Mathematical Modelling and Applications

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Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals

Received: 25 February 2022    Accepted: 22 March 2022    Published: 31 March 2022
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Abstract

Center for Disease control, informs that it takes two weeks after one is fully vaccinated for the body to build protection (immunity) against the virus that causes COVID-19. Moreover, no vaccine is hundred percent effective and that includes the COVID-19 vaccines. This implies that one can still contact and spread the virus for some days after getting vaccinated. In this paper, we formulated a model for COVID-19 transmission dynamics amongst the vaccinated individuals using differential equations. We analyzed all the parameters that are responsible for the disease spread and showed the effect of other social control measures, like the use of face masks in the public, on the spread of the virus. Numerical values of these parameters were derived from some acknowledged literatures, some calculated with the data from other literatures and others judiciously estimated. The disease reproduction number R0 was obtained and found that the disease will only spread if its value exceeds one. Numerical simulation was carried out on the model, using MATLAB to show the dynamics in the different compartments and the effect of these other social control measures on the disease spread among the vaccinated individuals. The result showed that in the absence of other social control measures, almost all the vaccinated persons will be infected and will be able to infect others especially within few days of receiving the COVID-19 vaccine.

DOI 10.11648/j.mma.20220701.12
Published in Mathematical Modelling and Applications (Volume 7, Issue 1, March 2022)
Page(s) 26-32
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

Mathematical Modelling, COVD-19 Transmission, COVID-19 Vaccines, Numerical Simulation, Reproduction Number

References
[1] Caroline Tien (March 03, 2022), How Long Will COVID-19 Vaccine-Induced Immunity Last? https://www.verywellhealth.com/length-of-covid-19-vaccine-immunity-5094857
[2] Center for Disease control (Sept. 10, 2021); Monitoring incidence of COVID-19 cases, Hospitalizations, and Deaths, by Vaccination Statues-13 U.S. jurisdictions, April 4 - July 17, 2021. https://www.cdc.gov
[3] Center for Disease control and prevention (May 27, 2021); How vaccines work (understanding how COVID-19 work) https://www.cdc.gov
[4] Christian Ogaugwu, Hammed Mogaji, Euphemia Ogaugwu, Uchechukwu Nebo, Hillary Okoh, Stanley Agbo and Andrew Agbon, (2020); Effect of weather on COVID-19 transmission and mortality in Lagos, Nigeria. Sciectifica, Volume 2020, Article ID 2562641, 6 pages https://doi.org/10.1155/2020/2562641
[5] Daniel Estrin (2020); Highly vaccinated Isreal is seeing a dramatic surge in new COVID Cases. https://www.npr.org/sections/goatsandsoda/2021/08/20/1029628471/highly-vaccinated-israel-is-seeing-a-dramatic-surge-in-new-covid-cases-heres-why
[6] Diekmann O., Heesterbeek J. A. P., (2000). Mathematical Epidemiology of infectious disease, Wiley series in Mathematical and Computational Biology, Wiley, Chichester.
[7] Gumel A. B., Ibio, E. A., Ngonghala, C. N, (2021); A premier on using mathematics to understand COVID-19 dynamics: modelling, analysis and simulations. Infectious Disease Modelling, 6, 148-168.
[8] Ibio, E. A., Ngonghala, C. N, Gumel, A. B. (2020); Willanimpact vaccine curtail the COVID-19 pandemic in US? Infectious Disease Modeling, 5, 510-524.
[9] NewYork-Presbyterian (2021). COVID-19 Vaccines and Immunity: How long does it take for the vaccines to provide protection; https://healthmatters.nyp.org
[10] Ngoghala, C. N., Ibio, E., Eikenberry, S., Scotch M., Macintyre, C. R., Bonds, M. H (2020); Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus. Mathematical Biosciences 325, 108364.
[11] Nita H. Shah, Jyoti Gupta (2014); Modelling of HIV-TB Co-Infection Transmission Dynamics, American Journal of Epidemiology and Infectious Disease, Vol. 2, No 1, 1-7.
[12] Sheena Cruickshank (12 Jan. 2021), COVID-19 Immunity: how long does it last? https://www.gavi.org/vaccineswork/covid-19-immunity-how-long-does-it-last? gclid=Cj0KCQjwuMuRBhCJARIsAHXdnqPvn0kDljBWtoNOxoiSmvptkjIlG-tz-9cvDF8gUXQ2CC53FA-kStMaAhQjEALw_wcB
[13] Simon A. Rella, Yuliya A. Kulikova, Fyodor A. Kondrashov (30 July, 2021): Rate of SARS-Cov-2 transmission and vaccination impact the fate of vaccine-resistant strains. Scientific Reports 11, 15729. https://www.nature.com
[14] WHO (21 Jan. 22) COVID-19 Advice for the public: Getting vaccinated. https://www.who.int/emergencies/diseases/novel-coronavirus-2019/covid-19-vaccines/advice
[15] World Health Organisation (15 February, 2022) COVID-19 Weekly Epidemiological Update Edition 79, https://reliefweb.int/sites/reliefweb.int/files/resources/ 20220215_Weekly_Epi_Update_79.pdf
Cite This Article
  • APA Style

    Christopher Chukwuma Asogwa, Stephen Ekwueme Aniaku, Emmanuel Chukwudi Mbah. (2022). Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals. Mathematical Modelling and Applications, 7(1), 26-32. https://doi.org/10.11648/j.mma.20220701.12

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    ACS Style

    Christopher Chukwuma Asogwa; Stephen Ekwueme Aniaku; Emmanuel Chukwudi Mbah. Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals. Math. Model. Appl. 2022, 7(1), 26-32. doi: 10.11648/j.mma.20220701.12

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    AMA Style

    Christopher Chukwuma Asogwa, Stephen Ekwueme Aniaku, Emmanuel Chukwudi Mbah. Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals. Math Model Appl. 2022;7(1):26-32. doi: 10.11648/j.mma.20220701.12

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  • @article{10.11648/j.mma.20220701.12,
      author = {Christopher Chukwuma Asogwa and Stephen Ekwueme Aniaku and Emmanuel Chukwudi Mbah},
      title = {Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals},
      journal = {Mathematical Modelling and Applications},
      volume = {7},
      number = {1},
      pages = {26-32},
      doi = {10.11648/j.mma.20220701.12},
      url = {https://doi.org/10.11648/j.mma.20220701.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20220701.12},
      abstract = {Center for Disease control, informs that it takes two weeks after one is fully vaccinated for the body to build protection (immunity) against the virus that causes COVID-19. Moreover, no vaccine is hundred percent effective and that includes the COVID-19 vaccines. This implies that one can still contact and spread the virus for some days after getting vaccinated. In this paper, we formulated a model for COVID-19 transmission dynamics amongst the vaccinated individuals using differential equations. We analyzed all the parameters that are responsible for the disease spread and showed the effect of other social control measures, like the use of face masks in the public, on the spread of the virus. Numerical values of these parameters were derived from some acknowledged literatures, some calculated with the data from other literatures and others judiciously estimated. The disease reproduction number R0 was obtained and found that the disease will only spread if its value exceeds one. Numerical simulation was carried out on the model, using MATLAB to show the dynamics in the different compartments and the effect of these other social control measures on the disease spread among the vaccinated individuals. The result showed that in the absence of other social control measures, almost all the vaccinated persons will be infected and will be able to infect others especially within few days of receiving the COVID-19 vaccine.},
     year = {2022}
    }
    

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    T1  - Mathematical Study of the Optimal Control of COVID-19 Transmission Amongst the Vaccinated Individuals
    AU  - Christopher Chukwuma Asogwa
    AU  - Stephen Ekwueme Aniaku
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    Y1  - 2022/03/31
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    N1  - https://doi.org/10.11648/j.mma.20220701.12
    DO  - 10.11648/j.mma.20220701.12
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
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    UR  - https://doi.org/10.11648/j.mma.20220701.12
    AB  - Center for Disease control, informs that it takes two weeks after one is fully vaccinated for the body to build protection (immunity) against the virus that causes COVID-19. Moreover, no vaccine is hundred percent effective and that includes the COVID-19 vaccines. This implies that one can still contact and spread the virus for some days after getting vaccinated. In this paper, we formulated a model for COVID-19 transmission dynamics amongst the vaccinated individuals using differential equations. We analyzed all the parameters that are responsible for the disease spread and showed the effect of other social control measures, like the use of face masks in the public, on the spread of the virus. Numerical values of these parameters were derived from some acknowledged literatures, some calculated with the data from other literatures and others judiciously estimated. The disease reproduction number R0 was obtained and found that the disease will only spread if its value exceeds one. Numerical simulation was carried out on the model, using MATLAB to show the dynamics in the different compartments and the effect of these other social control measures on the disease spread among the vaccinated individuals. The result showed that in the absence of other social control measures, almost all the vaccinated persons will be infected and will be able to infect others especially within few days of receiving the COVID-19 vaccine.
    VL  - 7
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, University of Nigeria, Nsukka, Nigeria

  • Department of Mathematics, University of Nigeria, Nsukka, Nigeria

  • Department of Mathematics, University of Nigeria, Nsukka, Nigeria

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