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

Richard Onyino Simwa; Nelson Lwoyelo Muhati; Lucy Chikamai

doi:doi:10.11648/j.sjams.20170505.11

| Peer-Reviewed

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

Richard Onyino Simwa, Nelson Lwoyelo Muhati, Lucy Chikamai

Published in Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 5)

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 5)
DOI	10.11648/j.sjams.20170505.11
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), 2017. 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

[1]	WHO (2013). Global tuberculosis report 2013.
[2]	Friedl and G, Churchyard GJ and Nardell E. (2007). Tuberculosis and HIV coinfection: current state of knowledge and research priorities. J Infect Dis 1 (9-6).
[3]	Naresh R., Tripathi A., Omar S., (2006). Modelling the spread of AIDS epidemic with vertical transmission. Applied Mathematics and Computation.
[4]	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.
[5]	Raviglione MD, Snider DE, Kochi A, (1995). Global epidemiology of tuberculosis: morbidity and mortality of a worldwide epidemic. Washington D. C.
[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).
[13]	Johnson N. L, Kotz S. (1970). Distributions in statistics: continuous univariate distributions. Vol 2. New York, NY: John Wiley and Sons, Inc.
[14]	Ong’ala J. O, Mugisha J. and Oleche P. (2014). A Probabilistic Estimation of the Basic Reproduction Number: A Case of Control Strategy of Pneumonia. Science Journal of Applied Mathematics and Statistics 2(2): 53-59.
[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.
[18]	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.
[19]	Diekmann O., Heesterbeek, J. A. P., Metz, J. A. P., (1990). On the definition and computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations. J Math Biol 365-382.
[20]	Hoffmann C. (2007). Opportunistic Infections: In Bernd Sebastian Kamps, ed. HIV Medicine. 15th Edition. Flying Publisher, Paris, p. 389-477.
[21]	Hyman J. M., Li J. and Stanley, E. A. (1999). The differential infectivity and staged progression models for the transmission of HIV. Math Biosci 155(2): 77-109.
[22]	Swartzman G. L and Kaluzny S. P. (1987). Ecological simulation primer. New York, NY: Macmillan Publishing Company.

Cite This Article

Plain Text BibTeX RIS

APA Style

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

Copy | Download

ACS 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

Copy | Download

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - 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
AU  - Richard Onyino Simwa
AU  - Nelson Lwoyelo Muhati
AU  - Lucy Chikamai
Y1  - 2017/08/22
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170505.11
DO  - 10.11648/j.sjams.20170505.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 169
EP  - 173
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170505.11
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.
VL  - 5
IS  - 5
ER  -

Copy | Download

Author Information

Richard Onyino Simwa

School of Mathematics, University of Nairobi, Nairobi, Kenya
Nelson Lwoyelo Muhati

Department of Mathematics, Kibabii University, Bungoma, Kenya
Lucy Chikamai

Department of Mathematics, Kibabii University, Bungoma, Kenya

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

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

Copy | Download

ACS 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

Copy | Download

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

TY  - JOUR
T1  - 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
AU  - Richard Onyino Simwa
AU  - Nelson Lwoyelo Muhati
AU  - Lucy Chikamai
Y1  - 2017/08/22
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170505.11
DO  - 10.11648/j.sjams.20170505.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 169
EP  - 173
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170505.11
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.
VL  - 5
IS  - 5
ER  -

Copy | Download