Skip to main content

Advertisement

SpringerLink
Log in

Control of COVID-19 transmission dynamics, a game theoretical approach

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We analyze a mathematical model of COVID-19 transmission control, which includes the interactions among different groups of the population: vaccinated, susceptible, exposed, infectious, super-spreaders, hospitalized and fatality, based on a system of ordinary differential equations, which describes compartment model of a disease and its treatment. The aim of the model is to predict the development disease under different types of treatment during some fixed time period. We develop a game theoretic approach and a dual dynamic programming method to formulate optimal conditions of the treatment for an administration of a vaccine. Next, we calculate numerically an optimal treatment.

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

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this published article (and its supplementary information files).

References

  1. Alam, M., Kabir, K.M.A., Tanimoto, J.: Based on mathematical epidemiology and evolutionary game theory, which is more effective: quarantine or isolation policy? J. Stat. Mech. Theory Exp. Online at stacks.iop.org/JSTAT/2020/033502. https://doi.org/10.1088/1742-5468/ab75ea

  2. Bauch, C.T.: Imitation dynamics predict vaccinating behaviour. Proc. R. Soc. B 272, 1669–1675 (2005). https://doi.org/10.1098/rspb.2005.3153

    Article  Google Scholar 

  3. Bauch, C.T., Bhattacharyya, S.: Evolutionary game theory and social learning can determine how vaccine scares unfold. PLoS Comput. Biol. 8(4), e1002452 (2012). https://doi.org/10.1371/journal.pcbi.1002452

    Article  MathSciNet  Google Scholar 

  4. Bauch, C.T., Earn, D.J.D.: Vaccination and the theory of games. PNAS 101, 13391–13394 (2004). https://doi.org/10.1073/pnas.0403823101

    Article  MathSciNet  MATH  Google Scholar 

  5. Britto, C., Foy, B.H., Wahl, B., Mehta, K., Shet, A., Menon, G.I.: Comparing COVID-19 vaccine allocation strategies in India: a mathematical modelling study. Int. J. Infect. Dis. 103, 431–438 (2021)

    Article  Google Scholar 

  6. Faranda, D., Alberti, T., Arutkin, M., Lembo, V., Lucarini, V.: Interrupting vaccination policies can greatly spread SARS-CoV-2 and enhance mortality from COVID-19 disease: the AstraZeneca case for France and Italy. Chaos 31, 041105 (2021). https://doi.org/10.1063/5.0050887

    Article  MathSciNet  Google Scholar 

  7. Galewska, E., Nowakowski, A.: A dual dynamic programming for multidimensional elliptic optimal control problems. Numer. Funct. Anal. Optim. 27, 279–289 (2006)

    Article  MathSciNet  Google Scholar 

  8. Gumel, A.B., Iboi, E.A., Ngonghala, C.N., Elbasha, E.H.: A primer on using mathematics to understand COVID-19 dynamics: modeling, analysis and simulations. Infect. Dis. Model. 6, 148–168 (2021)

    Google Scholar 

  9. Jentsch, P.C., Anand, M., Bauch, C.T.: Prioritising COVID-19 vaccination in changing social and epidemiological landscapes: a mathematical modelling study. Lancet Infect. Dis. 21, 1097–1106 (2021). https://doi.org/10.1016/S1473-30992100057-8

    Article  Google Scholar 

  10. Kabir, K.M.A., Tanimoto, J.: Evolutionary game theory modelling to represent the behavioural dynamics of economic shutdowns and shield immunity in the COVID-19 pandemic. R. Soc. Open Sci. 7, 201095 (2020). https://doi.org/10.1098/rsos.201095

    Article  Google Scholar 

  11. Machowska, D., Nowakowski, A.: Competition in defensive and offensive advertising strategies in a segmented market. Eur. J. Control 53, 98–108 (2020)

    Article  MathSciNet  Google Scholar 

  12. Krishna, M.V.: Mathematical modelling on diffusion and control of COVID-19. Infect. Dis. Model. 5, 588–597 (2020)

  13. Ndaïrou, F., Area, I., Nieto, J.J., Torres, F.M.D.: Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Soliton Fractals 135, 109846 (2020)

    Article  MathSciNet  Google Scholar 

  14. Nowakowska, I., Nowakowski, A.: A dual dynamic programming for minimax optimal control problems governed by parabolic equation. Optimization 60, 347–363 (2011)

    Article  MathSciNet  Google Scholar 

  15. Nowakowski, A.: The dual dynamic programming. Proc. Am. Math. Soc. 116, 1089–1096 (1992)

    Article  MathSciNet  Google Scholar 

  16. Nowakowski, A.: Sufficient optimality conditions for Dirichlet boundary control of wave equations SIAM. J. Control Optim. 47, 92–110 (2008)

  17. Pedro, S.A., Ndjomatchoua, F.T., Jentsch, P., Tchuenche, J.M., Anand, M., Bauch, C.T.: Conditions for a second wave of COVID-19 due to interactions between disease dynamics and social processes. Front. Phys. 8, 574514 (2020). https://doi.org/10.3389/fphy.2020.574514

    Article  Google Scholar 

  18. Tanimoto, J.: Fundamentals of Evolutionary Game Theory and its Applications. Springer, Tokyo (2015)

    Book  Google Scholar 

  19. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–148 (2002)

    Article  MathSciNet  Google Scholar 

  20. Wang, H., Yamamoto, N.: Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona. Math. Biosci. Eng. 17(5), 4891–4904 (2020)

    Article  MathSciNet  Google Scholar 

  21. Zeb, A., Alzahrani, E., Erturk, V.S., Zaman, G.: Model, Mathematical, for Coronavirus Disease 2019 (COVID-19) Containing Isolation Class. Hindawi BioMed Research International. Article ID 3452402 (2020). https://doi.org/10.1155/2020/3452402

  22. Zhao, S., Stone, L., Gao, D., Musa, S.S., Chong, M.K.C., He, D., Wang, M.H.: Imitation dynamics in the mitigation of the novel coronavirus disease (COVID-19) outbreak in Wuhan China from 2019 to 2020. Ann. Transl. Med. 8(7), 448 (2020). https://doi.org/10.21037/atm.2020.03.168

    Article  Google Scholar 

Download references

Acknowledgements

The authors did not receive support from any organization for the submitted work.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Sciences, University of Lodz, Banacha 22, 90-238, Lodz, Poland

    R. Matusik & A. Nowakowski

Authors
  1. R. Matusik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Nowakowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Matusik.

Ethics declarations

Conflict of interest

The authors have not disclosed any conflict of interest.

Supplementary material

In this section, we share for the convenience of our readers complete files created in MATLAB R2015b, which can help realize own calculations. Using files placed at the address http://math.uni.lodz.pl/radmat/article/ the readers can find best strategies, solving system of differential equations (1.1)–(1.8) and solving also (2.1)–(2.2) and (2.7)–(2.8). To check equalities (3.1)–(3.2) in the verification theorem 1, the spreadsheet was used.

Additional information

Publisher's Note

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

Appendix

Appendix

See Tables 4 and 5.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matusik, R., Nowakowski, A. Control of COVID-19 transmission dynamics, a game theoretical approach. Nonlinear Dyn 110, 857–877 (2022). https://doi.org/10.1007/s11071-022-07654-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07654-6

Keywords

Mathematics Subject Classification

Search

Navigation