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
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
 [1] Adams, A. P., Boneva, T. and Golin, C. R. (2020). The impact of the corona virus lockdown on mental health: evidence from the US, 142-148. [2] Ali, M and Pierre, A. Prediction of confinement effects on the number of COVID-19 outbreak in Algeria(2020). Mathematical Modelling of natural phenomena,) 3-13, https://doi.org/10.1051/mmnp [3] Bin, J., Tao, Y., Yuan, J. and Zhang, Y. The impact of social distancing and epicenter lockdown on the COVID-19 epidemic in mainland China: A data driven SEIQR model study(2020), MedRxiv, 1-12, https://doi.org/10.1101/2020.03.04.20031187 [4] Brauer, F., Carlos, C. C. Mathematical Models in population Biology and Epidemiology. (2012), Springer, 2nd edition, 345-405. [5] Centers for Disease Control and Prevention (2020). About COVID-19 [6] Centers for Disease Control and Preventions, (2020). Corona virus variant. https://www.cdc.gov/coronavirus/2019-ncov/index.html, accessed 15/04/2023. [7] Chiew, C. J., Lee, V. J. and Wilder-Smith, A. Can we contain the COVID-19 outbreak with the same measures as for sars? (2020). The lancet infectious diseases, 20(5): e102-e107. [8] Demographics (2021). https://www.worldeconomics.com/Demographics/life-Expectancy/Kenya.aspx., accessed 15/06/2023 [9] Edridge, A. W., Hoste, A. C., Kaczorowska, J. Corona virus protective immunity is short-lasting (2021). https://wwww.medrxiv.org.content/10.1101/2020.05.11.20086439v2 [10] Ferguson, N., Laydon, D., Nedjati, G.etal(2020). Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand Imperial College London, vol. 2020, no. 10, 77482 [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 [13] Kate, F. Mathematical Models of COVID-19(2021), Virtual Commons-Bridgewater State University, 2021. https://vc.bridgew.edu/honors-proj/470 [14] Matt, J. K and Pejman, R. Modelling Infectious Diseases in Humans and Animals(2008). Princeton University Press [15] Mitzi, N. and Prakash, N. (2022). How long does protective immunity against COVID-19 last after infection or vaccination? University of Carolina. https://www.google.com/amp/s/theconversation.com/amp/how-long-does-protective-immunity-against-COVID-19-last-after-infection-or-vaccination-two-immunologists-explain-177309, accessed 20/04/2023. [16] Murray, J. D. Mathematical Biology I. (1993), Springer Verlag, Berlin, 17, 315-330 [17] Pauline, D. Infectious Disease Modelling. (2017). Department of Mathematics and Statistics, University of Victoria, Victoria, BC. V8W 2Y2, Canada, 287-288. [18] WHO coronavirus (COVID-19) Dashboard (2023). https://covid19.who.int, accessed 22/07/2023 [19] Worldpopulationreview.com/countries/Kenya-population, accessed 15/06/2023 [20] Yongshi, Y., Fujun, P., Kai, G., Runsheng, W. The 2003 SARS pandemic and the 2020 novel coronavirus epidemic in China. (2020), Journal of Autoimmunity, 109 (2020) 102434, 1-16, https://doi.org/10.1016/j.jaut.2020.102434
• 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

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

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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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.},
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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.
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Author Information
• Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

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

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

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

• Sections