A Fractional Mixing Propagation Model of Computer Viruses and Countermeasures Involving Mittag-Leffler Type Kernel

  • Sümeyra Uçar
  • Necati Özdemir
  • Zakia HammouchEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1111)


Countermeasures are recognized as a remarkable effort to comprehend the computer virus problem and estimate its forthcoming actions. Countermeasure-Competing (CMC) strategy is a conception comprising viruses and countermeasures. The main point of this paper is to probe a mixing propagation model of computer viruses and countermeasures in the light of the newly fractional derivative introduced by Atangana and Baleanu. The existence and uniqueness of solutions for this fractionalized model is discussed by taking the fixed point theory into consideration.. The efficacious belongings of this fractional model are exhibited theoretically, confirmed by numerical graphics.


  1. 1.
    Murray, W.H.: The application of epidemiology to computer viruses. Comput. Secur. 7(2), 130–150 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mishra, B.K., Saini, D.K.: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 188(2), 1476–1482 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Toutonji, O.A., Yoo, S.M., Park, M.: Stability analysis of VEISV propagation modeling for network worm attack. Appl. Math. Model. 36(6), 2751–2761 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Yuan, H., Chen, G., Wu, J., Xiong, H.: Towards controlling virus propagation in information systems with point-to-group information sharing. Decis. Support Syst. 48(1), 57–68 (2009)CrossRefGoogle Scholar
  5. 5.
    Piqueira, J.R.C., Araujo, V.O.: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213(2), 355–360 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Han, X., Tan, Q.: Dynamical behavior of computer virus on Internet. Appl. Math. Comput. 217(6), 2520–2526 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ren, J., Yang, X., Yang, L.X., Xu, Y., Yang, F.: A delayed computer virus propagation model and its dynamics. Chaos, Solitons Fractals 45(1), 74–79 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Zhu, Q., Yang, X., Yang, L.X., Zhang, C.: Optimal control of computer virus under a delayed model. Appl. Math. Comput. 218(23), 11613–11619 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, L., Carley, K.M.: The impact of countermeasure propagation on the prevalence of computer viruses. IEEE Trans. Syst. Man Cybern. B Cybern. 34(2), 823–833 (2004)CrossRefGoogle Scholar
  10. 10.
    Baleanu, D., Guvenc, Z.B., Machado, J.A.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht (2010)zbMATHCrossRefGoogle Scholar
  11. 11.
    Atangana, A., Alkahtani, B.T.: Analysis of non-homogenous heat model with new trend of derivative with fractional order. Chaos, Solitons Fractals 89, 566–571 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Tyagi, S., Abbas, S., Hafayed, M.: Global Mittag-Leffler stability of complex valued fractional-order neural network with discrete and distributed delays. Rendiconti del Circolo Matematico di Palermo Series 2 65(3), 485–505 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Özdemir, N., Karadeniz, D., İskender, B.B.: Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373(2), 221–226 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Evirgen, F.: Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM. Int. J. Optim. Control Theor. Appl. (IJOCTA) 6(2), 75–83 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Avcı, D., Eroğlu, B.B., Özdemir, N.: Conformable heat equation on a radial symmetric plate. Therm. Sci. 21(2), 819–826 (2017)zbMATHCrossRefGoogle Scholar
  16. 16.
    Özdemir, N., Yavuz, M.: Numerical solution of fractional black-scholes equation by using the multivariate pade approximation. Acta Phys. Pol A 132, 1050–1053 (2016)CrossRefGoogle Scholar
  17. 17.
    Hammouch, Z., Mekkaoui, T.: Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex Intell. Syst. 4(4), 251–260 (2018)zbMATHCrossRefGoogle Scholar
  18. 18.
    Uçar, E., Özdemir, N., Altun, E.: Fractional order model of immune cells influenced by cancer cells. Math. Model. Nat. Phenom. 14(3), 308 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)CrossRefGoogle Scholar
  20. 20.
    Koca, I.: Analysis of rubella disease model with non-local and non-singular fractional derivatives. Int. J. Optim. Control Theor. Appl. (IJOCTA) 8(1), 17–25 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Avcı, D., Yetim, A.: Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain. J. Balı kesir Univ. Inst. Sci. Technol. 20(2), 382–395 (2018)Google Scholar
  22. 22.
    Gomez Aguilar, J.F.: Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Physica A 494, 52–75 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Baleanu, B., Fernandez, A.: On some new properties of fractional derivatives with Mittag-Leffler kernel. Commun. Nonlinear Sci. Numer. Simul. 59, 444–462 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yavuz, M., Özdemir, N., Baskonus, H.M.: Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel. Eur. Phys. J. Plus 133, 215 (2018)CrossRefGoogle Scholar
  25. 25.
    Morales-Delgadoa, V.F., Gomez-Aguilar, J.F., Taneco-Hernandez, M.A., Escobar-Jimenezc, R.F., Olivares-Peregrino, V.H.: Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel. J. Nonlinear Sci. Appl. 11(8), 994–1014 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Fernandez, A., Baleanu, D., Srivastava, H.M.: Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions. Commun. Nonlinear Sci. Numer. Simul. 67, 517–527 (2019)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Uçar, S., Uçar, E., Özdemir, N., Hammouch, Z.: Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative. Chaos, Solitons Fractals 118, 300–306 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Owolabi, K.M., Hammouch, Z.: Mathematical modeling and analysis of two-variable system with noninteger-order derivative. Chaos Interdisc. J. Nonlinear Sci. 29, 013145 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Owolabi, K.M., Hammouch, Z.: Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative. Physica A. 523, 1072–1090 (2019)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Jarad, F., Abdeljawad, T., Hammouch, Z.: On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative. Chaos, Solitons Fractals 117, 16–20 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhu, Q., Yang, X., Yang, L.X., Zhang, X.: A mixing propagation model of computer viruses and countermeasures. Nonlinear Dyn. 73(3), 1433–1441 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Baleanu, D., Jajarmi, A., Hajipour, M.: On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag-Leffler kernel. Nonlinear Dyn. 94(1), 397–414 (2018)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Sümeyra Uçar
    • 1
  • Necati Özdemir
    • 1
  • Zakia Hammouch
    • 2
    Email author
  1. 1.Department of Mathematics, Faculty of Arts and SciencesBalıkesir UniversityBalıkesirTurkey
  2. 2.Department of Mathematics, Faculty of Sciences ans TechniquesMoulay Ismail University of MeknesErrachidiaMorocco

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