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

Nandwa Khayo Geofrey; Makwata Harun; Muthiga Samuel; Njuguna Edward

doi:doi:10.11648/j.ijssam.20240902.11

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

Nandwa Khayo Geofrey^*

, Makwata Harun

, Muthiga Samuel, Njuguna Edward

Published in International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 2)

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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[11]	George, K., Isaac, M. W., Mary, W., Stanley, S. Mathematical modelling of COVID-19 Transmission in Kenya: A model with Reinfection Transmission mechanism. (2021), Hindawi Computational and Mathematical Methods in Medicine, 2021, 1-18, https://doi.org/10.1155/2021/5384481
[12]	Grantz, H. K., Lauer, A. S. and Qifang, B. The incubation period of corona virus disease 2019 (COVID- 19) from publicly reported cases estimation and application (2020)., Annals of Internal Medicine, 1-5, https://doi.org/10.7326/M20.0504
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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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TY  - JOUR
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
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240902.11
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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Author Information

Nandwa Khayo Geofrey

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

Contact Email

http://orcid.org/0009-0000-7604-530X
Makwata Harun

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

Contact Email

http://orcid.org/0000-0002-5768-1615
Muthiga Samuel

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

Contact Email
Njuguna Edward

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

Contact Email

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240902.11
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  -

Copy | Download