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), 2024. Published by Science Publishing Group

Keywords

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

References

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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