On Erlang mixture approximations for differential equations with distributed time delays

Abstract

In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that we 1) propose an Erlang mixture approximation of the kernel in the DDEs and 2) use the linear chain trick to transform the resulting approximate DDEs to ODEs. Furthermore, we prove that the approximation converges for continuous and bounded kernels and for specific choices of the coefficients if the number of terms increases sufficiently fast. We show that the approximate ODEs can be used to assess the stability of the steady states of the original DDEs and that the solution to the ODEs converges if the kernel is also exponentially bounded. Additionally, we propose an approach based on bisection and least-squares estimation for determining optimal parameter values in the approximation. Finally, we present numerical examples that demonstrate the accuracy and convergence rate obtained with the optimal parameters and the efficacy of the proposed approach for bifurcation analysis and Monte Carlo simulation. The numerical examples involve a modified logistic equation, chemotherapy-induced myelosuppression, and a point reactor kinetics model of a molten salt nuclear fission reactor.

Figures (8)

• Figure 1: Erlang kernels for $a = 1.5$ (left) and $a = 1$ (middle) and $m = 0, \ldots, 5$, and an Erlang mixture kernel (right) for $a = 1$, $M = 5$, and the following values of the coefficients. Blue: $c_m = 1/(M+1)$ for $m = 0, \ldots, M$. Red: $c_0 = 0$ and $c_m = 1/M$ for $m = 1, \ldots, M$. Yellow: $c_0 = c_M = 1/2$ and $c_m = 0$ for $m = 1, \ldots, M-1$.
• Figure 1: Convergence analysis for the modified logistic equation. The state (top left) and kernel (bottom left) errors for Erlang mixture approximations of different orders, $M$, obtained with the proposed least-squares approach and the reference approach based on the theoretical expressions for the coefficients. The bottom right plot shows the state and kernel errors against each other, and the top right plot shows the state error obtained with the numerical approaches for non-stiff and stiff DDEs for different time step sizes, $\Delta t$. The black solid lines in the left column are proportional to $1/(M+1)$, and the one in the top right is proportional to $\Delta t$.
• Figure 1: Bifurcation analysis with respect to the model parameter $\sigma$ (left column) and the kernel parameter $\mu_2$ (right column) for the modified logistic equation. First row: Eigenvalues. Second row: The largest real part of the eigenvalues. Third and bottom row: Simulations for selected parameter values (obtained with the numerical method described in Appendix \ref{['sec:numerical:simulation:non:stiff']}).
• Figure 1: An infinite-order Erlang mixture delta family for a fixed rate parameter, $a$, and different values of $t$ (left) and for fixed $t$ and different values of $a$ (right). For $t = 0$ in the left figure, the value of $\delta_a$ is 2 for $s \in [0, 0.5)$ (see also Corollary \ref{['lem:erlang:mixture:delta:t:equal:zero']}).
• Figure 2: Convergence analysis for the myelosuppression model. The maximum state error (top left) and the kernel error (bottom left) for Erlang mixture approximations of different orders, $M$, obtained with the least-squares approach and the reference approach based on theoretical values of the coefficients. The kernel error is shown for Guglielmi and Hairer's approach in the bottom right, and the true and approximate kernels are shown in the top right for $M = 128$ and $\epsilon_r = 10^{-11}$ (corresponding to 309 terms). The black solid line in the top left is proportional to $1/(M+1)$ whereas the black dashed line in the bottom left is proportional to $1/(M+1)^{3/4}$.

Theorems & Definitions (65)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Corollary 3.4
• Proof 1
• Corollary 3.5
• Corollary 3.6
• Proof 2
• Definition 4.1
• Proposition 4.2: Erlang mixture approximation