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Maximum likelihood inference for multivariate delay differential equation models.

Ahmed Adly Mahmoud1, Abdalla Rabie1, Sarat Chandra Dass2

  • 1Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt.

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Summary

A new maximum likelihood inference framework is developed for general delay differential equation models. This approach handles multiple delay parameters without restrictive assumptions, advancing statistical modeling for complex systems.

Keywords:
Delay differential equation (DDE)Delay differential equation models (DDEMs)Delayed pharmacokinetic modelsDelayed susceptible-infective-recovered (SIR) modelMaximum likelihood estimation (MLE)

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Area of Science:

  • Mathematical Modeling
  • Statistical Inference
  • Dynamical Systems

Background:

  • Delay differential equations (DDEs) are crucial for modeling systems with time delays.
  • Previous inference methods for DDEs often imposed restrictive assumptions on model structure.
  • Multivariate DDEs with multiple delays present significant inferential challenges.

Purpose of the Study:

  • To develop a flexible maximum likelihood inference framework for multivariate delay differential equation models.
  • To overcome limitations of previous methods by not assuming specific forms for the DDEs.
  • To enable the application of maximum likelihood inference to a broader class of DDE models.

Main Methods:

  • Development of a maximum likelihood inference framework for general DDEs.
  • Implementation of adaptive grid and gradient descent numerical algorithms.
  • Formulation of methods for estimating the information matrix and constructing confidence intervals.

Main Results:

  • A robust framework for maximum likelihood inference in multivariate DDEs with one or more delay parameters is established.
  • Numerical algorithms (adaptive grid, gradient descent) are developed for parameter estimation and information matrix calculation.
  • The framework is demonstrated on epidemic and pharmacokinetic models, showing its practical applicability.

Conclusions:

  • The developed framework provides a powerful tool for analyzing complex systems modeled by DDEs.
  • The method's generality allows for wider application in scientific research, including epidemiology and pharmacology.
  • This work advances the statistical inference capabilities for a significant class of dynamical systems.