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A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection

Received: 16 October 2019     Accepted: 28 November 2019     Published: 19 February 2020
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Abstract

In this paper, the dynamics of SVEIR model with saturated incidence force of infection and saturated vaccination function for Streptococcus pneumonia (that is, model that monitors the temporal transmission dynamics of the disease in the presence of preventive vaccine) was formulated and analyzed. The basic reproduction number that determines disease extinction and disease survival was revealed. The existing threshold conditions of all kinds of the equilibrium points are obtained and proved to be locally asymptotic stable for disease-free equilibrium using linearization method and Lyapunov functional method for Endemic equilibrium. Qualitative Analysis of the model was obtained and the positive of solution obtained. It was revealed that the model is positively –invariant and attracting. Thus the region is positively invariant. Hence, it is sufficient to consider the dynamics of the model (1) in the given region. In this region, the model can be considered as been epidemiologically and mathematically well-posed. The governing model was normalized and also Adomian Decomposition method was used to compute an approximate solution of the non-linear system of differential equations governing the model. Maple was used in carrying out the simulations (numerical solutions) of the model. Graphical results were presented and discussed to illustrate the solution of the problem. The achieved results reveal that the disease will die out within the community if the vaccination coverage is above the critical vaccination proportion. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease.

Published in Mathematical Modelling and Applications (Volume 5, Issue 1)
DOI 10.11648/j.mma.20200501.13
Page(s) 16-38
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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), 2020. Published by Science Publishing Group

Keywords

Mathematical Model, SVEIR Model, Streptococcus Pneumonia, Saturated Incidence Force Ofinfection

References
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    Opara Chiekezi Zephaniah, Uche-Iwe Ruth Nwaugonma, Inyama Simeon Chioma, Omame Adrew. (2020). A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection. Mathematical Modelling and Applications, 5(1), 16-38. https://doi.org/10.11648/j.mma.20200501.13

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    Opara Chiekezi Zephaniah; Uche-Iwe Ruth Nwaugonma; Inyama Simeon Chioma; Omame Adrew. A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection. Math. Model. Appl. 2020, 5(1), 16-38. doi: 10.11648/j.mma.20200501.13

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    AMA Style

    Opara Chiekezi Zephaniah, Uche-Iwe Ruth Nwaugonma, Inyama Simeon Chioma, Omame Adrew. A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection. Math Model Appl. 2020;5(1):16-38. doi: 10.11648/j.mma.20200501.13

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  • @article{10.11648/j.mma.20200501.13,
      author = {Opara Chiekezi Zephaniah and Uche-Iwe Ruth Nwaugonma and Inyama Simeon Chioma and Omame Adrew},
      title = {A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection},
      journal = {Mathematical Modelling and Applications},
      volume = {5},
      number = {1},
      pages = {16-38},
      doi = {10.11648/j.mma.20200501.13},
      url = {https://doi.org/10.11648/j.mma.20200501.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200501.13},
      abstract = {In this paper, the dynamics of SVEIR model with saturated incidence force of infection and saturated vaccination function for Streptococcus pneumonia (that is, model that monitors the temporal transmission dynamics of the disease in the presence of preventive vaccine) was formulated and analyzed. The basic reproduction number that determines disease extinction and disease survival was revealed. The existing threshold conditions of all kinds of the equilibrium points are obtained and proved to be locally asymptotic stable for disease-free equilibrium using linearization method and Lyapunov functional method for Endemic equilibrium. Qualitative Analysis of the model was obtained and the positive of solution obtained. It was revealed that the model is positively –invariant and attracting. Thus the region is positively invariant. Hence, it is sufficient to consider the dynamics of the model (1) in the given region. In this region, the model can be considered as been epidemiologically and mathematically well-posed. The governing model was normalized and also Adomian Decomposition method was used to compute an approximate solution of the non-linear system of differential equations governing the model. Maple was used in carrying out the simulations (numerical solutions) of the model. Graphical results were presented and discussed to illustrate the solution of the problem. The achieved results reveal that the disease will die out within the community if the vaccination coverage is above the critical vaccination proportion. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection
    AU  - Opara Chiekezi Zephaniah
    AU  - Uche-Iwe Ruth Nwaugonma
    AU  - Inyama Simeon Chioma
    AU  - Omame Adrew
    Y1  - 2020/02/19
    PY  - 2020
    N1  - https://doi.org/10.11648/j.mma.20200501.13
    DO  - 10.11648/j.mma.20200501.13
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
    SP  - 16
    EP  - 38
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20200501.13
    AB  - In this paper, the dynamics of SVEIR model with saturated incidence force of infection and saturated vaccination function for Streptococcus pneumonia (that is, model that monitors the temporal transmission dynamics of the disease in the presence of preventive vaccine) was formulated and analyzed. The basic reproduction number that determines disease extinction and disease survival was revealed. The existing threshold conditions of all kinds of the equilibrium points are obtained and proved to be locally asymptotic stable for disease-free equilibrium using linearization method and Lyapunov functional method for Endemic equilibrium. Qualitative Analysis of the model was obtained and the positive of solution obtained. It was revealed that the model is positively –invariant and attracting. Thus the region is positively invariant. Hence, it is sufficient to consider the dynamics of the model (1) in the given region. In this region, the model can be considered as been epidemiologically and mathematically well-posed. The governing model was normalized and also Adomian Decomposition method was used to compute an approximate solution of the non-linear system of differential equations governing the model. Maple was used in carrying out the simulations (numerical solutions) of the model. Graphical results were presented and discussed to illustrate the solution of the problem. The achieved results reveal that the disease will die out within the community if the vaccination coverage is above the critical vaccination proportion. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease.
    VL  - 5
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

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