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Volume 2017, Article ID 6148934, 14 pages
Research Article

Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays

1Fundamental and Applied Sciences Department, Faculty of Science and Information Technology, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak, Malaysia
2Department of Electrical and Electronic Engineering, Center for Intelligent Signal and Imaging Research (CISIR), Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak, Malaysia

Correspondence should be addressed to Ahmed A. Mahmoud; ge.ude.rahza@ylda

Received 11 May 2017; Revised 19 August 2017; Accepted 23 August 2017; Published 12 October 2017

Academic Editor: Fathalla A. Rihan

Copyright © 2017 Ahmed A. Mahmoud et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This article presents statistical inference methodology based on maximum likelihoods for delay differential equation models in the univariate setting. Maximum likelihood inference is obtained for single and multiple unknown delay parameters as well as other parameters of interest that govern the trajectories of the delay differential equation models. The maximum likelihood estimator is obtained based on adaptive grid and Newton-Raphson algorithms. Our methodology estimates correctly the delay parameters as well as other unknown parameters (such as the initial starting values) of the dynamical system based on simulation data. We also develop methodology to compute the information matrix and confidence intervals for all unknown parameters based on the likelihood inferential framework. We present three illustrative examples related to biological systems. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2014b.