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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5419-5450. doi: 10.3934/mbe.2019270
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Equivalences between age structured models and state dependent distributed delay differential equations

  • 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
  • Received: 30 November 2018 Accepted: 27 March 2019 Published: 11 June 2019

  • 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.

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

    Related Papers:

  • 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.


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