 Département de mathématiques et de statistique, Université de Montréal, 2920 chemin de la Tour, Montréal, H3T 1J4, Canada
3 Department of Physiology, McGill University, 3655 Promenade Sir-William-Osler, Montréal, H3G 1Y6, Canada

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Special Issues: Recent Advances in Mathematical Population Dynamics

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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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Keywords: delay differential equations; age structured populations; transit compartment models; linear chain technique; state dependent delays

Citation: Tyler Cassidy, Morgan Craig, Antony R. Humphries. Equivalences between age structured models and state dependent distributed delay differential equations. Mathematical Biosciences and Engineering, 2019, 16(5): 5419-5450. doi: 10.3934/mbe.2019270

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