Abstract
Human immunodeficiency virus (HIV) causes acquired immune deficiency syndrome (AIDS). The process of infection and mutation by HIV can be described by fifth-order ordinary differential equations. These equations can be reduced to third-order differential equations through the singular perturbation theory. The objective of this paper is to present a parameter estimation algorithm for this third-order HIV model, using two (out of three) state variables. We first show that the parameters of the HIV model are identifiable with these measurements. The structure of the proposed estimator parallels that of the full state feedback estimator with the unavailable state replaced with an estimated variable. We then prove that the resulting adaptive observer equipped with the so-called σ-modification can be tuned to be ultimately bounded under some conditions in terms of the concentration of uninfected CD4+ T cells. Furthermore, it is seen through computer simulations that an iterative application of the proposed algorithm is effective; the estimated parameters approach their true values, and the stability analysis of the ensuing HIV model leads to the results that are consistent with those obtained previously.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. A. Nowak, S. Bonhoeffer, G. M. Shaw, and R. M. May, “Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations,” Journal of Theoretical Biology, vol. 184, no. 2, pp. 203–217, January 1997.
M. A. Nowak and R. M. May, Virus Dynamics, Mathematical Principles of Immunology and Virology, Oxford University Press, 2000.
H. Chang and A. Astolfia, “Enhancement of the immune system in HIV dynamics by output feedback,” Automatica, 45, no. 7, pp. 1765–1770, 2009,.
H. T. Banks, T. Jang, H.-D. Kwon, “Feedback control of HIV antiviral therapy with long measurement time,” International Journal of Pure and Applied Mathematics, vol. 66, no. 4, pp. 461–485, 2011.
J. David, H. Tran, and H. T. Banks, “Receding horizon control of HIV,” Optimal Control Applications and Methods, vol. 32, no. 6, pp. 681–699, 2011.
H. Shim, S.-J. Han, C. C. Chung, S. W. Nam, and J. H. Seo, “Optimal scheduling of drug treatment for HIV infection: continuous dose control and receding horizon control,” International Journal of Control, Automation, and Systems, vol. 1, no. 3, pp. 282–288, 2003.
X. Xia and C. H. Moog, “Identifiability of nonlinear systems with applications to HIV/AIDS models,” IEEE Trans. on Automatic Control, vol. 48, no. 2, pp. 330–336, February 2003.
X. Xia, “Estimation of HIV/AIDS parameters,” Automatica, vol. 39, no. 11, pp. 1983–1988, July 2003.
Y. Huang, D. Liu, and H. Wu, “Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system,” Biometrics, vol. 62, no. 2, pp. 413–423, 2006.
H. Putter, S. H. Heisterkamp, J. M. A. Lange, and F. de Wolf, “A Bayesian approach to parameter estimation in HIV dynamical models,” Statistics in Medicine, vol. 21, no. 15, pp. 2199–2214, 2002.
H. Miao, C. Dykes, L. M. Demeter, J. Cavenaugh, S. Y. Park, A. S. Perelson, and H. Wu, “Modeling and estimation of kinetic parameters and replicative fitness of HIV-1 from flow-cytometry-based growth competition experiments,” Bulletin of Mathematical Biology, vol. 70, no. pp. 1749–1771, 2008.
H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, 2002.
R. S. Braithwaite, S. Shechter, C.-C. H. Chang, A. Schaefer, and M. S. Roberts, “Estimating the rate of accumulating drug resistance mutations in the HIV genome,” Value in Health: The Journal of the International Society for Pharmacoeconomics and Outcomes Research, vol. 10, no. 3, pp. 204–213, May 2007.
R. Marino and P. Tomei, Nonlinear Control Design, Prentice Hall, 1995.
Q. Zhang, “Adaptive observer for MIMO linear time-varying systems,” IEEE Trans. on Automatic Control, vol. 47, no. 3, pp. 525–529, March 2002.
J. H. Lee and T.-W Yoon, “An HIV model with CTL and drug-resistant mutants, and optimal drug scheduling,” Proc. of Information and Control Symposium (in Korean), pp. 135–137, May 2009.
S. Bonhoeffer, R. M. May, G. M. Shaw, and M. A. Nowak, “Virus dynamics and drug therapy,” Proc. of the National Academy of Science, vol. 94, no. 13, pp. 6971–6976, June 1997.
H.-D. Kwon, “Optimal treatment strategies derived from a HIV model with drug-resistant mutants,” Applied Mathematics and Computation, no. 188, pp. 1193–1204, 2007.
World Health Organization, Laboratory Guidelines for Enumerating CD4 T Lymphocytes, Regional Office for South-East Asia New Delhi, 2007.
R. Luo, M. J. Piovoso, J. Martinez-Picado, and R. Zurakowski, “Optimal antiviral switching to minimize resistance risk in HIV therapy,” PLoS One, vol. 6, no. 11, pp. 1–9, November 2011.
H. Wu, H. Zhu, H. Miao, and A. S. Perelson, “Parameter identifiability and estimation of HIV/AIDS dynamic models,” Bulletin of Mathematical Biology, vol. 70, no. 3, pp. 785–799, 2008.
S. T. Glad, “Solvability of differential algebraic equations and inequalities: an algorithm,” Proc. of European Control Conference, 1997.
G. Conte, C. H. Moog, and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting, Springer, London, 1999.
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.
M. Krstic, L. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, Wiley-Interscience, 1995.
A. Ilchmann, D. H. Owens, and D. Pratzel-Wolters, “Sufficient conditions for stability of linear timevarying systems,” Systems & Control Letters, vol. 9, no. 2, 1987.
S.-K. Kim, C.-N. Kim, and T.-W. Yoon, “Output feedback parameter estimation for a HIV model with mutants,” Proc. of the 31st IASTED International Conference on Modelling, Identification, and Control, pp. 399–405, February 2011.
Author information
Authors and Affiliations
Corresponding author
Additional information
Seok-Kyoon Kim received his B.S. degree from the Department of Electronics and IT Media Engineering at Seoul Tech. in 2004, and his Ph.D. degree in Electrical Engineering from Korea University in 2014. Since 2014, he is now working as a researcher for the Research Institute of Electrical Information Technology of SeoulTech. His research interests include model predictive control (MPC), adaptive control, nonlinear control, system identification, and applications of power electronics devices and power system stabilization.
Donghu Kim received his B.S. degree in Electrical Engineering from University of Ulsan in 2009 and his M.S. degree in Electrical, Electronic, Computer Engineering at Korea University in 2011. Since 2011, he is now working as a research engineer at System Engineering Group in Hyundai Autron Co., Seoul, Korea. His research interests include nonlinear systems, adaptive systems, parameter estimation, and vehicle dynamics.
Tae-Woong Yoon received his B.S. and M.S. degrees in Control Engineering from Seoul National University, in 1984 and 1986, and his D.Phil degree in Engineering Science from Oxford University. During 1986–1994, he worked as a researcher at the Korea Institute of Science and Technology (KIST). In 1995, he joined the faculty of Electrical Engineering, Korea University where he is now a Professor. His research interests include control/systems theory, e.g. on adaptive systems and model predictive control (MPC). As an educator, he emphasizes to his students the importance of mathematical/critical thinking and logical/clear technical writing.
Rights and permissions
About this article
Cite this article
Kim, SK., Kim, D. & Yoon, TW. Adaptive observer for estimating the parameters of an HIV model with mutants. Int. J. Control Autom. Syst. 13, 126–137 (2015). https://doi.org/10.1007/s12555-013-9018-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-013-9018-y