Research Article | | Peer-Reviewed

A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19

Received: 6 January 2024     Accepted: 30 January 2024     Published: 12 June 2024
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Abstract

In this paper a mathematical model that investigates how vaccination affects the dynamics of COVID-19 was considered. More particularly the model takes into account the waning rate of immunity after vaccination as well as administration of booster vaccine. Posititivity and boundedness of solutions of the model were proved. The disease free equilibrium of the model was determined and by using the next generation matrix method both the basic and effective reproduction numbers of the model were determined. Further, from the effective reproduction number, the minimum critical value of individuals to be vaccinated for containment of the diseases was determined. It was found that the value is less for a perfect vaccine compared to an imperfect vaccine. Numerical simulation of the model was done to determine how the parameters of interest in the study (waning rate of immunity, vaccination rate, administration of booster vaccine and efficacy of the vaccine) affect the effective reproduction number. The results show that increasing the rates of vaccination, administering booster vaccine will decrease the effective reproduction number while an increase in waning rate of immunity increases the effective reproduction number. The disease persist in the population due to the declining of immunity after vaccination which increases the effective reproduction number.

Published in International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 2)
DOI 10.11648/j.ijssam.20240902.11
Page(s) 20-29
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

Vaccination, Reproduction Number, COVID-19, Mathematical Model, Re-infection, Waning of Immunity

References
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  • APA Style

    Geofrey, N. K., Harun, M., Samuel, M., Edward, N. (2024). A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19. International Journal of Systems Science and Applied Mathematics, 9(2), 20-29. https://doi.org/10.11648/j.ijssam.20240902.11

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    ACS Style

    Geofrey, N. K.; Harun, M.; Samuel, M.; Edward, N. A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19. Int. J. Syst. Sci. Appl. Math. 2024, 9(2), 20-29. doi: 10.11648/j.ijssam.20240902.11

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    AMA Style

    Geofrey NK, Harun M, Samuel M, Edward N. A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19. Int J Syst Sci Appl Math. 2024;9(2):20-29. doi: 10.11648/j.ijssam.20240902.11

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  • @article{10.11648/j.ijssam.20240902.11,
      author = {Nandwa Khayo Geofrey and Makwata Harun and Muthiga Samuel and Njuguna Edward},
      title = {A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {9},
      number = {2},
      pages = {20-29},
      doi = {10.11648/j.ijssam.20240902.11},
      url = {https://doi.org/10.11648/j.ijssam.20240902.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240902.11},
      abstract = {In this paper a mathematical model that investigates how vaccination affects the dynamics of COVID-19 was considered. More particularly the model takes into account the waning rate of immunity after vaccination as well as administration of booster vaccine. Posititivity and boundedness of solutions of the model were proved. The disease free equilibrium of the model was determined and by using the next generation matrix method both the basic and effective reproduction numbers of the model were determined. Further, from the effective reproduction number, the minimum critical value of individuals to be vaccinated for containment of the diseases was determined. It was found that the value is less for a perfect vaccine compared to an imperfect vaccine. Numerical simulation of the model was done to determine how the parameters of interest in the study (waning rate of immunity, vaccination rate, administration of booster vaccine and efficacy of the vaccine) affect the effective reproduction number. The results show that increasing the rates of vaccination, administering booster vaccine will decrease the effective reproduction number while an increase in waning rate of immunity increases the effective reproduction number. The disease persist in the population due to the declining of immunity after vaccination which increases the effective reproduction number.},
     year = {2024}
    }
    

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    T1  - A Mathematical Model to Investigate How Vaccination Affect the Reproduction Number for COVID-19
    AU  - Nandwa Khayo Geofrey
    AU  - Makwata Harun
    AU  - Muthiga Samuel
    AU  - Njuguna Edward
    Y1  - 2024/06/12
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    DO  - 10.11648/j.ijssam.20240902.11
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 20
    EP  - 29
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20240902.11
    AB  - In this paper a mathematical model that investigates how vaccination affects the dynamics of COVID-19 was considered. More particularly the model takes into account the waning rate of immunity after vaccination as well as administration of booster vaccine. Posititivity and boundedness of solutions of the model were proved. The disease free equilibrium of the model was determined and by using the next generation matrix method both the basic and effective reproduction numbers of the model were determined. Further, from the effective reproduction number, the minimum critical value of individuals to be vaccinated for containment of the diseases was determined. It was found that the value is less for a perfect vaccine compared to an imperfect vaccine. Numerical simulation of the model was done to determine how the parameters of interest in the study (waning rate of immunity, vaccination rate, administration of booster vaccine and efficacy of the vaccine) affect the effective reproduction number. The results show that increasing the rates of vaccination, administering booster vaccine will decrease the effective reproduction number while an increase in waning rate of immunity increases the effective reproduction number. The disease persist in the population due to the declining of immunity after vaccination which increases the effective reproduction number.
    VL  - 9
    IS  - 2
    ER  - 

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