Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process
American Journal of Applied Mathematics
Volume 4, Issue 5, October 2016, Pages: 235-246
Received: Sep. 6, 2016; Accepted: Sep. 23, 2016; Published: Oct. 15, 2016
Views 3322      Downloads 159
Rotich Kiplimo Titus, Department of Center for Teacher Education, Moi University, Eldoret, Kenya
Article Tools
Follow on us
The spread of the Human Immunodeficiency Virus (HIV) and the resulting Acquired Immune Deficiency syndrome (AIDS) is a major health concern. Mathematical models are therefore commonly applied to understand the spread of the HIV epidemic. In this study, HIV dynamics is analyzed using a Stochastic Discrete-Time Markov Chain Mathematical Model. Demographic and epidemiological parameters that affect the model population dynamics were investigated. Well posedness of the model determined and the conditions for the existence and stability of disease-free and endemic equilibrium points proved, using the next generation matrix technique. The effect of various intervention strategies, were simulated by varying the parameters representing the possible strategies and comparing the respective values of the reproductive ratio R_0. The numerical simulation results using intervention transition matrix showed that vertical transmission is the most sensitive parameter standing at 0.6 followed by the use of HAART at 0.4. This indicates the strategy which requires much effort to avert progression of infected individual to AIDS.
Markov Chain, Reproductive Ratio, Stability, Transition Matrix
To cite this article
Rotich Kiplimo Titus, Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process, American Journal of Applied Mathematics. Vol. 4, No. 5, 2016, pp. 235-246. doi: 10.11648/j.ajam.20160405.15
Copyright © 2016 Authors retain the copyright of this article.
This article 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.
Adler, F. R., (1992) The effects of averaging on the basic reproduction ratio, Mathematical biosciences, 111, pp. 89-98.
Brauer, F., (1995) Models for diseases with vertical transmission and nonlinear population dynamics, Mathematical biosciences, 128, pp. 13-24.
Cao, H. and Tan, H., (2015) The discrete tuberculosis transmission model with treatment of latently infected individuals, Advances in Difference Equations, 2015, pp. 1.
Cooley, P., Hamill, D., Liner, E., Myers, L. and Van Der Horst, C., (1993) A linked risk group model for investigating the spread of HIV, Mathematical and computer modelling, 18, pp. 85-102.
Cooley, P. C., Hamill, D., Liner, E., Myers, L. and Van Der Horst, C., (1993) A linked risk group model for investigating the spread of HIV, Mathematical and computer modelling, 18, pp. 85-102.
Craig, B. A., Fryback, D. G., Klein, R. and Klein, B. E., (1999) A Bayesian approach to modelling the natural history of a chronic condition from observations with intervention, Statistics in medicine, 18, pp. 1355-1371.
Diekmann, O., Heesterbeek, J. and Metz, J. A., (1990) On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, Journal of mathematical biology, 28, pp. 365-382.
Dietz, K., (1975) Transmission and control of arbovirus diseases, Epidemiology, pp. 104-121.
Hethcote, H. W., (2000) The mathematics of infectious diseases, SIAM review, 42, pp. 599-653.
Hethcote, H. W., 1989Three basic epidemiological models, Applied mathematical ecology, Springer, pp. 119-144.
Hiligsmann, M., Ethgen, O., Bruyère, O., Richy, F., Gathon, H. J. and Reginster, J. Y., (2009) Development and validation of a Markov microsimulation model for the economic evaluation of treatments in osteoporosis, Value in health, 12, pp. 687-696.
Johnson, L., (2004) An Introduction to the mathematics of HIV/AIDS modelling, Capetown: Centre for Actuarial Research, University of Capetown, unpublished manuscript.
Keeling, M. J. and Rohani, P., 2008.Modeling infectious diseases in humans and animals, Princeton University Press, 2008.
Lekone, P. E. and Finkenstädt, B. F., (2006) Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62, pp. 1170-1177.
Mbabazi, D., (2008) Population Dynamic Type Models in HIV Infection, African Institute for Mathematical Science.
Mugisha, J. and Luboobi, L. S., (2003) Modelling the effect of vertical transmission in the dynamics of HIV/AIDS in an age-structured population, The South Pacific Journal of Natural and Applied Sciences, 21, pp. 82-90.
Naresh, R., Tripathi, A. and Omar, S., (2006) Modelling the spread of AIDS epidemic with vertical transmission, Applied Mathematics and Computation, 178, pp. 262-272.
Newell, M.-L., (1998) Mechanisms and timing of mother‐to‐child transmission of HIV‐1, Aids, 12, pp. 831-837.
Newman, M. E., (2002) Spread of epidemic disease on networks, Physical review E, 66, pp. 016128.
Palella Jr, F. J., Delaney, K. M., Moorman, A. C., Loveless, M. O., Fuhrer, J., Satten, G. A., Aschman, D. J. and Holmberg, S. D., (1998) Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, New England Journal of Medicine, 338, pp. 853-860.
Piot, P. and Bartos, M., 2002The epidemiology of HIV and AIDS, AIDS in Africa, Springer, pp. 200-217.
Schneider, M. F., Gange, S. J., Williams, C. M., Anastos, K., Greenblatt, R. M., Kingsley, L., Detels, R. and Muñoz, A., (2005) Patterns of the hazard of death after AIDS through the evolution of antiretroviral therapy: 1984–2004, Aids, 19, pp. 2009-2018.
Smith, H. L., Wang, L. and Li, M. Y., (2001) Global dynamics of an SEIR epidemic model with vertical transmission, SIAM Journal on Applied Mathematics, 62, pp. 58-69.
Van Den Driessche, P. and Watmough, J., (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180, pp. 29-48.
Venkataramanan, L. and Sigworth, F., (2002) Applying hidden Markov models to the analysis of single ion channel activity, Biophysical journal, 82, pp. 1930-1942.
Who, W. H. O., 2009. 2008 Report on the Global AIDS Epidemic, World Health Organization, 2009.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186