Bulletin of Mathematical Biology

, Volume 70, Issue 3, pp 785–799 | Cite as

Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models

  • Hulin Wu
  • Haihong Zhu
  • Hongyu Miao
  • Alan S. Perelson
Original Article

Abstract

We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and λ) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.

Keywords

Identifiability Inverse problem Statistical estimation Viral dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adanu, K., 2006. Optimizing the GARCH model—an application of two global and two local search methods. Comput. Econ. 28(3), 277–290. MATHCrossRefGoogle Scholar
  2. Conte, G., Moog, C.H., Perdon, A.M., 1999. Nonlinear Control Systems: An Algebraic Setting. Springer, London. MATHGoogle Scholar
  3. Diop, S., Fliess, M., 1991. On nonlinear observability. In: Proc. of the First Europ. Control Conf., Paris, Hermes, pp. 152–157. Google Scholar
  4. Glad, S.T., 1997. Solvability of differential algebraic equations and inequalities: an algorithm. In: European Control Conference, ECC97, Brussels. Google Scholar
  5. Ho, D.D., Neumann, A.U., Perelson, A.S., Chen, W., Leonard, J.M., Markowitz, N.M., 1995. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373, 123–126. CrossRefGoogle Scholar
  6. Huang, Y., Wu, H., 2006. A Bayesian approach for estimating antiviral efficacy in HIV dynamic models. J. Appl. Stat. 33, 155–174. MATHCrossRefMathSciNetGoogle Scholar
  7. Huang, Y., Rosenkranz, S.L., Wu, H., 2003. Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity. Math. Biosci. 184, 165–186. MATHCrossRefMathSciNetGoogle Scholar
  8. Huang, Y., Liu, D., Wu, H., 2006. Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamics. Biometrics 62, 413–423. MATHCrossRefMathSciNetGoogle Scholar
  9. Jeffrey, A.M., Xia, X., 2005. Identifiability of HIV/AIDS model. In: W.Y. Tan, H. Wu (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention. World Scientific, Singapore. Google Scholar
  10. Ljung, L., Glad, T., 1994. On global identifiability for arbitrary model parameterizations. Automatica 30(2), 265–276. MATHCrossRefMathSciNetGoogle Scholar
  11. Nowak, M.A., May, R.M., 2000. Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford. MATHGoogle Scholar
  12. Perelson, A.S., Nelson, P.W., 1999. Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41(1), 3–44. MATHCrossRefMathSciNetGoogle Scholar
  13. Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D., 1996. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586. CrossRefGoogle Scholar
  14. Perelson, A.S., Essunger, P., Cao, Y., Vesanen, M., Hurley, A., Saksela, K., Markowitz, M., Ho, D.D., 1997. Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387, 188–191. CrossRefGoogle Scholar
  15. Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S., 2000. Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol. 203, 285–301. CrossRefGoogle Scholar
  16. Storn, R., Price, K., 1997. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359. MATHCrossRefMathSciNetGoogle Scholar
  17. Tan, W.Y., Wu, H., 2005. Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention. World Scientific, Singapore. Google Scholar
  18. Tunali, T., Tarn, T.J., 1987. New results for identifiability of nonlinear systems. IEEE Trans. Autom. Contr. 32(2), 146–154. MATHCrossRefMathSciNetGoogle Scholar
  19. Wei, X., Ghosh, S.K., Taylor, M.E., et al., 1995. Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373, 117–122. CrossRefGoogle Scholar
  20. Wu, H., Ding, A.A., 1999. Population HIV-1 dynamics in vivo: applicable models and inferential tools for virological data from AIDS clinical trials. Biometrics 55, 410–418. MATHCrossRefGoogle Scholar
  21. Wu, H., Kuritzkes, D.R., McClernon, D.R., et al., 1999. Characterization of viral dynamics in Human Immunodeficiency Virus type 1-infected patients treated with combination antiretroviral therapy: relationships to host Factors, cellular restoration and virological endpoints. J. Infect. Diseas. 179(4), 799–807. CrossRefGoogle Scholar
  22. Xia, X., 2003. Estimation of HIV/AIDS parameters. Automatica 39, 1983–1988. MATHCrossRefGoogle Scholar
  23. Xia, X., Moog, C.H., 2003. Identifiability of nonlinear systems with applications to HIV/AIDS models. IEEE Trans. Autom. Contr. 48(2), 330–336. CrossRefMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • Hulin Wu
    • 1
  • Haihong Zhu
    • 1
  • Hongyu Miao
    • 1
  • Alan S. Perelson
    • 2
  1. 1.Department of Biostatistics and Computational BiologyUniversity of Rochester School of Medicine and DentistryRochesterUSA
  2. 2.Theoretical Biology and Biophysics Group, MS-K710Los Alamos National LaboratoryLos AlamosUSA

Personalised recommendations