Skip to main content
SpringerLink
Log in

ANALYSIS AND OPTIMAL CONTROL OF A VACCINATED PANDEMIC COVID-19 MODEL

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this article, we propose a modified SVIRS model that describes the evolution of the Covid-19 on a human population during a vaccination strategy. After describing our proposed model, we study the local and the global stability analysis of the model using Routh-Hurwitz criteria and by constructing the Lyapunov functions. In order to formulate an optimal control problem, we study the sensitivity analysis of the model parameters to determine the model robustness to parameter values, that is, to help us know the parameters that have a high impact on the reproduction number \(\mathscr {R}_{0}\) and to propose some appropriate control. The existence of the optimal control is investigated, and a characterization of the optimal control is given using the Pontryagin’s maximum principle. By this, we propose three control strategies that help the public health officials to implement programs to achieve collective immunity and thus come closer to reduce the spread of the SARS-CoV-2 virus. These strategies are as follows: awareness and prevention, administrative measures and awareness campaigns about the importance of vaccination and also the prevention measures, and finally, strengthening internal measures by controlling the proportion of immigrants coming to the country. In the end, some numerical simulations are performed to illustrate the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Notes

  1. https://www.hopkinsmedicine.org/health/conditions-and-diseases/coronavirus/breakthrough-infections-coronavirus-after-vaccination

  2. https://www.who.int/groups/strategic-advisory-group-of-experts-on-immunization/Covid-19-materials

  3. https://www.who.int/groups/strategic-advisory-group-of-experts-on-immunization/Covid-19-materials

References

  1. Hui, D.S., Azhar, E.I., Madani, T.A., Ntoumi, F., Kock, R., Dar, O., Ippolito, G., Mchugh, T.D., Memish, Z.A., Drosten, C., Zumla, A., Petersen, E.: The continuing 2019-ncov epidemic threat of novel coronaviruses to global health - the latest 2019 novel coronavirus outbreak in wuhan, china. International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases 91, 264–266 (2020)

    CAS  PubMed  Google Scholar 

  2. Wang, L., Zhou, Y., He, J., Zhu, B., Wang, F., Tang, L., Kleinsasser, M., Barker, D., Eisenberg, M.C., Song, P.X.: An epidemiological forecast model and software assessing interventions on the covid-19 epidemic in china. Journal of Data Science 18(3), 409–432 (2020)

    Google Scholar 

  3. Grifoni, A., Weiskopf, D., Ramirez, S.I., Mateus, J., Dan, J.M., Moderbacher, C.R., Rawlings, S.A., Sutherland, A., Premkumar, L., Jadi, R.S., et al.: Targets of t cell responses to sars-cov-2 coronavirus in humans with covid-19 disease and unexposed individuals. Cell 181(7), 1489–1501 (2020)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. WHO: Report of the who-china joint mission on coronavirus disease 2019 (covid-19). Encyclopedia Britannica. (2022)

  5. Rogers, K.: Covid-19 vaccine. World Health Organization (2020)

  6. Seto, W., Tsang, D., Yung, R., Ching, T., Ng, T., Ho, M., Ho, L., Peiris, J.: Effectiveness of precautions against droplets and contact in prevention of nosocomial transmission of severe acute respiratory syndrome (sars). The lancet 361(9368), 1519–1520 (2003)

    Article  CAS  Google Scholar 

  7. Al-Jasser, F.S., Nouh, R.M., Youssef, R.M.: Epidemiology and predictors of survival of mers-cov infections in riyadh region, 2014–2015. Journal of infection and public health 12(2), 171–177 (2019)

    Article  PubMed  Google Scholar 

  8. Thompson, W.W., Weintraub, E., Dhankhar, P., Cheng, P.-Y., Brammer, L., Meltzer, M.I., Bresee, J.S., Shay, D.K.: Estimates of us influenza-associated deaths made using four different methods. Influenza and other respiratory viruses 3(1), 37–49 (2009)

    Article  PubMed  PubMed Central  Google Scholar 

  9. Organization, W.H., et al.: Are the ebola outbreaks in nigeria and senegal over. http://www.who.int/mediacentre/news/ebola/14-october-2014/en (2014)

  10. Walensky, R.P., Del Rio, C.: From mitigation to containment of the covid-19 pandemic: putting the sars-cov-2 genie back in the bottle. Jama 323(19), 1889–1890 (2020)

    Article  CAS  PubMed  Google Scholar 

  11. Walker, P.G., Whittaker, C., Watson, O.J., Baguelin, M., Winskill, P., Hamlet, A., Djafaara, B.A., Cucunubá, Z., Olivera Mesa, D., Green, W., et al.: The impact of covid-19 and strategies for mitigation and suppression in low-and middle-income countries. Science 369(6502), 413–422 (2020)

    Article  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  12. Rabady, S.: No one is safe unless everyone is safe. Zeitschrift für Allgemeinmedizin 97(10), 385–385 (2021)

    Article  Google Scholar 

  13. Rogers, K.: World health organization: Advice for public. WHO Int (2020)

  14. Wang, X., Washington, D., Weber, G.F.: Complex systems analysis informs on the spread of covid-19. Epidemiologic Methods 10(s1), 20210019 (2021)

    Article  Google Scholar 

  15. Anderson, R.M., May, R.M.: Infectious diseases of humans: dynamics and control. Oxford university press (1991)

  16. Lin, Q., Zhao, S., Musa, S., Gao, D., Lou, Y., Yang, S., Musa, S., Wang, M., Cai, Y., Wang, W., et al.: A conceptual model of the outbreak of novel coronavirus (2019-ncov) in wuhan, china, with human reaction and holiday effects. Int. J. Infect. Dis. 93, 211–216 (2020)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  17. Peng, L., Yang, W., Zhang, D., Zhuge, C., Hong, L.: Epidemic analysis of covid-19 in china by dynamical modeling. arXiv preprint arXiv:2002.06563 (2020)

  18. López, L., Rodo, X.: A modified seir model to predict the covid-19 outbreak in spain and italy: simulating control scenarios and multi-scale epidemics. Results in Physics 21, 103746 (2021)

    Article  PubMed  Google Scholar 

  19. Zuo, C., Zhu, F., Ling, Y.: Analyzing covid-19 vaccination behavior using an seirm/v epidemic model with awareness decay. Frontiers in Public Health 10, 817749 (2022)

    Article  PubMed  PubMed Central  Google Scholar 

  20. Li, Y., Ge, L., Zhou, Y., Cao, X., Zheng, J.: Toward the impact of non-pharmaceutical interventions and vaccination on the covid-19 pandemic with time-dependent seir model. Frontiers in Artificial Intelligence 4, 648579 (2021)

    Article  PubMed  PubMed Central  Google Scholar 

  21. Yang, C., Yang, Y., Li, Y.: Assessing vaccination priorities for different ages and age-specific vaccination strategies of covid-19 using an seir modelling approach. Plos one 16(12), 0261236 (2021)

    Article  Google Scholar 

  22. Kar, T.K., Batabyal, A.: Stability analysis and optimal control of an sir epidemic model with vaccination. Biosystems 104(2-3), 127–135 (2011)

    Article  CAS  PubMed  Google Scholar 

  23. Bakare, E.A., Nwagwo, A., Danso-Addo, E.: Optimal control analysis of an sir epidemic model with constant recruitment. International Journal of Applied Mathematics Research 3(3), 273 (2014)

    Article  Google Scholar 

  24. Laarabi, H., Abta, A., Hattaf, K.: Optimal controliwliw of a delayed sirs epidemic model with vaccination and treatment. Acta biotheoretica 63(2), 87–97 (2015)

    Article  PubMed  Google Scholar 

  25. Samsuzzoha, M., Singh, M., Lucy, D.: Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. Applied Mathematical Modelling 37(3), 903–915 (2013)

    Article  MathSciNet  Google Scholar 

  26. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.: Sensitivity analysis in a dengue epidemiological model. In: Conference Papers in Mathematics, vol. 2013, pp. 1–7 (2013). Hindawi Limited

  27. Mahata, A., Paul, S., Mukherjee, S., Das, M., Roy, B.: Dynamics of caputo fractional order seirv epidemic model with optimal control and stability analysis. International Journal of Applied and Computational Mathematics 8(1), 28 (2022)

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  28. Kriss, J.L., Reynolds, L.E., Wang, A., Stokley, S., Cole, M.M., Harris, L.Q., Shaw, L.K., Black, C.L., Singleton, J.A., Fitter, D.L., et al.: Covid-19 vaccine second-dose completion and interval between first and second doses among vaccinated personsunited states, december 14, 2020- february 14, 2021. Morbidity and Mortality Weekly Report 70(11), 389 (2021)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Roser, M., Ritchie, H., Ortiz-Ospina, E., Hasell, J.: Coronavirus disease (covid-19)–statistics and research. Our World in data 4, 1–45 (2020)

  30. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences 180(1-2), 29–48 (2002)

    Article  MathSciNet  PubMed  Google Scholar 

  31. Ye, H., Gao, J., Ding, Y.: A generalized gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications 328(2), 1075–1081 (2007). https://doi.org/10.1016/j.jmaa.2006.05.061

    Article  MathSciNet  Google Scholar 

  32. Roskilly, T., Mikalsen, R.: Closed-loop stability. Marine Systems Identification, Modelling and Control, 97–122 (2015)

  33. Makinde, O.D., Okosun, K.O.: Impact of chemo-therapy on optimal control of malaria disease with infected immigrants. BioSystems 104(1), 32–41 (2011)

    Article  CAS  PubMed  Google Scholar 

  34. Chitnis, N., Cushing, J.M., Hyman, J.: Bifurcation analysis of a mathematical model for malaria transmission. SIAM Journal on Applied Mathematics 67(1), 24–45 (2006)

    Article  MathSciNet  Google Scholar 

  35. Bather, J.: Deterministic and stochastic optimal control. Wiley Online Library (1976)

  36. Lukes, D.L.: Differential equations: classical to controlled (1982)

  37. Pontryagin, L.S.: Mathematical theory of optimal processes. CRC press (1987)

Download references

Author information

Author notes

  1. Omar Balatif and Omar Kebiri contributed equally to this work.

Authors and Affiliations

  1. LARGESS Laboratory, Chouaïb Doukkali University, El Jadida, Morocco

    Sidi Mohamed Lalaoui Ben Cherif

  2. Laboratory of Fundamental Mathematics and their Applications, Department of Mathematics Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco

    Omar Balatif

  3. Department of Stochastics and its Applications, BTU Cottbus-Senftenberg, Konrad-Wachsmann-Allee 1, Cottbus, 03046, Brandenburg, Germany

    Omar Kebiri

Authors
  1. Sidi Mohamed Lalaoui Ben Cherif
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Balatif
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Omar Kebiri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sidi Mohamed Lalaoui Ben Cherif.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lalaoui Ben Cherif, S.M., Balatif, O. & Kebiri, O. ANALYSIS AND OPTIMAL CONTROL OF A VACCINATED PANDEMIC COVID-19 MODEL. J Math Sci (2024). https://doi.org/10.1007/s10958-024-06992-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10958-024-06992-7

Keywords

Search

Navigation