Skip to main content
SpringerLink
Log in

Discrete Gradient Methods for Solving SIRI Epidemic Model Numerically While Preserving First Integrals

  • Research Article-Physics
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

In this paper, we use discrete gradient methods for numerical solutions of mathematical models of epidemiology: a susceptible infected recovered infected (SIRI) model. The numerical results mimic the true and qualitatively correct behavior of the SIRI model. The first integrals (also known as the constant of motion) of the SIRI model obtained in literature using partial Lagrangian approach are well preserved by the discrete gradient method. The graphical results verify the advantages of the proposed schemes.

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.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bernoulli, D.: Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Histoire de l’Acad., Roy. Sci.(Paris) avec Mem pp. 1–45, (1760)

  2. Biazar, J.: Solution of the epidemic model by adomian decomposition method. Appl. Math. Comput. 173(2), 1101–1106 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Dahlby, M.; Owren, B.; Yaguchi, T.: Preserving multiple first integrals by discrete gradients. J. Phys. A: Math. Theor. 44(30), 305205 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Derrick, W.; Van Den Driessche, P.: A disease transmission model in a nonconstant population. J. Math. Biol. 31(5), 495–512 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Edwards, M.; Nucci, M.: Application of lie group analysis to a core group model for sexually transmitted diseases. J. Nonlinear Math. Phys. 13(2), 211–230 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Habib, Y.: Long-term behaviour of G-symplectic methods. Ph. D. thesis, university of Auckland, New Zealand., (2010)

  7. Hairer, E.; Lubich, C.; Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Irshad, W.; Habib, Y.; Farooq, M.U.: A complex lie-symmetry approach to calculate first integrals and their numerical preservation. Symmetry 11(1), 11 (2019)

    Article  MATH  Google Scholar 

  9. Kara, A.; Mahomed, F.: Noether-type symmetries and conservation laws via partial lagrangians. Nonlinear Dyn. 45(3–4), 367–383 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kara, A.; Mahomed, F.; Naeem, I.; Wafo Soh, C.: Partial noether operators and first integrals via partial lagrangians. Math. Methods Appl. Sci. 30(16), 2079–2089 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kermack, W.O.; McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc R Soc of london. Ser A, Contain. Pap. Math. Phys. Character 115(772), 700–721 (1927)

    MATH  Google Scholar 

  12. Korobeinikov, A.; Wake, G.C.: Lyapunov functions and global stability for sir, sirs, and sis epidemiological models. Appl. Math. Lett. 15(8), 955–960 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Levitas, V.I.; Roy, A.M.: Multiphase phase field theory for temperature- and stress-induced phase transformations. Phys. Rev. B 91, 174109 (2015)

    Article  Google Scholar 

  14. Levitas, V.I.; Roy, A.M.: Multiphase phase field theory for temperature-induced phase transformations: formulation and application to interfacial phases. Acta Mater. 105, 244–257 (2016)

    Article  Google Scholar 

  15. Levitas, V.I.; Roy, A.M.; Preston, D.L.: Multiple twinning and variant-variant transformations in martensite: phase-field approach. Phys. Rev. B 88(5), 054113 (2013)

    Article  Google Scholar 

  16. Mansfield, E.L.; Quispel, G.R.W.: On the construction of discrete gradients. Technical Report, University of Kent, Canterbury, UK (2009)

  17. McLachlan, R.I.; Quispel, G.; Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. Lond. Ser A: Math. Phys. Eng. Sci. 357(1754), 1021–1045 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Naz, R.; Mahomed, F.; Chaudhry, A.: A partial hamiltonian approach for current value hamiltonian systems. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3600–3610 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Naz, R.; Mahomed, K.; Naeem, I.: First integrals and exact solutions of the siri and tuberculosis models. Math. Methods Appl. Sci. 39(15), 4654–4666 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nucci, M.; Leach, P.: An integrable sis model. J. Math. Anal. Appl. 290(2), 506–518 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nucci, M.; Leach, P.: Lie integrable cases of the simplified multistrain/two-stream model for tuberculosis and dengue fever. J. Math. Anal. Appl. 333(1), 430–449 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nucci, M.C.: Using lie symmetries in epidemiology. Electron. J. Diff. Equ. Conf 12, 87–101 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Quispel, G.; Capel, H.: Solving odes numerically while preserving a first integral. Phys. Lett. A 218(3–6), 223–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rafei, M.; Daniali, H.; Domiri ganji., D.: Variational iteration method for solving the epidemic model and the prey and predator problem. Appl. Math. Comput. 186, 1701–1709 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Rafei, M.; Ganji, D.; Daniali, H.: Solution of the epidemic model by homotopy perturbation method. Appl. Math. Comput. 187(2), 1056–1062 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Sanz-Serna, J.M.: Runge-kutta schemes for Hamiltonian systems. BIT Numer. Math. 28(4), 877–883 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sanz-Serna, J.M.; Calvo, M.P.: Symplectic numerical methods for hamiltonian problems. Int. J. Mod. Phys. C 4(02), 385–392 (1993)

    Article  Google Scholar 

  28. Sanz-Serna, J.M.; Calvo, M.P.: Numerical Hamiltonian Problems. Chapman and Hall, London (1994)

    Book  MATH  Google Scholar 

Download references

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Author information

Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

    Farrukh Shehzad, Yousaf Habib & Atia Rashid

  2. Department of BS and H, College of E and ME, National University of Sciences and Technology (NUST), Islamabad, H-12, Pakistan

    Muhammad Umar Farooq

Authors
  1. Farrukh Shehzad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yousaf Habib
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Atia Rashid
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Muhammad Umar Farooq
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yousaf Habib.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shehzad, F., Habib, Y., Rashid, A. et al. Discrete Gradient Methods for Solving SIRI Epidemic Model Numerically While Preserving First Integrals. Arab J Sci Eng 46, 663–668 (2021). https://doi.org/10.1007/s13369-020-04726-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-020-04726-w

Keywords

Search

Navigation