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Using multi-delay discrete delay differential equations to accurately simulate models with distributed delays

Delayed processes are ubiquitous throughout biology. These delays may arise through maturation processes or as the result of complex multi-step networks, and mathematical models with distributed delays are increasingly used to capture the heterogeneity present in these delayed processes. Typically, these distributed delay differential equations are simulated by discretizing the distributed delay and using existing tools for the resulting multi-delay delay differential equations or by using an equivalent representation under additional assumptions on the delayed process. Here, we use the existing framework of functional continuous Runge-Kutta methods to confirm the convergence of this common approach. Our analysis formalizes the intuition that the least accurate numerical method dominates the error. We give a number of examples to illustrate the predicted convergence, derive a new class of equivalences between distributed delay and discrete delay differential equations, and give conditions for the existence of breaking points in the distributed delay differential equation. Finally, our work shows how recently reported multi-delay complexity collapse arises naturally from the convergence of equations with multiple discrete delays to equations with distributed delays, offering insight into the dynamics of the Mackey-Glass equation.

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