Skip to main content
SpringerLink
Log in

Global Stability of a Caputo Fractional SIRS Model with General Incidence Rate

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

We introduce a fractional order SIRS model with non-linear incidence rate. Existence of a unique positive solution to the model is proved. Stability analysis of the disease free equilibrium and positive fixed points are investigated. Finally, a numerical example is presented.

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

Similar content being viewed by others

References

  1. Agarwal, R.P., Baleanu, D., Nieto, J.J., Torres, D.F.M., Zhou, Y.: A survey on fuzzy fractional differential and optimal control nonlocal evolution equations. J. Comput. Appl. Math. 339, 3–29 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Almeida, R.: What is the best fractional derivative to fit data? Appl. Anal. Discrete Math. 11(2), 358–368 (2017)

    Article  MathSciNet  Google Scholar 

  4. Almeida, R.: Analysis of a fractional SEIR model with treatment. Appl. Math. Lett. 84, 56–62 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Almeida, R., Brito da Cruz, A.M.C., Martins, N., Monteiro, M.T.T.: An epidemiological MSEIR model described by the Caputo fractional derivative. Int. J. Dyn. Control 7(2), 776–784 (2019)

    Article  MathSciNet  Google Scholar 

  6. Almeida, R., Pooseh, S., Torres, D.F.M.: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015)

    Book  MATH  Google Scholar 

  7. Almeida, R., Tavares, D., Torres, D.F.M.: The Variable-Order Fractional Calculus of Variations. SpringerBriefs in Applied Sciences and Technology, Springer, Cham (2019)

    Book  MATH  Google Scholar 

  8. Bayour, B., Torres, D.F.M.: Complex-valued fractional derivatives on time scales. In: Differential and Difference Equations with Applications, Springer Proceedings in Mathematics and Statistics, vol. 164, pp 79–87. Springer, Cham (2016)

  9. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Article  Google Scholar 

  10. Carvalho, A.R.M., Pinto, C.M.A.: Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection. Commun. Nonlinear Sci. Numer. Simul. 61, 104–126 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carvalho, A.R.M., Pinto, C.M.A.: Immune response in HIV epidemics for distinct transmission rates and for saturated CTL response, Math. Model. Nat. Phenom. 14(3), 13, Art. 307 (2019)

  12. Carvalho, A.R.M., Pinto, C.M.A., Baleanu, D.: HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load. Adv. Differ. Equ. 2018(2), 22 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8, 211–221 (1989)

    Article  Google Scholar 

  14. DeAngelis, D.L., Goldsten, R.A., O’Neill, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)

    Article  Google Scholar 

  15. Debbouche, A., Nieto, J.J., Torres, D.F.M.: Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations. J. Optim. Theory Appl. 174(1), 7–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Doungmo Goufo, E.F., Maritz, R., Munganga, J.: Some properties of the Kermack–McKendrick epidemic model with fractional derivative and nonlinear incidence. Adv. Differ. Equ. 2014(278), 9 (2014)

    MathSciNet  MATH  Google Scholar 

  18. El-Saka, H.A.A.: The fractional-order SIR and SIRS epidemic models with variable population size. Math. Sci. Lett. 2, 195–200 (2013)

    Article  Google Scholar 

  19. El-Shahed, M., Alsaedi, A.: The fractional SIRC model and influenza A. Math. Probl. Eng. 2011, 9, Art. ID 480378 (2011)

  20. Hattaf, K., Yousfi, N., Tridane, A.: Stability analysis of a virus dynamics model with general incidence rate and two delays. Appl. Math. Comput. 221, 514–521 (2013)

    MathSciNet  MATH  Google Scholar 

  21. He, J.-H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115–123 (2000)

    MathSciNet  MATH  Google Scholar 

  22. He, J.-H., Elagan, S.K., Li, Z.B.: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376(4), 257–259 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. He, J.-H., Hu, Y.: On fractals Space-time and fractional calculus. Thermal Sci. 20(3), 773–777 (2016)

    Article  Google Scholar 

  24. He, J.-H., Li, Z.B.: Converting fractional differential equations into partial differential equations. Thermal Sci. 16(2), 331–334 (2012)

    Article  Google Scholar 

  25. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co. Inc, River Edge (2000)

    Book  MATH  Google Scholar 

  26. Huo, J., Zhao, H., Zhu, L.: The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics, part I. Proc. R. Soc. A 115, 700–721 (1927)

    MATH  Google Scholar 

  28. Khosravian-Arab, H., Torres, D.F.M.: Uniform approximation of fractional derivatives and integrals with application to fractional differential equations. Nonlinear Stud. 20, 533–548 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Li, Z.-B., He, J.-H.: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 15(5), 970–973 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332(1), 709–726 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, H.-Y., He, J.-H., Li, Z.-B.: Fractional calculus for nanoscale flow and heat transfer. Int. J. Numer. Methods Heat Fluid Flow 24(6), 1227–1250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lotfi, E.M., Mahrouf, M., Maziane, M., Silva, C.J., Torres, D.F.M., Yousfi, N.: A minimal HIV-AIDS infection model with general incidence rate and application to Morocco data. Stat. Optim. Inf. Comput. 7(3), 588–603 (2019)

    Article  MathSciNet  Google Scholar 

  33. Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. SpringerBriefs in Applied Sciences and Technology, Springer, Cham (2015)

    Book  MATH  Google Scholar 

  34. Mateus, J.P., Rebelo, P., Rosa, S., Silva, C.M., Torres, D.F.M.: Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete Contin. Dyn. Syst. Ser. S 11(6), 1179–1199 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Odibat, Z.M., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7(1), 27–34 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Odibat, Z.M., Momani, S.: Approximate solutions for boundary value problems of time-fractional wave equation. Appl. Math. Comput. 181(1), 767–774 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Odibat, Z., Momani, S.: An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inf. 26, 15–27 (2008)

    MATH  Google Scholar 

  38. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Okyere, E., Oduro, F.T., Amponsah, S.K., Dontwi, I.K., Frempong, N.K.: Fractional order SIR model with constant population. Br. J. Math. Comput. Sci. 14(2), 1–12 (2016)

    Article  Google Scholar 

  40. Özalp, N., Demirci, E.: A fractional order SEIR model with vertical transmission. Math. Comput. Model. 54(1–2), 1–6 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  42. Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press Inc, San Diego (1999)

    Google Scholar 

  43. Rachah, A., Torres, D.F.M.: Analysis, simulation and optimal control of a SEIR model for Ebola virus with demographic effects, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 67(1), 179–197 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Razminia, K., Razminia, A., Torres, D.F.M.: Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure. Appl. Math. Comput. 257, 374–380 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Rosa, S., Torres, D.F.M.: Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection. Chaos Solitons Fractals 117, 142–149 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rostamy, D., Mottaghi, E.: Numerical solution and stability analysis of a nonlinear vaccination model with historical effects. J. Math. Stat. Conf. Proc. p. 17. https://doi.org/10.15672/HJMS.20174720333 (2017)

  47. Salati, A.B., Shamsi, M., Torres, D.F.M.: Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 67, 334–350 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  49. Sierociuk, D., Skovranek, T., Macias, M., Podlubny, I., Petras, I., Dzielinski, A., Ziubinski, P.: Diffusion process modeling by using fractional-order models. Appl. Math. Comput. 257, 2–11 (2015)

    Google Scholar 

  50. Stanislavsky, A.A.: Fractional oscillator. Phys. Rev. E 70, 051103 (2004)

    Article  Google Scholar 

  51. Vargas-De-León, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 75–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, X., He, Y., Wang, M.: Chaos control of a fractional order modified coupled dynamos system. Nonlinear Anal. 71(12), 6126–6134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, K.-L., Liu, S.Y.: He’s fractional derivative and its application for fractional Fornberg–Whitham equation. Thermal Sci. 21(5), 2049–2055 (2017)

    Article  Google Scholar 

  54. West, B.J., Turalska, M., Grigolini, P.: Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics. New J. Phys. 17(4), 045009 (2015)

    Article  MATH  Google Scholar 

  55. Wojtak, W., Silva, C.J., Torres, D.F.M.: Uniform asymptotic stability of a fractional tuberculosis model, Math. Model. Nat. Phenom. 13(1), 10, Art. 9 (2018)

  56. Wu, X.-E., Liang, Y.-S.: Relationship between fractal dimensions and fractional calculus. Nonlinear Sci. Lett. A 8(1), 77–89 (2017)

    MathSciNet  Google Scholar 

  57. Wu, X., Tang, L., Zhong, T.: An iteration algorithm for fractal dimensions of a self-similar set. Nonlinear Sci. Lett. A 8(1), 117–120 (2017)

    Google Scholar 

  58. Yang, X.-J., Machado, J.A.T., Baleanu, D.: Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(4), 1740006 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by Fundação para a Ciência e a Tecnologia (FCT) within the R&D unit Centro de Investigação e Desenvolvimento em Matemática e Aplicações (CIDMA), Project UIDB/04106/2020. The authors are very grateful to two anonymous reviewers, for several critical remarks and precious suggestions.

Author information

Authors and Affiliations

  1. AMNEA Group, Department of Mathematics, Faculty of Sciences and Techniques, Moulay Ismail University, B. P. 509, Errachidia, Morocco

    Moulay Rchid Sidi Ammi & Mostafa Tahiri

  2. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

Authors
  1. Moulay Rchid Sidi Ammi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mostafa Tahiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Delfim F. M. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delfim F. M. Torres.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sidi Ammi, M.R., Tahiri, M. & Torres, D.F.M. Global Stability of a Caputo Fractional SIRS Model with General Incidence Rate. Math.Comput.Sci. 15, 91–105 (2021). https://doi.org/10.1007/s11786-020-00467-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-020-00467-z

Keywords

Mathematics Subject Classification

Search

Navigation