Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process

Rotich Kiplimo Titus

doi:doi:10.11648/j.ajam.20160405.15

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Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process

Rotich Kiplimo Titus

Published in American Journal of Applied Mathematics (Volume 4, Issue 5)

Received: 6 September 2016 Accepted: 23 September 2016 Published: 15 October 2016

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 4, Issue 5)
DOI	10.11648/j.ajam.20160405.15
Page(s)	235-246
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Markov Chain, Reproductive Ratio, Stability, Transition Matrix

References

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[8]	Dietz, K., (1975) Transmission and control of arbovirus diseases, Epidemiology, pp. 104-121.
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[15]	Mbabazi, D., (2008) Population Dynamic Type Models in HIV Infection, African Institute for Mathematical Science.
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[17]	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.
[18]	Newell, M.-L., (1998) Mechanisms and timing of mother‐to‐child transmission of HIV‐1, Aids, 12, pp. 831-837.
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[20]	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.
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APA Style

Rotich Kiplimo Titus. (2016). Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. American Journal of Applied Mathematics, 4(5), 235-246. https://doi.org/10.11648/j.ajam.20160405.15

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ACS Style

Rotich Kiplimo Titus. Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. Am. J. Appl. Math. 2016, 4(5), 235-246. doi: 10.11648/j.ajam.20160405.15

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AMA Style

Rotich Kiplimo Titus. Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. Am J Appl Math. 2016;4(5):235-246. doi: 10.11648/j.ajam.20160405.15

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@article{10.11648/j.ajam.20160405.15,
  author = {Rotich Kiplimo Titus},
  title = {Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {5},
  pages = {235-246},
  doi = {10.11648/j.ajam.20160405.15},
  url = {https://doi.org/10.11648/j.ajam.20160405.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160405.15},
  abstract = {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.},
 year = {2016}
}

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TY  - JOUR
T1  - Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process
AU  - Rotich Kiplimo Titus
Y1  - 2016/10/15
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160405.15
DO  - 10.11648/j.ajam.20160405.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 235
EP  - 246
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160405.15
AB  - 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.
VL  - 4
IS  - 5
ER  -

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Author Information

Rotich Kiplimo Titus

Department of Center for Teacher Education, Moi University, Eldoret, Kenya

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APA Style

Rotich Kiplimo Titus. (2016). Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. American Journal of Applied Mathematics, 4(5), 235-246. https://doi.org/10.11648/j.ajam.20160405.15

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ACS Style

Rotich Kiplimo Titus. Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. Am. J. Appl. Math. 2016, 4(5), 235-246. doi: 10.11648/j.ajam.20160405.15

Copy | Download

AMA Style

Rotich Kiplimo Titus. Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process. Am J Appl Math. 2016;4(5):235-246. doi: 10.11648/j.ajam.20160405.15

Copy | Download

@article{10.11648/j.ajam.20160405.15,
  author = {Rotich Kiplimo Titus},
  title = {Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {5},
  pages = {235-246},
  doi = {10.11648/j.ajam.20160405.15},
  url = {https://doi.org/10.11648/j.ajam.20160405.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160405.15},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modeling of the Spread of HIV/AIDS by Markov Chain Process
AU  - Rotich Kiplimo Titus
Y1  - 2016/10/15
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160405.15
DO  - 10.11648/j.ajam.20160405.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 235
EP  - 246
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160405.15
AB  - 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.
VL  - 4
IS  - 5
ER  -

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