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A novel technique to solve the modified epidemiological model of computer viruses

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Abstract

The aim of this paper is to present a simple and accurate method to estimate the approximate solution of non-linear epidemiological model of computer viruses. For this reason, the variational iteration method (VIM) is applied. Also, in order to show the efficiency of presented method, we compare the numerical results with the differential transform method (DTM) and the homotopy analysis transform method (HATM). Several graphs of residual error functions for various iterations are demonstrated. By applying these graphs we show the results of VIM are accurate in comparison with the other methods.

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References

  1. Abbasbandy, S.: Extended Newton’s method for a system of nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 170, 648–656 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Bota, C., Caruntu, B.: Approximate analytical solutions of nonlinear differential equations using the Least Squares Homotopy Perturbation Method. J. Math. Anal. Appl. 448(1), 401–408 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, S.-H.: Convergence of variational iteration method applied to two-point diffusion problems. Appl. Math. Model. 40(15–16), 6805–6810 (2016)

    Article  MathSciNet  Google Scholar 

  4. Cohen, F.: Computer viruses: theory and experiments. Comput. Secur. 6, 22–35 (1987)

    Article  Google Scholar 

  5. Fariborzi Araghi, M.A., Noeiaghdam, S.: Fibonacci-regularization method for solving Cauchy integral equations of the first kind. Ain Shams Eng J. 8, 363–369 (2017)

    Article  Google Scholar 

  6. Fariborzi Araghi, M.A., Noeiaghdam, S.: A novel technique based on the homotopy analysis method to solve the first kind Cauchy integral equations arising in the theory of airfoils. J. Interpolat. Approx. Sci. Comput. 1, 1–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fariborzi Araghi, M.A., Noeiaghdam, S.: Homotopy analysis transform method for solving generalized Abel’s fuzzy integral equations of the first kind, IEEE (2016). https://doi.org/10.1109/CFIS.2015.7391645

  8. Fariborzi Araghi, M.A., Noeiaghdam, S.: Homotopy regularization method to solve the singular Volterra integral equations of the first kind. Jordan J. Math. Stat. 11(1), 1–12 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Fariborzi Araghi, M.A., Behzadi, Sh: Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method. Comput. Methods Appl. Math. 9(4), 1–11 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Fariborzi Araghi, M.A., Fallahzadeh, A.: Dynamical Control of accuracy using the stochastic arithmetic to estimate the solution of the ordinary differential equations via Adomian decomposition method. Asian J. Math. Comput. Res. 8(2), 128–135 (2016)

    Google Scholar 

  11. Ghorbani, A., Bakherad, M.: A variational iteration method for solving nonlinear Lane-Emden problems. New Astron. 54, 1–6 (2017)

    Article  Google Scholar 

  12. Han, X., Tan, Q.: Dynamical behavior of computer virus on Internet. Appl. Math. Comput. 217, 2520–2526 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Kephart, J.O., Hogg, T., Huberman, B.A.: Dynamics of computational ecosystems. Phys. Rev. A 40(1), 404–421 (1998)

    Article  MathSciNet  Google Scholar 

  14. Khuri, S.A., Sayfy, A.: Generalizing the variational iteration method for BVPs: proper setting of the correction functional. Appl. Math. Lett. 68, 68–75 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lee, M.K., Hosseini Fouladi, M., Namasivayam, S.N.: Natural frequencies of thin rectangular plates using homotopy-perturbation method. Appl. Math. Model. 50, 524–543 (2017)

    Article  MathSciNet  Google Scholar 

  16. Martin, O.: A modified variational iteration method for the analysis of viscoelastic beams. Appl. Math. Model. 40(17–18), 7988–7995 (2016)

    Article  MathSciNet  Google Scholar 

  17. Mikaeilvand, N., Noeiaghdam, S.: Mean value theorem for integrals and its application on numerically solving of Fredholm integral equation of second kind with Toeplitz plus Hankel Kernel. Int. J. Ind. Math. 6, 351–360 (2014)

    Google Scholar 

  18. Mishra, B.K., Jha, N.: Fixed period of temporary immunity after run of anti-malicious software on computer nodes. Appl. Math. Comput. 190, 1207–1212 (2007)

    MATH  Google Scholar 

  19. Noeiaghdam, S., Zarei, E., Barzegar Kelishami, H.: Homotopy analysis transform method for solving Abel’s integral equations of the first kind. Ain Shams Eng. J. 7, 483–495 (2016)

    Article  Google Scholar 

  20. Noeiaghdam, S.: Numerical solution of \(N\)-th order Fredholm integro-differential equations by integral mean value theorem method. Int. J. Pure Appl. Math. 99(3), 277–287 (2015)

    Article  Google Scholar 

  21. Piqueira, J.R.C., de Vasconcelos, A.A., Gabriel, C.E.C.J., Araujo, V.O.: Dynamic models for computer viruses. Comput. Secur. 27, 355–359 (2008)

    Article  Google Scholar 

  22. Piqueira, J.R.C., Araujo, V.O.: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213, 355–360 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Ren, J., Yang, X., Yang, L., Xu, Y., Yang, F.: A delayed computer virus propagation model and its dynamics. Chaos Soliton Fractal 45, 74–79 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ren, J., Yang, X., Zhu, Q., Yang, L., Zhang, C.: A novel computer virus model and its dynamics. Nonlinear Anal. Real 3, 376–384 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Suleman, M., Lu, D., He, J.H., Farooq, U., Noeiaghdam, S., Chandio, F.A.: Elzaki projected differential transform method for fractional order system of linear and nonlinear fractional partial differential equation. Fractals. https://doi.org/10.1142/S0218348X1850041X (in press) (2018)

  26. Wazwaz, A.M.: A comparison between the variational iteration method and Adomian decomposition method. J. Comput. Appl. Math. 207, 129–136 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wazwaz, A.M.: The variational iteration method for analytic treatment for linear and nonlinear ODEs. Appl. Math. Comput. 212, 120–134 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Wazwaz, A.M.: The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Appl. Math. Comput. 188, 485–491 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Wazwaz, A.M.: A study on linear and nonlinear Schrodinger equations by the variational iteration method. Chaos Solitons Fractals 37, 1136–1142 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wierman, J.C., Marchette, D.J.: Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction. Comput. Stat. Data Anal. 45, 3–23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yuan, H., Chen, G.: Network virus-epidemic model with the point-to-group information propagation. Appl. Math. Comput. 206, 357–367 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Zarei, E., Noeiaghdam, S.: Solving generalized Abel’s integral equations of the first and second kinds via Taylor-collocation method. arXiv:1804.08571

  33. Zhu, Q., Yang, X., Ren, J.: Modeling and analysis of the spread of computer virus. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5117–5124 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Noeiaghdam, S. A novel technique to solve the modified epidemiological model of computer viruses. SeMA 76, 97–108 (2019). https://doi.org/10.1007/s40324-018-0163-3

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