Computing Transition Intensity Using Time-Homogeneous Markov Jump Process: Case of South African HIV/AIDS Disposition

Authors: A. Bayaga

Abstract:

This research provides a technical account of estimating Transition Probability using Time-homogeneous Markov Jump Process applying by South African HIV/AIDS data from the Statistics South Africa. It employs Maximum Likelihood Estimator (MLE) model to explore the possible influence of Transition Probability of mortality cases in which case the data was based on actual Statistics South Africa. This was conducted via an integrated demographic and epidemiological model of South African HIV/AIDS epidemic. The model was fitted to age-specific HIV prevalence data and recorded death data using MLE model. Though the previous model results suggest HIV in South Africa has declined and AIDS mortality rates have declined since 2002 – 2013, in contrast, our results differ evidently with the generally accepted HIV models (Spectrum/EPP and ASSA2008) in South Africa. However, there is the need for supplementary research to be conducted to enhance the demographic parameters in the model and as well apply it to each of the nine (9) provinces of South Africa.

Keywords: AIDS mortality rates, Epidemiological model, Time-homogeneous Markov Jump Process, Transition Probability, Statistics South Africa.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1338850

References:

[1] S. W. Duffy, N. E. Day, L. Tarbar, H. H. Chen, “Models of breast tumour progression: Some age-specific results,” Journal of the National Cancer Institute, vol 22, no. 93, pp. 94-97, 1997.
[2] J. Bergenthum, L. Rüschendorf, “Comparison of semimartingales and Lévy processes,” Annals of Probability, vol, 35, pp. 228–254, 2007a.
[3] S. Lee, J. Ko, X. Tan,. I. Patel, R. Balkrishnan, J., Chang, “Markov chain modelling analysis of HIV/AIDS progression: A race-based forecast in the United States,” Indian J Pharm Sci (serial online), vol 76, pp. 107-15, 2014. Available from: http://www.ijpsonline.com/text.asp?2014/76/2/107/131519.
[4] Stats SA Statistics South Africa. Mortality and causes of death in South Africa, 2010: Findings from death notification. Available: http://www.statssa.gov.za/publications2/P03093/P030932010.pdf. Accessed 12 Dec 2013.
[5] R. E. Dorrington, “The ASSA2000 suite of models. Actuarial Society of South Africa Convention, 2000. Somerset West, South Africa,” Available: www.assa.org.za/default.asp?id=1000000086, 2004. Accessed 25 February 2015.
[6] Department of Health. “Concordat and moratorium on genetics and insurance.” Available online at http://www.dh.gov.uk., 2005. Accessed January 2015.
[7] R. E. Dorrington, D. Bourne, D.,. Bradshaw, R. Laubscher, I. M Timæus,. “The impact of HIV/AIDS on adult mortality in South Africa. Burden of Disease Research Unit, Medical Research Council,” Available: http://www.mrc.ac.za/bod/complete.pdf, 2001. Accessed 30 July 2012.
[8] R. E. Dorrington, “Alternative South African mid-year estimates, 2013,” Centre for Actuarial Research. Available: commerce.uct.ac.za/Research_Units/CARE/Monographs/Monographs/ Mono13.pdf, 2013. Accessed 19 Nov 2013.
[9] Stats SA. “Statistician General’s results launch presentation – Census 2011,” Available: www. statssa.gov.za/Census2011/Products/SG_Presentation.pdf, 2011. Accessed January 2015.
[10] O. Shisana, T, Rehle L. C. Simbayi, “South African national HIV prevalence, incidence, behaviour and communication survey, 2008: A turning tide among teenagers?” Human Sciences Research Council. Available: http://www.hsrcpress.ac.za, 2009. Accessed 9 January 2015.
[11] H. Daduna, R. Szekli, “Dependence ordering for Markov processes on partially ordered spaces,” Journal of Applied Probability, vol 43, pp.793–814, 2006.
[12] A. S. Macdonald, “Genetics and insurance: What we have learned so far?” Scandinavian Actuarial Journal, vol 324, no, 348, pp. 27 – 28, 2003b.
[13] A. S. Macdonald, H. R. Waters, and C. T. Wekwete “A model for coronary heart disease and stroke, with applications to critical illness insurance underwriting I: The model,” North American Actuarial Journal, vol 13, no, 40, pp. 40-48, 2005a.
[14] B. Maughan-Brown, A. S. Venkataramani N, Nattrass J. Seekings A. W. Whiteside, “A cut above the rest: traditional male circumcision and HIV risk among Xhosa men in 134. Cape Town, South Africa,” Journal of Acquired Immune Deficiency Syndromes, vol. 58, pp. 499-505, 2011.
[15] F. Nyabadza Z. Mukandavire, S. D. Hove-Musekwa, “Modelling the HIV/AIDS epidemic trends in South Africa: Insights from a simple mathematical model. Nonlinear Analysis,” Real World Applications, vol 12, pp. 2091-2104, 2011.
[16] R. S. McClelland, S. M. Graham, B. A. Richardson, “Treatment with antiretroviral therapy is not associated with increased sexual risk behavior in Kenyan female sex workers,” AIDS, vol 24, pp. 891-7, 2010.
[17] D. J. McQuoid-Mason, “Is the mass circumcision drive in KwaZulu- Natal involving neonates and children less than 16 years of age legal? What should doctors do?” South African Medical Journal, vol 103, pp. 283-4, 2013.
[18] N. McGrath, L. Richter, and M. L. Newell, “Sexual risk after HIV diagnosis: a comparison of pre-ART individuals with CD4>500 cells/μl and ART-eligible individuals in a HIV treatment and care programme in rural KwaZulu-Natal, South Africa,” Journal of the International AIDS Society, vol 16, pp. 18048- 1809, 2013.
[19] J. McNeil, “A history of official government HIV/AIDS policy in South Africa,” Available: www.sahistory.org.za/topic/history-officialgovernment- hivaids-policy-south-africa, 2012. Accessed January 2015.
[20] Actuarial Society of South Africa, “ASSA2008 AIDS and Demographic Model,” Available: http://aids.actuarialsociety.org.za, 2008. Accessed 5 April 2011.
[21] L. A. Shafer, R. N. Nsubuga, R White,. “Antiretroviral therapy and sexual behaviour in Uganda: a cohort study,” AIDS, vol, 25, pp. 671-8, 2011.
[22] J. Kimani, R. Kaul N. J. Nagelkerke “Reduced rates of HIV acquisition during unprotected sex by Kenyan female sex workers predating population declines in HIV prevalence,” AIDS vol 22, pp. 131-7, 2008.
[23] R. Kaul, C. R. Cohen, D. Chege, “Biological factors that may contribute to regional and racial disparities in HIV prevalence,” American Journal of Reproductive Immunology, vol 65, pp. 317-24, 2011.
[24] W, He S. Neil, H. Kulkarni Duffy, “Antigen receptor for chemokines mediates trans-infection of HIV-1 from red blood cells to target cells and affects HIV-AIDS susceptibility,” Cell Host and Microbe vol, 4, pp. 52- 62, 2008.
[25] J. Lajoie, J. Hargrove L. S. Zijenah, J. H. Humphrey, B. J. Ward. and M. Roger, “Genetic variants in nonclassical major histocompatibility complex class I human leukocyte antigen (HLA)-E and HLA-G molecules are associated with susceptibility to heterosexual acquisition of HIV-1,” Journal of Infectious Diseases. Vol, 193, pp. 298-301, 2006.