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

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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[6] Diekman,O., Heesterbeek, J.A.P. and Metz, J.A.P. (1990).On the definition and Computation of the basic reproduction ratio R_0 in the model of infectious disease in Heterogeneous populations. Journal of Mathematical Biology. 2(1):265-382.
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    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
  • School of CoCSE, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania

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

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

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