Skip to main content

Advertisement

SpringerLink
Log in

A fractional calculus based model for the simulation of an outbreak of dengue fever

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

Abstract

We propose a new mathematical model for the simulation of the dynamics of a dengue fever outbreak. Our model differs from the classical model in that it involves nonlinear differential equations of fractional, not integer, order. Using statistics from the 2009 outbreak of the disease in the Cape Verde islands, we demonstrate that our model is capable of providing numerical results that agree very well with the real data.

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

Similar content being viewed by others

References

  1. Agrawal, O.P., Defterli, O., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16, 1967–1976 (2010)

    Article  MathSciNet  Google Scholar 

  2. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012)

    MATH  Google Scholar 

  3. Caminade, C., Medlock, J.M., Ducheyne, E., McIntyre, K.M., Leach, S., Baylis, M., Morse, A.P.: Suitability of European climate for the Asian tiger mosquito Aedes albopictus: recent trends and future scenarios. J. R. Soc. Interface (2012). doi:10.1098/rsif.2012.0138

    Google Scholar 

  4. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Demirci, E., Unal, A., Özalp, N.: A fractional order SEIR model with density dependent death rate. Hacet. J. Math. Stat. 40, 287–295 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Diethelm, K.: Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution. J. Eur. Syst. Autom. 42, 665–676 (2008)

    Google Scholar 

  7. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  8. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4+ T-cells. Math. Comput. Model. 50, 386–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nishiura, H.: Mathematical and statistical analyses of the spread of dengue. In: Dengue Bulletin, vol. 30, pp. 51–67. World Health Organization, New Delhi (2006)

    Google Scholar 

  12. Pooseh, S., Rodrigues, H.S., Torres, D.F.M.: Fractional derivatives in dengue epidemics. In: Simos, T.E., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) Numerical Analysis and Applied Mathematics, ICNAAM, pp. 739–742. American Institute of Physics, Melville (2011)

    Google Scholar 

  13. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M., Zinober, A.: Dengue disease, basic reproduction number and control. Int. J. Comput. Math. 89, 334–346 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tenreiro Machado, J.A., Kiryakova, V., Mainardi, F.: A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 13, 329–334 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Tenreiro Machado, J.A., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. World Health Organization: Dengue—Guidelines for Diagnosis, Treatment, Prevention and Control. WHO, Geneva (2009). New edition. http://whqlibdoc.who.int/publications/2009/9789241547871_eng.pdf. Accessed 2 February 2012

Download references

Acknowledgement

I would like to thank Professor Delfim F. M. Torres for allowing me to access the statistical data from the 2009 Cape Verde outbreak that he had obtained from the government of Cape Verde.

Author information

Authors and Affiliations

  1. Institut Computational Mathematics, Technische Universität Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Kai Diethelm

Authors
  1. Kai Diethelm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kai Diethelm.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diethelm, K. A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn 71, 613–619 (2013). https://doi.org/10.1007/s11071-012-0475-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0475-2

Keywords

Search

Navigation