Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis

Shimelis Bekele Zerefe; Temesgen Tibebu Mekonnen

doi:doi:10.11648/j.ajam.20200803.15

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Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis

Shimelis Bekele Zerefe, Temesgen Tibebu Mekonnen

Published in American Journal of Applied Mathematics (Volume 8, Issue 3)

Received: 22 April 2020 Accepted: 15 May 2020 Published: 29 May 2020

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Abstract

In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 3)
DOI	10.11648/j.ajam.20200803.15
Page(s)	123-144
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), 2020. Published by Science Publishing Group

Keywords

Drug-sensitive Tuberculosis, Drug Resistance Tuberculosis, Effective Reproduction Number, Sensitivity Analysis, Numerical Analysis

References

[1]	A. Jabbari and C. Castilo-Chaviz, A two-strain TB mode with multiple latent stages mathematical biosciences and engineering, 2016.
[2]	B. K. Mishra, Mathematical model on pulmonary and multi-drug-resistant tuberculosis patients with vaccination, Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi 835 2015.
[3]	C. P. Bhunu, A two strain tuberculosis transmission model with therapy and quarantine, 2010.
[4]	D. Meressa Achieving high treatment success for multidrugresistant TB in Africa: initiation and scale-up of MDR TB care in Ethiopia—an observational cohort study, http://thorax.bmj.com/ on November 25, 2015.
[5]	F. B. Agusto, Mathematical Model of MDR-TB and XDR-TB with Isolation and Lost to Follow-Up, 2015.
[6]	A. Deribew, Tuberculosis Burden in Ethiopia from 1990 to 2016: Evidence from the Global Burden of Diseases, https://www.researchgate.net/publication/328602797, 2018.
[7]	FDREM-H Ethiopian Public Health Institute, 2014, https://www.who.int/globalchange/resources/wash-toolkit/review-of-policy-documents-on-climate-change-wash-and-public-health-in-ethiopia.pdf?ua=1.
[8]	Federal Democratic Republic of Ethiopia Ministry of Health (EMH), 2018, https://www.medbox.org/national-strategic-plan-tuberculosis-and-leprosy-control-2006-2013-ec-201314-2020/download.pdf.
[9]	Y. Yu, Dynamic model of tuberculosis considering multi-drug resistance and their applications, 2018.
[10]	A. A. Yeketi, A Compartmental Model on the Effect of Quarantine on MDR-TB, International Journal of Mathematics and Computer Science, 2019.
[11]	Implementation Guideline for GeneXpert MTB/RIF Assay in Ethiopia, 2014. https://www.medbox.org/et-guidelines-hiv-tb/implementation-guideline-for-gene-xpert-mtbrif-assay-in-ethiopia/preview.
[12]	M. Al-arydah, Modelling the impact of treatment on tuberculosis transmission dynamics, International Journal of Biomathematics and Systems Biology, 2015.
[13]	S. Eshetie, Tuberculosis treatment outcomes in Ethiopia from 2003 to 2016, and impact of HIV coinfection and prior drug exposure: A systematic review and meta-analysis, 2018.
[14]	Z. B. Shimelis and T. T. Temesgen Dynamics of a Tuberculosis Model with Vaccination and Dual Treatments: a Mathematical model Analysis, e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 15, Issue 4 Ser. II (Jul { Aug 2019), PP 47-60 www.iosrjournals.org, IOSR-JM.
[15]	Z. Shuai, Global stability of infectious disease models using Lyapunov functions, society for industrial and applied mathematics, 2013.
[16]	S. A. Gershgorin, Gershgorin's Circle Theorem. http://en.wikipedia.org, 1931.
[17]	Federal Democratic Republic of Ethiopia Ministry of Health national, https://www.afro.who.int/publications/national-strategic-plan-tuberculosis-and-leprosy-control-2006-2013-ec-201314-2020.
[18]	Ethiopia Demographics Profile, 2019, https://www.indexmundi.com/ethiopia/demographics_profile.html.
[19]	World Health Organization, Global tuberculosis report, https://www.who.int/tb/publications/global_report/gtbr2017_main_text.pdf.
[20]	B. Bewketu: Modeling Tuberculosis transmission dynamics in children and adults in the presence of vaccination (2011).
[21]	M. Maliyani, P. M. Mwamtobe, S. D. Hove-Musekwa and J. M. Tchuenche: Modelling the role of diagnosis, Treatment and Health education on Multi-Drug resistant tuberculosis dynamics International Journal of Biomathematics and Systems Biology, 2015.
[22]	Z. Gashu, The Yield of Community-Based “Retrospective” Tuberculosis Contact Investigation in a High Burden Setting in Ethiopia, DOI:10.1371/journal.pone.0160514, 2016.
[23]	A. B. Gumel, Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis, mathematical biosciences and engineering, 2008.
[24]	T. Girum, Epidemiology of multidrug-resistant tuberculosis (MDR-TB) in Ethiopia: a systematic review and meta-analysis of the prevalence, determinants and treatment outcome, Travel Medicine and Vaccines (2018).
[25]	D. Okuonghae, and A. Korobeinikov, Dynamics of Tuberculosis: The Effect ofDirectObservation Therapy Strategy (DOTS) in Nigeria, MathematicalModeling of NaturalPhenomena, 2 (2007), 101-113.
[26]	N. Chitnis, J. Hyman and J. Cusching, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology 70 (2008) 1272 1296.

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APA Style

Shimelis Bekele Zerefe, Temesgen Tibebu Mekonnen. (2020). Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. American Journal of Applied Mathematics, 8(3), 123-144. https://doi.org/10.11648/j.ajam.20200803.15

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ACS Style

Shimelis Bekele Zerefe; Temesgen Tibebu Mekonnen. Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. Am. J. Appl. Math. 2020, 8(3), 123-144. doi: 10.11648/j.ajam.20200803.15

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AMA Style

Shimelis Bekele Zerefe, Temesgen Tibebu Mekonnen. Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. Am J Appl Math. 2020;8(3):123-144. doi: 10.11648/j.ajam.20200803.15

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@article{10.11648/j.ajam.20200803.15,
  author = {Shimelis Bekele Zerefe and Temesgen Tibebu Mekonnen},
  title = {Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {123-144},
  doi = {10.11648/j.ajam.20200803.15},
  url = {https://doi.org/10.11648/j.ajam.20200803.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200803.15},
  abstract = {In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.},
 year = {2020}
}

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TY  - JOUR
T1  - Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis
AU  - Shimelis Bekele Zerefe
AU  - Temesgen Tibebu Mekonnen
Y1  - 2020/05/29
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200803.15
DO  - 10.11648/j.ajam.20200803.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 123
EP  - 144
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200803.15
AB  - In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.
VL  - 8
IS  - 3
ER  -

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Author Information

Shimelis Bekele Zerefe

Department of Mathematics, Addis Ababa Science and Technology University, Addis Ababa, Ethiopia
Temesgen Tibebu Mekonnen

Department of Mathematics, Debre Berhan University, Debre Berhan, Ethiopia

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Plain Text BibTeX RIS

APA Style

Shimelis Bekele Zerefe, Temesgen Tibebu Mekonnen. (2020). Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. American Journal of Applied Mathematics, 8(3), 123-144. https://doi.org/10.11648/j.ajam.20200803.15

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ACS Style

Shimelis Bekele Zerefe; Temesgen Tibebu Mekonnen. Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. Am. J. Appl. Math. 2020, 8(3), 123-144. doi: 10.11648/j.ajam.20200803.15

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AMA Style

Shimelis Bekele Zerefe, Temesgen Tibebu Mekonnen. Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis. Am J Appl Math. 2020;8(3):123-144. doi: 10.11648/j.ajam.20200803.15

Copy | Download

@article{10.11648/j.ajam.20200803.15,
  author = {Shimelis Bekele Zerefe and Temesgen Tibebu Mekonnen},
  title = {Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {123-144},
  doi = {10.11648/j.ajam.20200803.15},
  url = {https://doi.org/10.11648/j.ajam.20200803.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200803.15},
  abstract = {In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis
AU  - Shimelis Bekele Zerefe
AU  - Temesgen Tibebu Mekonnen
Y1  - 2020/05/29
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200803.15
DO  - 10.11648/j.ajam.20200803.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 123
EP  - 144
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200803.15
AB  - In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.
VL  - 8
IS  - 3
ER  -

Copy | Download