Maximum likelihood inference for multivariate delay differential equation models

• Published in: Scientific reports Vol. 15; no. 1; pp. 25229 - 22
• Main Authors: Mahmoud, Ahmed Adly, Rabie, Abdalla, Dass, Sarat Chandra, Alqasem, Ohud A., Mekiso, Getachew Tekle, Hussam, Eslam, Gemeay, Ahmed M.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 12.07.2025; Nature Publishing Group; Nature Portfolio
• Subjects: 639/705/1041; 639/705/531; 639/705/794; Algorithms; Approximation; Carbohydrates; Delay differential equation (DDE); Delay differential equation models (DDEMs); Delayed pharmacokinetic models; Delayed susceptible-infective-recovered (SIR) model; Differential equations; Dynamical systems; Humanities and Social Sciences; Mathematical models; Maximum likelihood estimation (MLE); Methods; multidisciplinary; Normal distribution; Optimization; Parameter estimation; Pharmacokinetics; Science; Science (multidisciplinary); Delay differential equation models (DDEMs); Delayed susceptible-infective-recovered (SIR) model; Maximum likelihood estimation (MLE); Delayed pharmacokinetic models; Delay differential equation (DDE)
• ISSN: 2045-2322; 2045-2322
• DOI: 10.1038/s41598-025-07227-8

The maximum likelihood inference framework for delay differential equation models in the multivariate settings is developed. The number of delay parameters is assumed to be one or more. This study does not make any restrictive assumptions on the form of the underlying delay differential equations which was one of the limitations of some of the previous work. Thus, the maximum likelihood inference framework can be applied to general delay differential equation models with multiple delay parameters. To obtain the maximum likelihood estimator and estimate of the information matrix, two numerical algorithms are developed: (i) the adaptive grid and (ii) the gradient descent algorithms. Two examples of multivariate delay differential equation models related to the epidemic and pharmacokinetic models, respectively, are presented in this paper. For the unknown parameters, standard errors and confidence intervals are constructed, and formulas and techniques for producing the information matrix are developed. The code and computations are developed with the help of the mathematical software MATLAB.