Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination

Stephen Edward; Dmitry Kuznetsov; Silas Mirau

doi:doi:10.11648/j.acm.20140304.16

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Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination

Stephen Edward, Dmitry Kuznetsov, Silas Mirau

Published in Applied and Computational Mathematics (Volume 3, Issue 4)

Received: 30 July 2014 Accepted: 8 August 2014 Published: 20 August 2014

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Abstract

In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e<1 and unstable when R_e>1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.

Published in	Applied and Computational Mathematics (Volume 3, Issue 4)
DOI	10.11648/j.acm.20140304.16
Page(s)	150-162
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), 2014. Published by Science Publishing Group

Keywords

Modeling, Sensitivity, Treatment, Vaccination, Epidemiology

References

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APA Style

Stephen Edward, Dmitry Kuznetsov, Silas Mirau. (2014). Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Applied and Computational Mathematics, 3(4), 150-162. https://doi.org/10.11648/j.acm.20140304.16

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ACS Style

Stephen Edward; Dmitry Kuznetsov; Silas Mirau. Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Appl. Comput. Math. 2014, 3(4), 150-162. doi: 10.11648/j.acm.20140304.16

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AMA Style

Stephen Edward, Dmitry Kuznetsov, Silas Mirau. Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Appl Comput Math. 2014;3(4):150-162. doi: 10.11648/j.acm.20140304.16

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@article{10.11648/j.acm.20140304.16,
  author = {Stephen Edward and Dmitry Kuznetsov and Silas Mirau},
  title = {Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {150-162},
  doi = {10.11648/j.acm.20140304.16},
  url = {https://doi.org/10.11648/j.acm.20140304.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.16},
  abstract = {In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.},
 year = {2014}
}

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TY  - JOUR
T1  - Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination
AU  - Stephen Edward
AU  - Dmitry Kuznetsov
AU  - Silas Mirau
Y1  - 2014/08/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.16
DO  - 10.11648/j.acm.20140304.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 150
EP  - 162
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.16
AB  - In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.
VL  - 3
IS  - 4
ER  -

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Author Information

Stephen Edward

School of CoCSE, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania
Dmitry Kuznetsov

School of CoCSE, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania
Silas Mirau

School of CoCSE, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania

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Plain Text BibTeX RIS

APA Style

Stephen Edward, Dmitry Kuznetsov, Silas Mirau. (2014). Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Applied and Computational Mathematics, 3(4), 150-162. https://doi.org/10.11648/j.acm.20140304.16

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ACS Style

Stephen Edward; Dmitry Kuznetsov; Silas Mirau. Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Appl. Comput. Math. 2014, 3(4), 150-162. doi: 10.11648/j.acm.20140304.16

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AMA Style

Stephen Edward, Dmitry Kuznetsov, Silas Mirau. Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Appl Comput Math. 2014;3(4):150-162. doi: 10.11648/j.acm.20140304.16

Copy | Download

@article{10.11648/j.acm.20140304.16,
  author = {Stephen Edward and Dmitry Kuznetsov and Silas Mirau},
  title = {Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {150-162},
  doi = {10.11648/j.acm.20140304.16},
  url = {https://doi.org/10.11648/j.acm.20140304.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.16},
  abstract = {In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination
AU  - Stephen Edward
AU  - Dmitry Kuznetsov
AU  - Silas Mirau
Y1  - 2014/08/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.16
DO  - 10.11648/j.acm.20140304.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 150
EP  - 162
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.16
AB  - In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.
VL  - 3
IS  - 4
ER  -

Copy | Download