Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease

Birke Seyoum Desta; Purnachandra Rao Koya

doi:doi:10.11648/j.ajam.20190701.13

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Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease

Birke Seyoum Desta, Purnachandra Rao Koya

Published in American Journal of Applied Mathematics (Volume 7, Issue 1)

Received: 18 June 2018 Accepted: 28 April 2019 Published: 29 May 2019

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Abstract

In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEI_CI_AR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the solutions are proved to be positive as well as bounded. The basic reproduction number R_O of the model is derived using the next generation matrix method. Further, the equilibrium points of the model are identified and mathematical analysis pertaining to their stability is conducted using Routh – Hurtiz criteria. It is shown that the disease free equilibrium point is locally and globally stable If R_O＜1. On the other hand, the endemic equilibrium point is proved to be stable if R_O＞1. Also, the numerical simulation study of the model is carried out using ode45 of MATLAB: Rung – Kutta order four. It is observed that, if the vaccination and treatment rates are increased then the infective population size decreases and evenfall to zero over time. Hence, it is concluded that the use of vaccination and treatment at the highest possible rates is essential so as to control the spread hepatitis B virus.

Published in	American Journal of Applied Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.ajam.20190701.13
Page(s)	13-20
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

Hepatitis B Virus, Vaccination, Treatment, Mathematical Model, Basic Reproduction Number, Stability Analysis, Numerical Simulation

References

[1]	U. S. Department of Health and Human Services, center for Disease control and prevention, 2016.
[2]	World Health Organization, Hepatitis B virus: epidemiology and transmission risks, 2015.
[3]	Tahir Khana, Gul Zamana and M., Ikhlaq Chohanb, The transmission dynamic and optimal control of acute and chronic hepatitis, Journal of Biological Dynamics, 11: 1, 172 − 189, 2017.
[4]	Saher, F., Rahman, K., Quresh, J. A., Irshad, M. and Iqbal, H. M., Investigation of an Inflammatory Viral Disease HBV in Cardiac Patients through Polymerase Chain Reaction, Advances in Bioscience and Biotechnology, 3, 417 −422. http: //dx.doi.org/10.4236/abb.2012.324059., (2012).
[5]	Hindia Nurhasen, The Transmission Dynamics and Optimal Control of Hepatitis B in Ethiopia Using SVEIRS Model, Addis Ababa University, October, 12, 2017.
[6]	Moneim I. A. Al-Ahmed M. and Mosa G. A., Stochastic and Monte Carlo Simulation for the Spread of the Hepatitis B, Australian Journal of Basic and Applied Sciences, (2009) 3, 1607 − 1615.
[7]	Saint Paul, Minnesota, www.immunize.org.www.vaccineinformation.org www.immunize.org/catg.d/p4205.pdf 651 − 647 − 9009.
[8]	I. A. Moneim, H. A. Khalil, Modeling and Simulation of the Spread of HBV Disease with Infectious Latent, Applied Mathematics, 2015, 6, 745−753.
[9]	Sacrifice Nana-Kyere, Joseph Ackora-Prah, Eric Okyere, Seth Marmah, Tuah Afram, Hepatitis B Optimal Control Model with Vertical Transmission Applied Mathematics, Applied Mathematics 7 (1): 5 − 13, 2017.
[10]	WHO, Hepatitis B Factsheet, (2001).
[11]	Tadele Tesfa Tegegne, Purnachandra Rao Koya and Temesgen Tibebu Mekonnen, Modeling and Simulation study of the Dynamics of HIV/AIDS due to Heterosexual and Vertical Transmissions, Hawassa, ETHIOPIA, 20015.
[12]	Molalegn Ayana, Modeling the Impact of Infective Immigrants on the Dynamics and Spread of Zika virus, Hawassa, Ethiopia June, 2017.
[13]	Carlos Castillo-Chavez, Zhilan Feng and Wenzhang Huang” On the computation of and its role on global stability” February, 2001.

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Birke Seyoum Desta, Purnachandra Rao Koya. (2019). Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. American Journal of Applied Mathematics, 7(1), 13-20. https://doi.org/10.11648/j.ajam.20190701.13

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ACS Style

Birke Seyoum Desta; Purnachandra Rao Koya. Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. Am. J. Appl. Math. 2019, 7(1), 13-20. doi: 10.11648/j.ajam.20190701.13

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AMA Style

Birke Seyoum Desta, Purnachandra Rao Koya. Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. Am J Appl Math. 2019;7(1):13-20. doi: 10.11648/j.ajam.20190701.13

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@article{10.11648/j.ajam.20190701.13,
  author = {Birke Seyoum Desta and Purnachandra Rao Koya},
  title = {Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {13-20},
  doi = {10.11648/j.ajam.20190701.13},
  url = {https://doi.org/10.11648/j.ajam.20190701.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.13},
  abstract = {In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEICIAR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the solutions are proved to be positive as well as bounded. The basic reproduction number RO of the model is derived using the next generation matrix method. Further, the equilibrium points of the model are identified and mathematical analysis pertaining to their stability is conducted using Routh – Hurtiz criteria. It is shown that the disease free equilibrium point is locally and globally stable If RO＜1. On the other hand, the endemic equilibrium point is proved to be stable if RO＞1. Also, the numerical simulation study of the model is carried out using ode45 of MATLAB: Rung – Kutta order four. It is observed that, if the vaccination and treatment rates are increased then the infective population size decreases and evenfall to zero over time. Hence, it is concluded that the use of vaccination and treatment at the highest possible rates is essential so as to control the spread hepatitis B virus.},
 year = {2019}
}

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TY - JOUR
T1 - Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease
AU - Birke Seyoum Desta
AU - Purnachandra Rao Koya
Y1 - 2019/05/29
PY - 2019
N1 - https://doi.org/10.11648/j.ajam.20190701.13
DO - 10.11648/j.ajam.20190701.13
T2 - American Journal of Applied Mathematics
JF - American Journal of Applied Mathematics
JO - American Journal of Applied Mathematics
SP - 13
EP - 20
PB - Science Publishing Group
SN - 2330-006X
UR - https://doi.org/10.11648/j.ajam.20190701.13
AB - In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEICIAR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the solutions are proved to be positive as well as bounded. The basic reproduction number RO of the model is derived using the next generation matrix method. Further, the equilibrium points of the model are identified and mathematical analysis pertaining to their stability is conducted using Routh – Hurtiz criteria. It is shown that the disease free equilibrium point is locally and globally stable If RO＜1. On the other hand, the endemic equilibrium point is proved to be stable if RO＞1. Also, the numerical simulation study of the model is carried out using ode45 of MATLAB: Rung – Kutta order four. It is observed that, if the vaccination and treatment rates are increased then the infective population size decreases and evenfall to zero over time. Hence, it is concluded that the use of vaccination and treatment at the highest possible rates is essential so as to control the spread hepatitis B virus.
VL - 7
IS - 1
ER -

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Author Information

Birke Seyoum Desta

Department of Mathematics, Hawassa University, Hawassa, Ethiopia
Purnachandra Rao Koya

Department of Mathematics, Hawassa University, Hawassa, Ethiopia

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APA Style

Birke Seyoum Desta, Purnachandra Rao Koya. (2019). Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. American Journal of Applied Mathematics, 7(1), 13-20. https://doi.org/10.11648/j.ajam.20190701.13

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ACS Style

Birke Seyoum Desta; Purnachandra Rao Koya. Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. Am. J. Appl. Math. 2019, 7(1), 13-20. doi: 10.11648/j.ajam.20190701.13

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AMA Style

Birke Seyoum Desta, Purnachandra Rao Koya. Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease. Am J Appl Math. 2019;7(1):13-20. doi: 10.11648/j.ajam.20190701.13

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@article{10.11648/j.ajam.20190701.13,
  author = {Birke Seyoum Desta and Purnachandra Rao Koya},
  title = {Modified Model and Stability Analysis of the Spread of Hepatitis B Virus Disease},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {13-20},
  doi = {10.11648/j.ajam.20190701.13},
  url = {https://doi.org/10.11648/j.ajam.20190701.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.13},
  abstract = {In this study the infected population is classified into two categories viz., chronic and acute and thus developed a five compartmental SEICIAR model. Also, both vaccination and treatments are included and studied their impact on the spread of hepatitis B virus. The present model is biologically meaningful and mathematically well posed since the solutions are proved to be positive as well as bounded. The basic reproduction number RO of the model is derived using the next generation matrix method. Further, the equilibrium points of the model are identified and mathematical analysis pertaining to their stability is conducted using Routh – Hurtiz criteria. It is shown that the disease free equilibrium point is locally and globally stable If RO＜1. On the other hand, the endemic equilibrium point is proved to be stable if RO＞1. Also, the numerical simulation study of the model is carried out using ode45 of MATLAB: Rung – Kutta order four. It is observed that, if the vaccination and treatment rates are increased then the infective population size decreases and evenfall to zero over time. Hence, it is concluded that the use of vaccination and treatment at the highest possible rates is essential so as to control the spread hepatitis B virus.},
 year = {2019}
}

Copy | Download