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The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie's Algorithm

Received: 8 October 2019     Accepted: 28 October 2019     Published: 4 November 2019
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Abstract

Mathematical modeling of disease has been an indispensable tool in accounting for disease transmission dynamics as well as disease spread. Epidemiological disease models have been used to explain the dynamics of HIV/AIDS in the population from the early 1900s. The models developed however faced considerable challenges ranging from inaccurate representation of natural data for deterministic models, to methods of forecasting such as statistical extrapolation which assumes that current conditions will prevail which is not always the case. Despite the spread of HIV/AIDS having been explored widely, not much literature is available on the Gillespie Algorithm based SIR model. This algorithm is able to give a statistically correct of the course of a disease with initial conditions to begin with and propensity functions to update the system. The purpose of this paper is to build on the concept of Gillespie's Algorithm based SIR models by developing a stochastic SIR model to simulate disease evolution in the population setting. The values produced through simulation by the model developed in this paper using a tau value as the time step of the model were compared to HIV/AIDS data from 1985 to 2018, given by NACC. We conclude that the simulated model reflects reality.

Published in International Journal of Data Science and Analysis (Volume 5, Issue 6)
DOI 10.11648/j.ijdsa.20190506.12
Page(s) 117-122
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), 2019. Published by Science Publishing Group

Keywords

Stochastic, Simulation, Deterministic, SIR Model, Continuous-Time Markov Chain, Gillespie's Algorithm Models

References
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[2] W. A. M. S. R. T. D. O. T. R. S. J. V. N. L. P. N. B. J. F. K. D. P. W. S. M. o. Paxton, "Relative resistance to HIV--1 infection of CD4 lymphocytes from persons who remain uninfected despite multiple high--risk sexual exposures," Nature medicine, vol. 2, no. 4, p. 412, 1996.
[3] W. A. N. G. R. K. H. J. L. U. O. C.-p. Z. N. W. A. N. G. Jun-jie, "Dynamic mathematical models of HIV/AIDS transmission in China," Chinese medical journal, vol. 123, no. 15, p. 2120, 2010.
[4] A. I. D. S. Council, "Working Committee Office UN Theme Group on AIDS in China," A joint assessment of HIV/AIDS prevention, treatment and care in China. Beijing: Ministry of Health, China, 2007.
[5] H. S. Rodrigues, "Application of SIR epidemiological model: new trends," arXiv preprint arXiv: 1611.02565, 2016.
[6] L. S. J. A. B. T. R. A. E. a. H. D. R. Bao, "Modelling national HIV/AIDS epidemics: revised approach in the UNAIDS Estimation and Projection Package 2011," BMJ Publishing Group Limited, no. 88, pp. 3-18, 2012.
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[12] R. P. Dobrow, "Introduction to stochastic processes with R," 2016.
[13] W. O. M. A. G. Kermack, "A contribution to the mathematical theory of epidemics," Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, vol. 115, no. 772, pp. 700-721, 1927.
[14] W. O. M. A. G. Kermack, "Contributions to the mathematical theory of epidemics. II.—The problem of endemicity," Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, vol. 138, no. 834, pp. 55-83, 1932.
[15] W. O. M. A. G. Kermack, "Contributions to the mathematical theory of epidemics. III.—Further studies of the problem of endemicity," Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 141, no. 843, pp. 94-122, 1933.
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Cite This Article
  • APA Style

    Kavyu Mary Kamina, Samuel Mwalili, Anthony Wanjoya. (2019). The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie's Algorithm. International Journal of Data Science and Analysis, 5(6), 117-122. https://doi.org/10.11648/j.ijdsa.20190506.12

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    ACS Style

    Kavyu Mary Kamina; Samuel Mwalili; Anthony Wanjoya. The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie's Algorithm. Int. J. Data Sci. Anal. 2019, 5(6), 117-122. doi: 10.11648/j.ijdsa.20190506.12

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    AMA Style

    Kavyu Mary Kamina, Samuel Mwalili, Anthony Wanjoya. The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie's Algorithm. Int J Data Sci Anal. 2019;5(6):117-122. doi: 10.11648/j.ijdsa.20190506.12

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  • @article{10.11648/j.ijdsa.20190506.12,
      author = {Kavyu Mary Kamina and Samuel Mwalili and Anthony Wanjoya},
      title = {The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie's Algorithm},
      journal = {International Journal of Data Science and Analysis},
      volume = {5},
      number = {6},
      pages = {117-122},
      doi = {10.11648/j.ijdsa.20190506.12},
      url = {https://doi.org/10.11648/j.ijdsa.20190506.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20190506.12},
      abstract = {Mathematical modeling of disease has been an indispensable tool in accounting for disease transmission dynamics as well as disease spread. Epidemiological disease models have been used to explain the dynamics of HIV/AIDS in the population from the early 1900s. The models developed however faced considerable challenges ranging from inaccurate representation of natural data for deterministic models, to methods of forecasting such as statistical extrapolation which assumes that current conditions will prevail which is not always the case. Despite the spread of HIV/AIDS having been explored widely, not much literature is available on the Gillespie Algorithm based SIR model. This algorithm is able to give a statistically correct of the course of a disease with initial conditions to begin with and propensity functions to update the system. The purpose of this paper is to build on the concept of Gillespie's Algorithm based SIR models by developing a stochastic SIR model to simulate disease evolution in the population setting. The values produced through simulation by the model developed in this paper using a tau value as the time step of the model were compared to HIV/AIDS data from 1985 to 2018, given by NACC. We conclude that the simulated model reflects reality.},
     year = {2019}
    }
    

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    AU  - Kavyu Mary Kamina
    AU  - Samuel Mwalili
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    AB  - Mathematical modeling of disease has been an indispensable tool in accounting for disease transmission dynamics as well as disease spread. Epidemiological disease models have been used to explain the dynamics of HIV/AIDS in the population from the early 1900s. The models developed however faced considerable challenges ranging from inaccurate representation of natural data for deterministic models, to methods of forecasting such as statistical extrapolation which assumes that current conditions will prevail which is not always the case. Despite the spread of HIV/AIDS having been explored widely, not much literature is available on the Gillespie Algorithm based SIR model. This algorithm is able to give a statistically correct of the course of a disease with initial conditions to begin with and propensity functions to update the system. The purpose of this paper is to build on the concept of Gillespie's Algorithm based SIR models by developing a stochastic SIR model to simulate disease evolution in the population setting. The values produced through simulation by the model developed in this paper using a tau value as the time step of the model were compared to HIV/AIDS data from 1985 to 2018, given by NACC. We conclude that the simulated model reflects reality.
    VL  - 5
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Author Information
  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

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