A Stochastic Model for the HIV/AIDS Dynamic Evolution

Di Biase, Giuseppe; D'Amico, Guglielmo; Di Girolamo, Arturo; Janssen, Jacques; Iacobelli, Stefano; Tinari, Nicola; Manca, Raimondo

doi:https://doi.org/10.1155/2007/65636

Mathematical Problems in Engineering

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Dynamics and Control in Sciences and Engineering

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Volume 2007 | Article ID 065636 | https://doi.org/10.1155/2007/65636

A Stochastic Model for the HIV/AIDS Dynamic Evolution

Giuseppe Di Biase,¹Guglielmo D'Amico,²Arturo Di Girolamo,³Jacques Janssen,⁴Stefano Iacobelli,⁵Nicola Tinari,⁵and Raimondo Manca⁶

Academic Editor: José Manoel Balthazar

Received28 Sept 2006

Revised28 Feb 2007

Accepted03 Jun 2007

Published14 Aug 2007

Abstract

This paper analyses the HIV/AIDS dynamic evolution as defined by CD4 levels, from a macroscopic point of view, by means of homogeneous semi-Markov stochastic processes. A large number of results have been obtained including the following conditional probabilities: an infected patient will be in state j after a time t given that she/he entered at time 0 (starting time) in state i; that she/he will survive up to a time t, given the starting state; that she/he will continue to remain in the starting state up to time t; that she/he reach stage j of the disease in the next transition, if the previous state was i and no state change occurred up to time t. The immunological states considered are based on CD4 counts and our data refer to patients selected from a series of 766 HIV-positive intravenous drug users.

References

P. Levy, “Processus semi-markoviens,” in Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. 3, pp. 416–426, Erven P. Noordhoff N.V., Groningen, The Netherlands, 1956.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. L. Smith, “Regenerative stochastic processes,” Proceedings of the Royal Society of London. Series A., vol. 232, pp. 6–31, 1955.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. A. Howard, Dynamic Probabilistic Systems. Vol. I: Markov Models, John Wiley & Sons, New York, NY, USA, 1971.
View at: Google Scholar | Zentralblatt MATH
R. A. Howard, Dynamic Probabilistic Systems. Vol. II: Semi-Markov and Decision Processes, John Wiley & Sons, New York, NY, USA, 1971.
View at: Google Scholar | Zentralblatt MATH
N. Limnios and G. Oprişan, Semi-Markov Processes and Reliability, Statistics for Industry and Technology, Birkhäuser, Boston, Mass, USA, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Janssen and R. Manca, “Applied Semi-Markov Processes,” Springer, New York, NY, USA, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Di Biase, J. Janssen, and R. Manca, “Future pricing through homogeneous semi-Markov processes,” Applied Stochastic Models in Business and Industry, vol. 21, no. 3, pp. 241–249, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Janssen and R. Manca, “A realistic non-homogeneous stochastic pension fund model on scenario basis,” Scandinavian Actuarial Journal, no. 2, pp. 113–137, 1997.
View at: Google Scholar | Zentralblatt MATH
G. D'Amico, J. Janssen, and R. Manca, “Homogeneous semi-Markov reliability models for credit risk management,” Decisions in Economics and Finance, vol. 28, no. 2, pp. 79–93, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
O. Davidov, “The steady state probabilities for a regenerative semi-Markov processes with application to prevention and screening,” Applied Stochastic Models and Data Analysis, vol. 15, no. 1, pp. 55–63, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
O. Davidov and M. Zelen, “Designing cancer prevention trials: a stochastic approach,” Statistics in Medicine, vol. 19, no. 15, pp. 1983–1995, 2000.
View at: Publisher Site | Google Scholar
S. W. Lagakos, C. J. Sommer, and M. Zelen, “Semi-Markov models for partially censored data,” Biometrika, vol. 65, no. 2, pp. 311–317, 1978.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. A. Satten and M. R. Sternberg, “Fitting semi-Markov models to interval-censored data with unknown initiation times,” Biometrics, vol. 55, no. 2, pp. 507–513, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. R. Sternberg and G. A. Satten, “Discrete-time nonparametric estimation for semi-Markov models of chain-of-events data subject to interval-censoring and truncation,” Biometrics, vol. 55, no. 2, pp. 514–522, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M J. Sweeting, D. De Angelis, and O. O. Aalen, “Bayesian back-calculation using a multi-state model with application to HIV,” Statistics in Medicine, vol. 24, no. 24, pp. 3991–4007, 2005.
View at: Publisher Site | Google Scholar | PubMed | MathSciNet
Y. Foucher, E. Mathieu, P. Saint-Pierre, J.-F. Durand, and J.-P. Daurès, “A semi-Markov model based on generalized Weibull distribution with an illustration for HIV disease,” Biometrical Journal, vol. 47, no. 6, pp. 825–833, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
P. Joly and D. Commenges, “A penalized likelihood approach for a progressive three-state model with censored and truncated data: application to AIDS,” Biometrics, vol. 55, no. 3, pp. 887–890, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. D. Gill, “Nonparametric estimation based on censored observations of a Markov renewal process,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 53, no. 1, pp. 97–116, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. K. Andersen, Ø. Borgan, R. D. Gill, and N. Keiding, Statistical Models Based on Counting Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. Ouhbi and N. Limnios, “Nonparametric estimation for semi-Markov processes based on its hazard rate functions,” Statistical Inference for Stochastic Processes, vol. 2, no. 2, pp. 151–173, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. Limnios and B. Oubhi, “Nonparametric estimation for semi-Markov processes based on $k$ -sample paths with application to reliability,” in Proceedings of the 11th Symposium on Applied Stochastic Models and Data Analysis (ASMDA '05), J. Janssen and P. Lenca, Eds., pp. 1061–1068, Brest, France, May 2005.
View at: Google Scholar
D. M. Dabrowska and W. T. Ho, “Estimation in a semiparametric modulated renewal process,” Statistica Sinica, vol. 16, no. 1, pp. 93–119, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Iacobelli, A. Ullrich, and N. Tinari et al., “The 90K tumor-associated antigen and clinical progression in human immunodeficiency virus infection,” Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology, vol. 10, no. 4, pp. 450–456, 1995.
View at: Google Scholar
E. Çinlar, “Introduction to Stochastic Processes,” Prentice-Hall, Englewood Cliffs, NJ, USA, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Çinlar, “Markov renewal theory: a survey,” Management Science, vol. 21, no. 7, pp. 727–752, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Pyke, “Markov renewal processes: definitions and preliminary properties,” Annals of Mathematical Statistics, vol. 32, pp. 1231–1242, 1961.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Corradi, J. Janssen, and R. Manca, “Numerical treatment of homogeneous semi-Markov processes in transient case—a straightforward approach,” Methodology and Computing in Applied Probability, vol. 6, no. 2, pp. 233–246, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Blasi, J. Janssen, and R. Manca, “Numerical treatment of homogeneous and non-homogeneous semi-Markov reliability models,” Communications in Statistics. Theory and Methods, vol. 33, no. 3, pp. 697–714, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Osaki, Stochastic System Reliability Modeling, vol. 5 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. P. Gelmann, M. Popovic, and D. Blayney et al., “Proviral DNA of a retrovirus, human T-cell leukemia virus, in two patients with AIDS,” Science, vol. 220, no. 4599, pp. 862–865, 1983.
View at: Publisher Site | Google Scholar
R. M. Anderson, G. F. Medley, R. M. May, and A. M. Johnson, “A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 3, no. 4, pp. 229–263, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. A. Levy, “Pathogenesis of human immunodeficiency virus infection,” Microbiological Reviews, vol. 57, no. 1, pp. 183–289, 1993.
View at: Google Scholar
Centres for Disease Control & Prevention, “1993 revised classification system for HIV infection and expanded surveillance case definition for AIDS among adolescents and adults,” MMWR Recommendations and Reports, vol. 41, no. RR-17, pp. 1–19, 1992.
View at: Google Scholar
H. W. Jaffe and A. R. Lifson, “Acquisition and transmission of HIV,” Infectious Diseases Clinic of North America, vol. 2, no. 2, pp. 299–306, 1988.
View at: Google Scholar
UNAIDS/WHO, “AIDS Epidemic Update,” December 2006, http://www.unaids.org/en/HIV_data/epi2006/default.asp.
View at: Google Scholar
S. Casari, A. Donasi, and G. Paraninfo et al., “Prognostic factors correlated with survival in AIDS patients,” European Journal of Epidemiology, vol. 15, no. 8, pp. 691–698, 1999.
View at: Publisher Site | Google Scholar
P. E. Sax, “Updated DHHS treatment guidelines,” AIDS Clinical Care, vol. 18, no. 12, p. 105, 2006.
View at: Google Scholar

Copyright

Copyright © 2007 Giuseppe Di Biase 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.

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