Markov modeling of HIV/AIDS progression was done under the assumption that the state holding time (waiting time) had a constant hazard. This paper discusses the properties of the hazard function of the Exponential distributions and its modifications namely; Parameter proportion hazard (PH) and Accelerated failure time models (AFT) and their effectiveness in modeling the state holding time in Markov modeling of HIV/AIDS progression with and without risk factors. Patients were categorized by gender and age with female gender being the baseline. Data simulated using R software was fitted to each model, and the model parameters were estimated. The estimated P and Z values were then used to test the null hypothesis that the state waiting time data followed an Exponential distribution. Model identification criteria; Akaike information criteria (AIC), Bayesian information criteria (BIC), log-likelihood (LL), and R2 were used to evaluate the performance of the models. For the Survival Regression model, P and Z values supported the non-rejection of the null hypothesis for mixed gender without interaction and supported the rejection of the same for mixed gender with interaction term and males aged 50 - 60 years. Both Parameters supported the non-rejection of the null hypothesis in the rest of the age groups. For Gender male with interaction both P and Z values supported rejection in all the age groups except the age group 20 - 30 years. For Cox Proportional hazard and AFT models, both P and Z values supported the non-rejection of the null hypothesis across all age groups. The P-values for the three models supported different decisions for and against the Null hypothesis with AFT and Cox values supporting similar decisions in most of the age groups. Among the models considered, the regression assumption provided a superior fit based on (AIC), (BIC), (LL), and R2 Model identification criteria. This was particularly evident in age and gender subgroups where the data exhibited non-proportional hazards and violated the assumptions required for the Cox Proportional Hazard model. Moreover, the simplicity of the regression model, along with its ability to capture essential state transitions without over fitting, made it a more appropriate choice.
References
[1]
Arias, R., Angeles, K.D., Maleki, S. and Ahangar, R.R. (2022) Mathematical Modeling of the HIV-AIDS Epidemic. Open Access Library Journal, 9, 1-15. https://doi.org/10.4236/oalib.1107972
[2]
Ahangar, R.R. (2022) Computation, Modeling, and Simulation of HIV-AIDS Epidemics with Vaccination. JournalofAppliedMathematicsandPhysics, 10, 1066-1082. https://doi.org/10.4236/jamp.2022.104073
[3]
May, R.M. and Anderson, R.M. (1989) The Transmission Dynamics of Human Immunodeficiency Virus (HIV). In: Levin, S.A., Hallam, T.G. and Gross, L.J., Eds., Applied Mathematical Ecology, Springer, 263-311. https://doi.org/10.1007/978-3-642-61317-3_12
[4]
Brookmeyer, R. and Gail, M.H. (1994) AIDS Epidemiology: A Quantitative Approach. Oxford University Press.
[5]
Moyo, E., Shakalima, J.C., Chambashi, G., Muchinga, J. and Matindih, L.K. (2021) Modelling HIV/AIDS Cases in Zambia: A Comparative Study of the Impact of Mandatory HIV Testing. OpenJournalofStatistics, 11, 409-419. https://doi.org/10.4236/ojs.2021.113025
[6]
Blaizot, S., Riche, B., Maman, D., Mukui, I., Kirubi, B., Etard, J., et al. (2015) Estimation and Short-Term Prediction of the Course of the HIV Epidemic Using Demographic and Health Survey Methodology-Like Data. PLOSONE, 10, e0130387. https://doi.org/10.1371/journal.pone.0130387
[7]
Auger, I., Thomas, P., De Gruttola, V., Morse, D., Moore, D., Williams, R., et al. (1988) Incubation Periods for Paediatric AIDS Patients. Nature, 336, 575-577. https://doi.org/10.1038/336575a0
[8]
Longini Jr., I.M., Clark, W.S., Gardner, L.I. and Brundage, J.F. (1991) The Dynamics of CD4+ T-Lymphocyte Decline in HIV-Infected Individuals: A Markov Modeling Approach. JAIDS Journal of Acquired Immune Deficiency Syndromes, 4, 1141-1147.
[9]
Plettenberg, A., Brockmeyer, N.H., Haastert, B., Michalik, C., Dupke, S., Schewe, K., et al. (2011) Impact of Earlier HAART Initiation on the Immune Status and Clinical Course of Treated Patients on the Basis of Cohort Data of the German Competence Network for HIV/Aids. Infection, 39, 3-12. https://doi.org/10.1007/s15010-010-0070-8
[10]
Longini, I.M., Clark, W.S., Byers, R.H., Ward, J.W., Darrow, W.W., Lemp, G.F., et al. (1989) Statistical Analysis of the Stages of HIV Infection Using a Markov Model. StatisticsinMedicine, 8, 831-843. https://doi.org/10.1002/sim.4780080708
[11]
Grover, G., Gadpayle, A.K., Swain, P.K. and Deka, B. (2013) A Multistate Markov Model Based on CD4 Cell Count for HIV/AIDS Patients on Antiretroviral Therapy (Art). InternationalJournalofStatisticsinMedicalResearch, 2, 144-151. https://doi.org/10.6000/1929-6029.2013.02.02.08
[12]
Lee, S., Ko, J., Tan, X., Patel, I., Balkrishnan, R. and Chang, J. (2014) Markov Chain Modelling Analysis of HIV/AIDS Progression: A Race-Based Forecast in the United States. Indian Journal of Pharmaceutical Sciences, 76, 107-124.
[13]
Redfield, R.R., Wright, D.C. and Tramont, E.C. (1986) The Walter Reed Staging Classification for HTLV-III/LAV Infection. NewEnglandJournalofMedicine, 314, 131-132. https://doi.org/10.1056/nejm198601093140232
[14]
Hendriks, J.C., Satten, G.A., Longini, I.M., van Druten, H.A.M., Schellekens, P.T.A., Coutinho, R.A., et al. (1996) Use of Immunological Markers and Continuous-Time Markov Models to Estimate Progression of HIV Infection .... AIDS, 10, 649-656. https://doi.org/10.1097/00002030-199606000-00011
[15]
Satten, G.A. and Longini, I.M. (1996) Markov Chains with Measurement Error: Estimating the ‘True’ Course of a Marker of the Progression of Human Immunodeficiency Virus Disease. AppliedStatistics, 45, 275-309. https://doi.org/10.2307/2986089
[16]
Howard, R.A. (1960) Dynamic Programming and Markov Processes. Wiley.
[17]
Mullins, C.D. and Weisman, E.S. (1996) A Simplified Approach to Teaching Markov Models. AmericanJournalofPharmaceuticalEducation, 60, 42-47. https://doi.org/10.1016/s0002-9459(24)04555-8
[18]
Ying Lu, and Stitt, F.W. (1994) Using Markov Processes to Describe the Prognosis of HIV-1 Infection. MedicalDecisionMaking, 14, 266-272. https://doi.org/10.1177/0272989x9401400309
[19]
Boyd, M.A. and Lau, S. (1998) An Introduction to Markov Modeling: Concepts and Uses. Annual Reliability and Maintainability Symposium 1998, Anaheim, 19-22 January 1998, 1-28. https://ntrs.nasa.gov/api/citations/20020050518/downloads/20020050518.pdf
[20]
Hazen, G.B. (1992) Stochastic Trees: A New Technique for Temporal Medical Decision Modeling. MedicalDecisionMaking, 12, 163-178. https://doi.org/10.1177/0272989x9201200302
[21]
Sonnenberg, F.A. and Beck, J.R. (1993) Markov Models in Medical Decision Making: A Practical Guide. MedicalDecisionMaking, 13, 322-338. https://doi.org/10.1177/0272989x9301300409
[22]
Beck, J.R. and Pauker, S.G. (1983) The Markov Process in Medical Prognosis. MedicalDecisionMaking, 3, 419-458. https://doi.org/10.1177/0272989x8300300403
[23]
Welch, H.G. (1996) Estimating Treatment Benefits for the Elderly: The Effect of Competing Risks. AnnalsofInternalMedicine, 124, 577-584. https://doi.org/10.7326/0003-4819-124-6-199603150-00007
[24]
Taylor-Smith, K., Tweya, H., Harries, A., Schoutene, E. and Jahn, A. (2010) Gender Differences in Retention and Survival on Antiretroviral Therapy of HIV-1 Infected Adults in Malawi. MalawiMedicalJournal, 22, 49-56. https://doi.org/10.4314/mmj.v22i2.58794
[25]
Piette, J.D., Intrator, O., Zierler, S., Mor, V. and Stein, M.D. (1992) An Exploratory Analysis of Survival with AIDS Using a Nonparametric Tree-Structured Approach. Epidemiology, 3, 310-318. https://doi.org/10.1097/00001648-199207000-00006
[26]
Beck, J.R., Kassirer, J.P. and Pauker, S.G. (1982) A Convenient Approximation of Life Expectancy (the “Deale”). TheAmericanJournalofMedicine, 73, 883-888. https://doi.org/10.1016/0002-9343(82)90786-0
[27]
Huster, W.J., Brookmeyer, R. and Self, S.G. (1989) Modelling Paired Survival Data with Covariates. Biometrics, 45, 145-156. https://doi.org/10.2307/2532041
[28]
Gadpayle, A.K., Kumar, N., Duggal, A., Rewari, B.B. and Ravi, V. (2012) Survival Trend and Prognostic Outcome of AIDS Patients according to Age, Sex, Stages, and Mode of Transmission—A Retrospective Study at ART Centre of a Tertiary Care Hospital. JIACM, 13, 291-298.
[29]
Pezzotti, P., Phillips, A.N., Dorrucci, M., Lepri, A.C., Galai, N., Vlahov, D., et al. (1996) Category of Exposure to HIV and Age in the Progression to AIDS: Longitudinal Study of 1199 People with Known Dates of Seroconversion. BMJ, 313, 583-586. https://doi.org/10.1136/bmj.313.7057.583
[30]
Melnick, S.L. (1994) Survival and Disease Progression According to Gender of Patients with HIV Infection: The Terry Beirn Community Programs for Clinical Re-search on AIDS. JAMA, 272, 1915-1921. https://doi.org/10.1001/jama.1994.03520240043039
[31]
Schackman, B.R., Goldie, S.J., Weinstein, M.C., Losina, E., Zhang, H. and Freedberg, K.A. (2001) Cost-Effectiveness of Earlier Initiation of Antiretroviral Therapy for Uninsured HIV-Infected Adults. AmericanJournalofPublicHealth, 91, 1456-1463. https://doi.org/10.2105/ajph.91.9.1456
[32]
Meditz, A.L., MaWhinney, S., Allshouse, A., Feser, W., Markowitz, M., Little, S., et al. (2011) Sex, Race, and Geographic Region Influence Clinical Outcomes Following Primary HIV-1 Infection. TheJournalofInfectiousDiseases, 203, 442-451. https://doi.org/10.1093/infdis/jiq085
[33]
Daykin, C.D., Clark, P.N.S., Eves, M.J., Haberman, S., Le Grys, D.J., Lockyer, J., et al. (1988) The Impact of HIV Infection and AIDS on Insurance in the United Kingdom. JournaloftheInstituteofActuaries, 115, 727-837. https://doi.org/10.1017/s0020268100042943
[34]
Werner, A. and Levy, J.A. (1993) Human Immunodeficiency Virus Type 1 Envelope Gp120 Is Cleaved after Incubation with Recombinant Soluble CD4. JournalofVirology, 67, 2566-2574. https://doi.org/10.1128/jvi.67.5.2566-2574.1993
[35]
Rosenberg, Z.F. and Fauci, A.S. (1989) Minireview: Induction of Expression of HIV in Latently or Chronically Infected Cells. NewEnglandJournalofMedicine, 314, 131-132.
[36]
Ausín Olivera, M.C., Wiper, M.P. and Lillo Rodríguez, R.E. (2001) Bayesian Estimation for the M/G/1 Queue Using a Phase-Type Approximation. Journal of Statistical Planning and Inference, 118, 83-101.
[37]
Bharucha-Reid, A.T. (1960) On Random Solutions of Fredholm Integral Equations. BulletinoftheAmericanMathematicalSociety, 66, 104-109. https://doi.org/10.1090/s0002-9904-1960-10417-x
[38]
Mwirigi, N., Sewe, S., Wainaina, M. and Simwa, R. (2022) Weibull Distribution as the Choice Model for State-Specific Failure Rates in HIV/AIDS Progression. MathematicsandStatistics, 10, 588-602. https://doi.org/10.13189/ms.2022.100315
[39]
D’Amico, G., Di Biase, G., Janssen, J. and Manca, R. (2011) HIV Evolution: A Quantification of the Effects Due to Age and to Medical Progress. Informatica, 22, 27-42. https://doi.org/10.15388/informatica.2011.312
[40]
Mwirigi, N., Simwa, P.R., Wainaina, D.M. and Sewe, D.S. (2022) Bayesian Model Averaging in Modeling of State Specific Failure Rates in HIV/AIDS Progression. MathematicsandStatistics, 10, 782-798. https://doi.org/10.13189/ms.2022.100409
