Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection

M. O. Ibrahim; S. A. Egbetade

doi:doi:10.11648/j.ajaic.20170101.12

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Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection

M. O. Ibrahim, S. A. Egbetade

Published in American Journal of Applied and Industrial Chemistry (Volume 1, Issue 1)

Received: 11 August 2015 Accepted: 28 August 2015 Published: 1 April 2017

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Abstract

In this paper, we extend the model of Blower et al. [1] by incorporating certain infection terms such as vaccinated individuals, treatment rate, waning rate and efficacy rate.A bifurcation analysis is performed on the vaccination model by applying a bifurcation method based on the use of center manifold theory.We determine threshold values and derive sufficient conditions for both forward and backward bifurcations.Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. The obtained results show the possibility of occurrence of forward and backward bifurcations even when the basic reproduction number is less than one so that it is now possible for the disease to exist. These results suggest the need for more study on the qualitative biological mechanisms responsible for backward bifurcation.

Published in	American Journal of Applied and Industrial Chemistry (Volume 1, Issue 1)
DOI	10.11648/j.ajaic.20170101.12
Page(s)	5-9
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

Mathematical Models, Tuberculosis, Bifurcation, Vaccination, Center Manifold Theory, Stability

References

[1]	Blower, S.M., McLean, A.R., Porco, T.C., Small, P.M., Hopewell, P.C., Sanchez, M.A. andMoss, A.R. (1995).The intrinsic transmission dynamics of tuberculosis epidemics.Nat. Med. 1(8), 815-821.
[2]	Kribs-Zaleta, C.M. and Velasco-Hernandez, J.X. (2000).A simple vaccination model withmultiple endemic states.Math. Biosci. 164(2), 183-201.
[3]	Huang, W., Cooke, K.L. and Castillo-Chavez, C. (1992).Stability and bifurcation for amultiple-group model for the dynamics of HIV/AIDS transmission.SIAM J.Appl. Math. 52, 835-854.
[4]	Greenhalgh, D. and Griffiths, M. (2009).Backward bifurcation, equilibrium and stabilityphenomena in a three-stage extended BRSV epidemic model.J. Math. Biosci.59, 1-36.
[5]	Buonomo, B. and Lacitignola, D. (2011).On the backward bifurcation of a vaccinationmodel with nonlinear incidence.Nonlinear Analysis: Modeling and Control 16(1),30-46.
[6]	Kribs-Zaleta, C.M. (1999). Core recruitment effects in SIS models with constant totalpopulations.Math. Biosci. 160(2), 109-158.
[7]	Wang, W. (2006).Backward bifurcation of an epidemic model with treatment.Math. Biosci. 201, 58-71.
[8]	Sharomi, O., Podder, C.N., Gumel, A.B., Elbasha, E.H. and Watmough, J. (2007).Role ofincidence function in vaccine-induced backward bifurcation in some HIV models.Mathematical Biosciences 210, 436-463.
[9]	Sophia, R. and Jang, J. (2008).Backward bifurcation in a discrete SIS modelwith vaccination.Journal of Biological Systems 16(4), 479-487.
[10]	Wan, H. and Zhu, H. (2010). The backward bifurcation in compartmental models for WestNile Virus.Mathematical Biosciences 227, 20-28.
[11]	Bowong, S. and Kurths, J. (2012).Modeling and analysis of the transmission dynamics oftuberculosis without and with seasonality.Nonlinear Dynamics 67, 2027-2051.
[12]	Li, J. and Cui, N. (2013). Bifurcation and chaotic behaviour of a discrete-time SIS model.Discrete Dynamics in Nature and Society. http://dx.doi.prg/10.1155/2013/705601
[13]	Castillo-Chavez, C. and Song, B. (2004).Dynamical models of tuberculosis and theirapplications.Math. Biosci. Engr. 1, 361-404.
[14]	van den Driessche, P. and Watmough, J. (2002).Reproduction numbers and sub-thresholdendemic equilibria for compartmental models of disease transmission.Math. Biosci.180, 29-48.

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M. O. Ibrahim, S. A. Egbetade. (2017). Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. American Journal of Applied and Industrial Chemistry, 1(1), 5-9. https://doi.org/10.11648/j.ajaic.20170101.12

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ACS Style

M. O. Ibrahim; S. A. Egbetade. Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. Am. J. Appl. Ind. Chem. 2017, 1(1), 5-9. doi: 10.11648/j.ajaic.20170101.12

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AMA Style

M. O. Ibrahim, S. A. Egbetade. Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. Am J Appl Ind Chem. 2017;1(1):5-9. doi: 10.11648/j.ajaic.20170101.12

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@article{10.11648/j.ajaic.20170101.12,
  author = {M. O. Ibrahim and S. A. Egbetade},
  title = {Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection},
  journal = {American Journal of Applied and Industrial Chemistry},
  volume = {1},
  number = {1},
  pages = {5-9},
  doi = {10.11648/j.ajaic.20170101.12},
  url = {https://doi.org/10.11648/j.ajaic.20170101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaic.20170101.12},
  abstract = {In this paper, we extend the model of Blower et al. [1] by incorporating certain infection terms such as vaccinated individuals, treatment rate, waning rate and efficacy rate.A bifurcation analysis is performed on the vaccination model by applying a bifurcation method based on the use of center manifold theory.We determine threshold values and derive sufficient conditions for both forward and backward bifurcations.Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. The obtained results show the possibility of occurrence of forward and backward bifurcations even when the basic reproduction number is less than one so that it is now possible for the disease to exist. These results suggest the need for more study on the qualitative biological mechanisms responsible for backward bifurcation.},
 year = {2017}
}

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TY  - JOUR
T1  - Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection
AU  - M. O. Ibrahim
AU  - S. A. Egbetade
Y1  - 2017/04/01
PY  - 2017
N1  - https://doi.org/10.11648/j.ajaic.20170101.12
DO  - 10.11648/j.ajaic.20170101.12
T2  - American Journal of Applied and Industrial Chemistry
JF  - American Journal of Applied and Industrial Chemistry
JO  - American Journal of Applied and Industrial Chemistry
SP  - 5
EP  - 9
PB  - Science Publishing Group
SN  - 2994-7294
UR  - https://doi.org/10.11648/j.ajaic.20170101.12
AB  - In this paper, we extend the model of Blower et al. [1] by incorporating certain infection terms such as vaccinated individuals, treatment rate, waning rate and efficacy rate.A bifurcation analysis is performed on the vaccination model by applying a bifurcation method based on the use of center manifold theory.We determine threshold values and derive sufficient conditions for both forward and backward bifurcations.Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. The obtained results show the possibility of occurrence of forward and backward bifurcations even when the basic reproduction number is less than one so that it is now possible for the disease to exist. These results suggest the need for more study on the qualitative biological mechanisms responsible for backward bifurcation.
VL  - 1
IS  - 1
ER  -

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Author Information

M. O. Ibrahim

Department of Mathematics, University of Ilorin, Ilorin, Nigeria
S. A. Egbetade

Department of Mathematics and Statistics, The Polytechnic, Ibadan, Nigeria

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APA Style

M. O. Ibrahim, S. A. Egbetade. (2017). Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. American Journal of Applied and Industrial Chemistry, 1(1), 5-9. https://doi.org/10.11648/j.ajaic.20170101.12

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ACS Style

M. O. Ibrahim; S. A. Egbetade. Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. Am. J. Appl. Ind. Chem. 2017, 1(1), 5-9. doi: 10.11648/j.ajaic.20170101.12

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AMA Style

M. O. Ibrahim, S. A. Egbetade. Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection. Am J Appl Ind Chem. 2017;1(1):5-9. doi: 10.11648/j.ajaic.20170101.12

Copy | Download

@article{10.11648/j.ajaic.20170101.12,
  author = {M. O. Ibrahim and S. A. Egbetade},
  title = {Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection},
  journal = {American Journal of Applied and Industrial Chemistry},
  volume = {1},
  number = {1},
  pages = {5-9},
  doi = {10.11648/j.ajaic.20170101.12},
  url = {https://doi.org/10.11648/j.ajaic.20170101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaic.20170101.12},
  abstract = {In this paper, we extend the model of Blower et al. [1] by incorporating certain infection terms such as vaccinated individuals, treatment rate, waning rate and efficacy rate.A bifurcation analysis is performed on the vaccination model by applying a bifurcation method based on the use of center manifold theory.We determine threshold values and derive sufficient conditions for both forward and backward bifurcations.Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. The obtained results show the possibility of occurrence of forward and backward bifurcations even when the basic reproduction number is less than one so that it is now possible for the disease to exist. These results suggest the need for more study on the qualitative biological mechanisms responsible for backward bifurcation.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection
AU  - M. O. Ibrahim
AU  - S. A. Egbetade
Y1  - 2017/04/01
PY  - 2017
N1  - https://doi.org/10.11648/j.ajaic.20170101.12
DO  - 10.11648/j.ajaic.20170101.12
T2  - American Journal of Applied and Industrial Chemistry
JF  - American Journal of Applied and Industrial Chemistry
JO  - American Journal of Applied and Industrial Chemistry
SP  - 5
EP  - 9
PB  - Science Publishing Group
SN  - 2994-7294
UR  - https://doi.org/10.11648/j.ajaic.20170101.12
AB  - In this paper, we extend the model of Blower et al. [1] by incorporating certain infection terms such as vaccinated individuals, treatment rate, waning rate and efficacy rate.A bifurcation analysis is performed on the vaccination model by applying a bifurcation method based on the use of center manifold theory.We determine threshold values and derive sufficient conditions for both forward and backward bifurcations.Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. The obtained results show the possibility of occurrence of forward and backward bifurcations even when the basic reproduction number is less than one so that it is now possible for the disease to exist. These results suggest the need for more study on the qualitative biological mechanisms responsible for backward bifurcation.
VL  - 1
IS  - 1
ER  -

Copy | Download