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The dynamics of Zika virus with Caputo fractional derivative

1 Department of mathematics, City university of Science and Information Technology, Peshawar, KP, Pakistan
2 Department of mathematics, University of Peshawar, KP, Pakistan
3 Department of mathematics, Abdul wali khan university, Mardan, KP, Pakistan

Special Issues: New trends of numerical and analytical methods with application to real world models for instance RLC with new nonlocal operators

In the present paper, we investigate a fractional model in Caputo sense to explore the dynamics of the Zika virus. The basic results of the fractional Zika model are presented. The local and global stability analysis of the proposed model is obtained when the basic reproduction reproduction number is less or greater than 1. To show the global stability of the fractional Zika model, we use the Lyapunov function theory in fractional environment. Further, we simulate the fractional Zika model to present the graphical results for different values of fractional order and model parameters.
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Keywords Zika virus model; stability analysis; generalized mean value theorem; Lyapunov function; Caputo derivative; numerical results

Citation: Muhammad Altaf Khan, Saif Ullah, Muhammad Farhan. The dynamics of Zika virus with Caputo fractional derivative. AIMS Mathematics, 2019, 4(1): 134-146. doi: 10.3934/Math.2019.1.134

References

  • 1. Zika virus, World Health Organization. Available from: http://www.who.int/mediacentre/factsheets/zika/en/.
  • 2. G. Calvet, R. S. Aguiar, A. S. O. Melo, et al. Detection and sequencing of Zika virus from amniotic fluid of fetuses with microcephaly in Brazil: a case study, Lancet infect dis., 16 (2016), 653-660.    
  • 3. T. A. Perkins, A. S. Siraj, C. W. Ruktanonchai, et al. Model-based projections of Zika virus infections in childbearing women in the Americas, Nat. Microbiol., 1 (2016), 16126.
  • 4. A. J. Kucharski, S. Funk, R. M. M. Eggo, et al. Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013-14 French Polynesia outbreak, PLoS Neglect. Trop. D., 10 (2016), 38588.
  • 5. E. Bonyah and K. O. Okosun, Mathematical modeling of Zika virus, Asian Pacific Journal of Tropical Disease, 6 (2016), 637-679.
  • 6. E. Bonyah, M. A. Khan, K. O. Okosun, et al. A theoretical model for Zika virus transmission, Plos one, 12 (2017), 1-26.
  • 7. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Elsevier, 1998.
  • 8. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, 1993.
  • 9. A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.    
  • 10. M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-13.
  • 11. K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5 (2016), 1643.
  • 12. K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Soliton. Fract., 103 (2017), 544-554.    
  • 13. K. M. Owolabi, A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Soliton. Fract., 111 (2018), 119-127.    
  • 14. E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 116 (2018), 121-125.    
  • 15. E. Bas, R. Ozarslan, D. Baleanu, Comparative simulations for solutions of fractional Sturm-Liouville problems with non-singular operators, Adv. Differ. Equ-NY, 2018 (2018), 350.
  • 16. E. Bas, The Inverse Nodal problem for the fractional diffusion equation, Acta Sci-Technol, 37 (2015), 251-257.    
  • 17. E. Bas, F. Metin, Fractional singular Sturm-Liouville operator for Coulomb potential, Adv. Differ. Equ-NY, 2013 (2013), 300.
  • 18. S. Ullah, M. A. Khan and M. F. Farooq, A new fractional model for the dynamics of Hepatitis B virus using Caputo-Fabrizio derivative, European Physical Journal Plus, 133 (2018), 237.
  • 19. M. A. Khan, S. Ullah and M. F. Farooq, A new fractional model for tuberculosis with relapse via AtanganaBaleanu derivative, Chaos Soliton. Fract., 116 (2018), 227-238.
  • 20. M. A. Khan, S. Ullah and M. F. Farooq, A fractional model for the dynamics of TB virus, Chaos Soliton. Fract., 116 (2018), 63-71.    
  • 21. B. S. T. Alkahtani, A. Atangana, I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, The Journal of Nonlinear Sciences and Applications, 10 (2017), 3191-3200.    
  • 22. H. Elsaka and E. Ahmed, A fractional order network model for ZIKA, BioRxiv, 2016.
  • 23. D. Hadi, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynam., 67 (2012), 2433-2439.
  • 24. V. D. L. Cruz, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci., 24 (2015), 75-85.    
  • 25. Z. M. Odibat and N. T. Shawagfeh, Generalized Taylors formula, Appl. Math. Comput., 186 (2007), 286-293.
  • 26. W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709-726.
  • 27. H. A. Antosiewicz, J. K. Hale, Studies in Ordinary Differential Equations, In: Englewood Cliffs (N. J.) by Mathematical Association of America, 1977.
  • 28. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Bellman Prize in Mathematical Biosciences, 180 (2002), 29-48.    

 

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