Numerical methods and hypoexponential approximations for gamma distributed delay differential equations.

Area of Science:

• Mathematical modeling
• Numerical analysis
• Computational mathematics

Background:

• Gamma distributed delay differential equations (DDEs) are crucial in various modeling applications.
• Existing numerical methods often rely on approximations like the Erlang distribution, which can lead to inaccuracies.
• There is a need for more accurate and robust numerical techniques for gamma distributed DDEs.

Purpose of the Study:

• To develop and validate novel numerical methods for gamma distributed DDEs.
• To address the limitations of the common Erlang approximation in DDE modeling.
• To introduce a more accurate hypoexponential approximation for gamma distributed DDEs.

Main Methods:

• Development of a fourth-order accurate functional continuous Runge-Kutta (FCRK) method for direct integration of gamma distributed DDEs.
• Derivation of a hypoexponential approximation for gamma distributed DDEs, enabling solution via standard ordinary differential equation (ODE) solvers.
• Comparison of solutions obtained from FCRK, hypoexponential approximation, and the traditional Erlang approximation using reference solutions.

Main Results:

• The FCRK method demonstrates fourth-order convergence and high accuracy for gamma distributed DDEs.
• The common Erlang approximation can yield qualitatively different solutions compared to the underlying gamma distributed DDE.
• The proposed hypoexponential approximation provides a more accurate representation of gamma distributed DDEs without the limitations of the Erlang method.
• The hypoexponential approximation was successfully applied to statistical inference on synthetic epidemiological data.

Conclusions:

• The developed FCRK method and hypoexponential approximation offer significant improvements for modeling with gamma distributed DDEs.
• The hypoexponential approximation is a superior alternative to the Erlang approximation for accurately solving gamma distributed DDEs.
• These new methods enhance the reliability of DDE-based modeling in fields like epidemiology.