Science Journal of Applied Mathematics and Statistics

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On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment

Received: 11 July 2017    Accepted: 21 July 2017    Published: 22 August 2017
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Abstract

Human Immunodeficiency Virus (HIV) and Mycobacterium Tuberculosis (TB) infections are two major world’s public health problems especially in developing countries. Worldwide, 13% of TB cases are estimated to be co-infected with HIV and about a third of 33 million people living with HIV are infected with the bacterium that causes TB. Deterministic models are derived and applied to estimate the basic reproduction number of HIV and TB co-infection as a single output value by treating each of the parameter input as a constant value. In this paper the basic reproduction number is modeled as a random variable, then the probability that there will bean epidemic, is derived and computed. In particular it is shown that for the sub-Saharan region, the probability of the epidemic occurring is 7.9%, as expected since an epidemic is generally a rare event. This research, thus develops the methodology for the computation of the probability of occurrence of an epidemic, which is useful for the public health policy formulation.

DOI 10.11648/j.sjams.20170505.11
Published in Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 5, October 2017)
Page(s) 169-173
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), 2024. Published by Science Publishing Group

Keywords

Normal Distribution, Basic Reproduction Number, Probability of an Epidemic, HIV/AID Sand TB Co-Infection Modeling, Stochastic Simulations, Ordinary Differential Equations

References
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[6] Rosas-Taraco A. G, Arce-Mendoza A. Y, Caballero-Olin G, and Salinas-Carmona M. C. (2006). Mycobacterium tuberculosis upregulates coreceptors CCR5 and CXCR4 while HIV modulates CD14 favoring concurrent infection. AIDS Res Hum Retroviruses 22(1):45-51.
[7] AIDS control and prevention (AIDSCAP) (1996). The status and Trends of the global HIV/AID Spandemic.
[8] Sharma S. K, Mohan A, Kadhiravan T. (2005). HIV-TB co-infection: epidemiology, diagnosis and management. Indian J Med Res 121(4):550-567.
[9] WHO (2015). Global Tuberculosis Report. http://www.who.int/tb/areas-ofwork/tb-hiv/en/
[10] Adewale S. O., Olopade I. A., Adeniran G. A. and Ajao S. O. (2015). Mathematical modelling and sensitivity analysis of HIV-TB co-infection. Journal of advances in Mathematics vol. 11, no. 8.
[11] AIDS control and prevention (AIDSCAP) (1996). The status and Trends of the global HIV/AID Spandemic.
[12] Fatmawati and Hengki T. An Optimal Treatment Control of TB/HIV Co-infection (2016). International Journal of Mathematics and Mathematical Sciences. Volume 2016, ID8261208 (OR http://dx.doi.org/10.1155/2016/8261208).
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[15] Sanchez M. and Blower S. (1997). Uncertainty and Sensitivity Analysis of the Basic Reproductive Rate: Tuberculosis as an Example. American Journal of Epidemiology. Vol. 12(145) pp. 1127-1137.
[16] Van den Driesche P. and Watmough J. (2005). Reproduction numbers andsub-threshold endemic equilibria for the compartmental models of disease transmission. Math Biosci 180: 29-48.
[17] Dye C., Garnett, G. P., Sleeman, K. and Williams, B. G. (1998). Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Directly observed short-course therapy. Lancet.
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Author Information
  • School of Mathematics, University of Nairobi, Nairobi, Kenya

  • Department of Mathematics, Kibabii University, Bungoma, Kenya

  • Department of Mathematics, Kibabii University, Bungoma, Kenya

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    Richard Onyino Simwa, Nelson Lwoyelo Muhati, Lucy Chikamai. (2017). On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment. Science Journal of Applied Mathematics and Statistics, 5(5), 169-173. https://doi.org/10.11648/j.sjams.20170505.11

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    Richard Onyino Simwa; Nelson Lwoyelo Muhati; Lucy Chikamai. On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment. Sci. J. Appl. Math. Stat. 2017, 5(5), 169-173. doi: 10.11648/j.sjams.20170505.11

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    AMA Style

    Richard Onyino Simwa, Nelson Lwoyelo Muhati, Lucy Chikamai. On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment. Sci J Appl Math Stat. 2017;5(5):169-173. doi: 10.11648/j.sjams.20170505.11

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  • @article{10.11648/j.sjams.20170505.11,
      author = {Richard Onyino Simwa and Nelson Lwoyelo Muhati and Lucy Chikamai},
      title = {On Derivation of the Probability of Occurrence of an Epidemic with Application to HIV/AIDS Spread Given Tuberculosis Co-infections in the Presence of Treatment},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {5},
      number = {5},
      pages = {169-173},
      doi = {10.11648/j.sjams.20170505.11},
      url = {https://doi.org/10.11648/j.sjams.20170505.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.sjams.20170505.11},
      abstract = {Human Immunodeficiency Virus (HIV) and Mycobacterium Tuberculosis (TB) infections are two major world’s public health problems especially in developing countries. Worldwide, 13% of TB cases are estimated to be co-infected with HIV and about a third of 33 million people living with HIV are infected with the bacterium that causes TB. Deterministic models are derived and applied to estimate the basic reproduction number of HIV and TB co-infection as a single output value by treating each of the parameter input as a constant value. In this paper the basic reproduction number is modeled as a random variable, then the probability that there will bean epidemic, is derived and computed. In particular it is shown that for the sub-Saharan region, the probability of the epidemic occurring is 7.9%, as expected since an epidemic is generally a rare event. This research, thus develops the methodology for the computation of the probability of occurrence of an epidemic, which is useful for the public health policy formulation.},
     year = {2017}
    }
    

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    AB  - Human Immunodeficiency Virus (HIV) and Mycobacterium Tuberculosis (TB) infections are two major world’s public health problems especially in developing countries. Worldwide, 13% of TB cases are estimated to be co-infected with HIV and about a third of 33 million people living with HIV are infected with the bacterium that causes TB. Deterministic models are derived and applied to estimate the basic reproduction number of HIV and TB co-infection as a single output value by treating each of the parameter input as a constant value. In this paper the basic reproduction number is modeled as a random variable, then the probability that there will bean epidemic, is derived and computed. In particular it is shown that for the sub-Saharan region, the probability of the epidemic occurring is 7.9%, as expected since an epidemic is generally a rare event. This research, thus develops the methodology for the computation of the probability of occurrence of an epidemic, which is useful for the public health policy formulation.
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