Predictability Analysis on HIV/AIDS System using Hurst Exponents
Methods of contemporary mathematical physics such
as chaos theory are useful for analyzing and understanding the
behavior of complex biological and physiological systems. The three
dimensional model of HIV/AIDS is the basis of active research since
it provides a complete characterization of disease dynamics and the
interaction of HIV-1 with the immune system. In this work, the
behavior of the HIV system is analyzed using the three dimensional
HIV model and a chaotic measure known as the Hurst exponent.
Results demonstrate that Hurst exponents of CD4, CD8 cells and
viral load vary nonlinearly with respect to variations in system
parameters. Further, it was observed that the three dimensional HIV
model can accommodate both persistent (H>0.5) and anti-persistent
(H<0.5) dynamics of HIV states. In this paper, the objectives of the
study, methodology and significant observations are presented in
detail.
[1] Xia, X.: HIV/AIDS modeling and control. Automatica. 39, 1983-1988
(2004)
[2] Filter, R.A., Xia, X., Gray, C.M.: Dynamic HIV/AIDS parameter
estimation with application to a vaccine readiness study in Southern
Africa. IEEE Trans. Biomed. Engin. 52, 784-791 (2005)
[3] de Souza, F.M.C.: Modeling the dynamics of HIV-1 and CD4 and CD8
lymphocytes. IEEE Engin. Med. Biol. 18, 21-24 (1999)
[4] Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho,
D.D.: HIV-1 dynamics in vivo: Virion clearance rate, infected cell lifespan,
and viral generation time. Science. 271, 1582-1586 (1996)
[5] Wu, H., Ding, A.A., De Gruttola, V.: Estimation of HIV dynamic
parameters. Statistics in Medicine. 17, 2463-2485 (1998)
[6] Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics
in vivo. SIAM Review. 41, 3-44 (1999)
[7] Ngarakana-Gwasira, E.T., Bhunu, C.P., Mushayabasa, S., Hove-
Musekwa, S.D., Garira, W., Tchuenche, J.M.: Exploring the effects of
parameter heterogeneity on the intrinsic dynamics of HIV/AIDS in
heterosexual settings. International Journal of Biomathematics. 4, 75-92
(2011)
[8] Huang, Y., Rosenkranz, S.L., Wu, H.: Modeling HIV dynamics and
antiviral responses with consideration of time-varying drug exposures,
sensitivities and adherence. Mathematical Biosciences. 184, 165-186
(2003)
[9] Huang, Y., Liu, D., Wu, H.: Hierarchical Bayesian methods for
estimation of parameters in a longitudinal HIV dynamic system.
Biometrics. 62, 413-423 (2006)
[10] Xia, X., Moog, C.H.: Identifiability of nonlinear systems with
applications to HIV/AIDS models. IEEE transactions on automatic
control. 48, 330-336 (2003)
[11] Miao, H., Dykes, C., Demeter, L.M., Hulin, Wu.: Differential Equation
Modeling of HIV Viral Fitness Experiments: Model Identification,
Model Selection, and Multimodel Inference. Biometrics. 65, 292-300
(2009)
[12] Bortz, D.M., Nelson, P.W.: Sensitivity Analysis of a Nonlinear Lumped
Parameter Model of HIV Infection Dynamics. Bulletin of Mathematical
Biology. 66, 1009-1026 (2004)
[13] David, J., Tran, H., Banks, H.T.: HIV Model Analysis and Estimation
Implementation Under Optimal Control Based Treatment Strategies.
International Journal of Pure and Applied Mathematics. 3, 357-392
(2009)
[14] Naresh, R., Sharma, D., Tripathi, A.: Modelling the effect of risky
sexual behaviour on the spread of HIV/AIDS. International Journal of
Applied Mathematics and Computation. 1, 132-147 (2009)
[15] Haiping, Ye., Ding, Y.: Nonlinear Dynamics and Chaos in a Fractional-
Order HIV Model. Mathematical Problems in Engineering Volume
2009, Article ID 378614 (2009)
[16] Al-Sheikh, S., Musali, F., Alsolami, M.: Stability Analysis of an
HIV/AIDS Epidemic Model with Screening. International Mathematical
Forum. 6, 3251-3273 (2011)
[17] Lavielle, M., Samson, A., Karina Fermin, A., Mentré, F.: Maximum
likelihood estimation of long-term HIV dynamic models and antiviral
response. Biometrics. 67, 250-259 (2011)
[18] Ho, C.Y-F., Ling, B.W-K.: Initiation of HIV Therapy. International
Journal of Bifurcation and Chaos. 20, 1279-1292 (2010)
[19] Klonowski, W.: From conformons to human brains: an informal
overview of nonlinear dynamics and its applications in biomedicine.
Nonlinear Biomedical Physics. 1, 5 (2007)
[20] Goldbeter, A.: Computational Approaches to Cellular Rhythms. Nature.
420, 238-245 (2002)
[21] Savi, M.A.: Chaos and Order in Biomedical Rhythms. J. of the Braz.
Soc. of Mech. Sci. & Eng. XXVII, 157-169 (2005)
[22] Das, A., Das, P., Roy, A.B.: Applicability of Lyapunov Exponent in
EEG data analysis. Complexity International. 9, 1-8 (2002)
[23] Iwasa, Y., Michor, F., Nowak, M.A.: Virus evolution within patients
increases pathogenicity. J. Theor. Biol. 232, 17-26 (2005)
[24] Gilchrist, M.A., Coombs, D., Perelson, A.S.: Optimizing within-host
viral fitness: Infected cell lifespan and virion production rate. J. Theor.
Biol. 229, 281-288 (2004)
[25] Jafelice, R.M., Barros, L.C., Bassanezi, R.C., Gomide, F.: Fuzzy setbased
model to compute the life expectancy of HIV infected
populations. IEEE Ann. Meeting of the Fuzzy Information Processing.
27-30 June 2004, NAFIPS, 314-318 (2004)
[26] Ge, S.S., Tian, Z., Lee, T.H.: Nonlinear Control of a Dynamic Model of
HIV-1. IEEE transactions on biomedical engineering. 52, 353-361
(2005)
[27] Hurst, H.E., Black, R.P., Simaika, Y.M.: Long-Term Storage: An
Experimental Study. Constable - London. (1965)
[28] Edmonds, A.N.: Time Series Prediction Using Supervised Learning and
Tools from Chaos Theory. Ph.D Thesis (1996)
[29] Jose, M.V., Govezensky, T., Bobadilla, J.R.: Fractional Brownian
motion in DNA sequences of bacterial chromosomes: a renormalization
group approach. Revista Mexicana De Fi'Sica. 56, 69-74 (2010)
[30] Taqqu, M.S., Teverovsky, V., Willinger, W.: Estimators for long range
dependence: an empirical study. Fractals. 3, 785 - 788 (1995)
[31] Bârbulescu, A., Serban, C., Maftei, C.: Evaluation of Hurst exponent for
precipitation time series. Latest Trends On Computers. 2, 590-595
(2010)
[32] Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall.
(1994)
[33] Kamalanand, K., Mannar Jawahar, P.: Estimation of HIV/AIDS
Parameters using Jumping Frogs Optimization. International Journal of
Medical Discovery. 4 (2012)G. O. Young, "Synthetic structure of
industrial plastics (Book style with paper title and editor)," in Plastics,
2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15-
64.
[1] Xia, X.: HIV/AIDS modeling and control. Automatica. 39, 1983-1988
(2004)
[2] Filter, R.A., Xia, X., Gray, C.M.: Dynamic HIV/AIDS parameter
estimation with application to a vaccine readiness study in Southern
Africa. IEEE Trans. Biomed. Engin. 52, 784-791 (2005)
[3] de Souza, F.M.C.: Modeling the dynamics of HIV-1 and CD4 and CD8
lymphocytes. IEEE Engin. Med. Biol. 18, 21-24 (1999)
[4] Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho,
D.D.: HIV-1 dynamics in vivo: Virion clearance rate, infected cell lifespan,
and viral generation time. Science. 271, 1582-1586 (1996)
[5] Wu, H., Ding, A.A., De Gruttola, V.: Estimation of HIV dynamic
parameters. Statistics in Medicine. 17, 2463-2485 (1998)
[6] Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics
in vivo. SIAM Review. 41, 3-44 (1999)
[7] Ngarakana-Gwasira, E.T., Bhunu, C.P., Mushayabasa, S., Hove-
Musekwa, S.D., Garira, W., Tchuenche, J.M.: Exploring the effects of
parameter heterogeneity on the intrinsic dynamics of HIV/AIDS in
heterosexual settings. International Journal of Biomathematics. 4, 75-92
(2011)
[8] Huang, Y., Rosenkranz, S.L., Wu, H.: Modeling HIV dynamics and
antiviral responses with consideration of time-varying drug exposures,
sensitivities and adherence. Mathematical Biosciences. 184, 165-186
(2003)
[9] Huang, Y., Liu, D., Wu, H.: Hierarchical Bayesian methods for
estimation of parameters in a longitudinal HIV dynamic system.
Biometrics. 62, 413-423 (2006)
[10] Xia, X., Moog, C.H.: Identifiability of nonlinear systems with
applications to HIV/AIDS models. IEEE transactions on automatic
control. 48, 330-336 (2003)
[11] Miao, H., Dykes, C., Demeter, L.M., Hulin, Wu.: Differential Equation
Modeling of HIV Viral Fitness Experiments: Model Identification,
Model Selection, and Multimodel Inference. Biometrics. 65, 292-300
(2009)
[12] Bortz, D.M., Nelson, P.W.: Sensitivity Analysis of a Nonlinear Lumped
Parameter Model of HIV Infection Dynamics. Bulletin of Mathematical
Biology. 66, 1009-1026 (2004)
[13] David, J., Tran, H., Banks, H.T.: HIV Model Analysis and Estimation
Implementation Under Optimal Control Based Treatment Strategies.
International Journal of Pure and Applied Mathematics. 3, 357-392
(2009)
[14] Naresh, R., Sharma, D., Tripathi, A.: Modelling the effect of risky
sexual behaviour on the spread of HIV/AIDS. International Journal of
Applied Mathematics and Computation. 1, 132-147 (2009)
[15] Haiping, Ye., Ding, Y.: Nonlinear Dynamics and Chaos in a Fractional-
Order HIV Model. Mathematical Problems in Engineering Volume
2009, Article ID 378614 (2009)
[16] Al-Sheikh, S., Musali, F., Alsolami, M.: Stability Analysis of an
HIV/AIDS Epidemic Model with Screening. International Mathematical
Forum. 6, 3251-3273 (2011)
[17] Lavielle, M., Samson, A., Karina Fermin, A., Mentré, F.: Maximum
likelihood estimation of long-term HIV dynamic models and antiviral
response. Biometrics. 67, 250-259 (2011)
[18] Ho, C.Y-F., Ling, B.W-K.: Initiation of HIV Therapy. International
Journal of Bifurcation and Chaos. 20, 1279-1292 (2010)
[19] Klonowski, W.: From conformons to human brains: an informal
overview of nonlinear dynamics and its applications in biomedicine.
Nonlinear Biomedical Physics. 1, 5 (2007)
[20] Goldbeter, A.: Computational Approaches to Cellular Rhythms. Nature.
420, 238-245 (2002)
[21] Savi, M.A.: Chaos and Order in Biomedical Rhythms. J. of the Braz.
Soc. of Mech. Sci. & Eng. XXVII, 157-169 (2005)
[22] Das, A., Das, P., Roy, A.B.: Applicability of Lyapunov Exponent in
EEG data analysis. Complexity International. 9, 1-8 (2002)
[23] Iwasa, Y., Michor, F., Nowak, M.A.: Virus evolution within patients
increases pathogenicity. J. Theor. Biol. 232, 17-26 (2005)
[24] Gilchrist, M.A., Coombs, D., Perelson, A.S.: Optimizing within-host
viral fitness: Infected cell lifespan and virion production rate. J. Theor.
Biol. 229, 281-288 (2004)
[25] Jafelice, R.M., Barros, L.C., Bassanezi, R.C., Gomide, F.: Fuzzy setbased
model to compute the life expectancy of HIV infected
populations. IEEE Ann. Meeting of the Fuzzy Information Processing.
27-30 June 2004, NAFIPS, 314-318 (2004)
[26] Ge, S.S., Tian, Z., Lee, T.H.: Nonlinear Control of a Dynamic Model of
HIV-1. IEEE transactions on biomedical engineering. 52, 353-361
(2005)
[27] Hurst, H.E., Black, R.P., Simaika, Y.M.: Long-Term Storage: An
Experimental Study. Constable - London. (1965)
[28] Edmonds, A.N.: Time Series Prediction Using Supervised Learning and
Tools from Chaos Theory. Ph.D Thesis (1996)
[29] Jose, M.V., Govezensky, T., Bobadilla, J.R.: Fractional Brownian
motion in DNA sequences of bacterial chromosomes: a renormalization
group approach. Revista Mexicana De Fi'Sica. 56, 69-74 (2010)
[30] Taqqu, M.S., Teverovsky, V., Willinger, W.: Estimators for long range
dependence: an empirical study. Fractals. 3, 785 - 788 (1995)
[31] Bârbulescu, A., Serban, C., Maftei, C.: Evaluation of Hurst exponent for
precipitation time series. Latest Trends On Computers. 2, 590-595
(2010)
[32] Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall.
(1994)
[33] Kamalanand, K., Mannar Jawahar, P.: Estimation of HIV/AIDS
Parameters using Jumping Frogs Optimization. International Journal of
Medical Discovery. 4 (2012)G. O. Young, "Synthetic structure of
industrial plastics (Book style with paper title and editor)," in Plastics,
2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15-
64.
@article{"International Journal of Medical, Medicine and Health Sciences:63344", author = "K. Kamalanand and P. Mannar Jawahar", title = "Predictability Analysis on HIV/AIDS System using Hurst Exponents", abstract = "Methods of contemporary mathematical physics such
as chaos theory are useful for analyzing and understanding the
behavior of complex biological and physiological systems. The three
dimensional model of HIV/AIDS is the basis of active research since
it provides a complete characterization of disease dynamics and the
interaction of HIV-1 with the immune system. In this work, the
behavior of the HIV system is analyzed using the three dimensional
HIV model and a chaotic measure known as the Hurst exponent.
Results demonstrate that Hurst exponents of CD4, CD8 cells and
viral load vary nonlinearly with respect to variations in system
parameters. Further, it was observed that the three dimensional HIV
model can accommodate both persistent (H>0.5) and anti-persistent
(H", keywords = "HIV/AIDS, mathematical model, chaos theory, Hurst
exponent", volume = "6", number = "8", pages = "406-5", }
