Abstract
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.
Highlights
Age structured population models have been used extensively in mathematical biology throughout the past 90 years [McKendrick, 1925; Trucco, 1965]. These age structured models describe the progression of individuals through an ageing process by using partial differential equations (PDEs), that can, in certain cases, be reduced to a delay differential equation (DDE) [Craig et al, 2016; Metz and Diekmann, 1986; Smith, 1993]
We show how the age structured PDE can be reduced to a state dependent distributed DDE
We show how a model of reticulocyte production can be reduced to a renewal equation whose dynamics are completely characterized by a simple system of ordinary differential equations
Summary
Age structured population models have been used extensively in mathematical biology throughout the past 90 years [McKendrick, 1925; Trucco, 1965] (see [Metz and Diekmann, 1986] for a review). To derive a state dependent distributed DDE, we consider a general age structured model with a variable ageing rate. For specific densities KA(t), we show equivalence between the state dependent distributed DDE and state-dependent discrete DDEs with one or two delays or a finite dimensional systems of ordinary differential equations (ODEs) These equivalences arise from the explicit consideration of the ageing process modelled by the distributed DDEs. By applying the linear chain technique to the age variable, instead of the time variable, we are able to establish the desired equivalences.
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