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Complexity
Volume 2017 (2017), Article ID 6148934, 14 pages
https://doi.org/10.1155/2017/6148934
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.

Linked References

  1. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993. View at MathSciNet
  2. J. J. Batzel and H. T. Tran, “Stability of the human respiratory control system. I. Analysis of a two-dimensional delay state-space model,” Journal of Mathematical Biology, vol. 41, no. 1, pp. 45–79, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. T. Kalmár-Nagy, G. Stépán, and F. C. Moon, “Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations,” Nonlinear Dynamics, vol. 26, no. 2, pp. 121–142, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2013. View at MathSciNet
  5. S. P. Ellner, B. E. Kendall, S. N. Wood, E. McCauley, and C. J. Briggs, “Inferring mechanism from time-series data: Delay-differential equations,” Physica D: Nonlinear Phenomena, vol. 110, no. 3-4, pp. 182–194, 1997. View at Publisher · View at Google Scholar · View at Scopus
  6. S. N. Wood, “Partially specified ecological models,” Ecological Monographs, vol. 71, no. 1, pp. 1–25, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. L. Wang and J. Cao, “Estimating parameters in delay differential equation models,” Journal of Agricultural, Biological, and Environmental Statistics, vol. 17, no. 1, pp. 68–83, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. Mehrkanoon, S. Mehrkanoon, and J. A. Suykens, “Parameter estimation of delay differential equations: an integration-free LS-SVM approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp. 830–841, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  9. R. A. Fisher, “On the mathematical foundations of theoretical statistics,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 222, no. 594-604, pp. 309–368, 1922. View at Publisher · View at Google Scholar
  10. R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 22, no. 5, pp. 700–725, 1925. View at Publisher · View at Google Scholar
  11. P. J. Huber, “The behavior of maximum likelihood estimates under nonstandard conditions,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1967.
  12. A. Wald, “Note on the consistency of the maximum likelihood estimate,” Annals of Mathematical Statistics, vol. 20, no. 4, pp. 595–601, 1949. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Selected papers of Hirotugu Akaike, pp. 199–213, Springer, 1998. View at Google Scholar · View at MathSciNet
  14. E. L. Lehmann and G. Casella, Theory of Point Estimation, Science & Business Media, 1998. View at MathSciNet
  15. A. Spanos, Probability Theory and Statistical Inference, Cambridge University Press, Cambridge, UK, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. E. Hutchinson, “Circular causal systems in ecology,” Annals of the New York Academy of Sciences, vol. 50, no. 4, pp. 221–246, 1948. View at Publisher · View at Google Scholar · View at Scopus
  17. R. D. Braddock and P. van den Driessche, “On a two-lag differential delay equation,” Australian Mathematical Society. Journal. Series B. Applied Mathematics, vol. 24, no. 3, pp. 292–317, 1982/83. View at Publisher · View at Google Scholar · View at MathSciNet
  18. K. L. Cooke and J. A. Yorke, “Some equations modelling growth processes and gonorrhea epidemics,” Mathematical Biosciences, vol. 16, pp. 75–101, 1973. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. Beuter, J. Bélair, C. Labrie, and J. Bélair, “Feedback and delays in neurological diseases: A modeling study using gynamical systems,” Bulletin of Mathematical Biology, vol. 55, no. 3, pp. 525–541, 1993. View at Publisher · View at Google Scholar · View at Scopus
  20. J. Bélair and S. A. Campbell, “Stability and bifurcations of equilibria in a multiple-delayed differential equation,” SIAM Journal on Applied Mathematics, vol. 54, no. 5, pp. 1402–1424, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Bélair, M. C. Mackey, and J. M. Mahaffy, “Age-structured and two-delay models for erythropoiesis,” Mathematical Biosciences, vol. 128, no. 1-2, pp. 317–346, 1995. View at Publisher · View at Google Scholar · View at Scopus
  22. J. K. Hale and W. Z. Huang, “Global geometry of the stable regions for two delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 344–362, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. M. Mahaffy, K. M. Joiner, and P. J. Zak, “A geometric analysis of stability regions for a linear differential equation with two delays,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 5, no. 3, pp. 779–796, 1995. View at Publisher · View at Google Scholar · View at MathSciNet