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Lyapunov Functions for Investigating Stability Properties of a Fractional-Order Computer Virus Propagation Model

  • Hoang, Manh Tuan [1]
    1. [1] FPT University

      FPT University

      Vietnam

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
  • Idioma: inglés
  • DOI: 10.1007/s12346-021-00516-3
  • Enlaces
  • Resumen

    • The aim of this work is to formulate and study a fractional-order computer virus propagation model, which is derived from the Caputo fractional derivative and a recognized system of ordinary differential equations. Firstly, standard comparison results for fractional differential equations are used to establish the positivity and boundedness of solutions of the model. Secondly, we propose a simple and unified approach to investigate stability properties including the local and global asymptotic stability and uniform stability of the fractional-order model. This approach is based on the construction of appropriate Lyapunov functions in combination with the fractional order Barbalat’s lemma. Consequently, the stability properties and dynamics of the model are established rigorously. Lastly, a set of numerical examples is performed to support and illustrate the theoretical results. The examples show that the numerical results are consistent with theoretical ones.

  • Referencias bibliográficas
    • 1. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80, 11–27...
    • 2. Agarwal, Ravi P., O‘Regan, D., Hristova, S.: Stability of Caputo fractional differential equations by Lyapunov functions. Appl. Math. 60,...
    • 3. Aguila-Camacho, N., Duarte-Mermoud, A.M., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer....
    • 4. Allen, L.J.S.: An Introduction to Mathematical Biology. Prentice Hall, Pearson (2007)
    • 5. Almeida, R.: Analysis of a fractional SEIR model with treatment. Appl. Math. Lett. 84, 56–62 (2018)
    • 6. Atangana, A., Baleanu, D.: New fractional derivative with non-local and non-singular kernel. Thermal Sci. 20, 757–763 (2016)
    • 7. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2021)
    • 8. Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)
    • 9. Baleanu, D., Agarwal, R.V.: Fractional calculus in the sky. Adv. Diff. Eqn. 117,(2021)
    • 10. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13, 529–539 (1967)
    • 11. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Diff. Appl. 1, 73–85 (2015)
    • 12. Chu, X., Xu, L., Hu, H.: Exponential quasi-synchronization of conformable fractional-order complex dynamical networks. Chaos Solitons...
    • 13. Cross, G.W.: Three types of matrix stability. Linear Algebra Appl. 20, 253–263 (1978)
    • 14. Dang, Q.A., Hoang, M.T., Tran, D.H.: Global dynamics of a computer virus propagation model with feedback controls. J. Comput. Sci. Cyber....
    • 15. Dang, Q.A., Hoang, M.T.: Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model. Int. J. Dyn....
    • 16. Dang, Q.A., Hoang, M.T.: Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses. J....
    • 17. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo...
    • 18. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, A.J., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov...
    • 19. Fatima, U., Baleanu, D., Ahmed, N., Azam, S., Raza, A., Rafiq, M., Rehman, M.A.: Numerical study of computer virus reaction diffusion...
    • 20. Fernandez, A., Baleanu, D.: Classes of operators in fractional calculus: a case study. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7341
    • 21. Gan, C., Yang, X., Liu,W., Zhu, Q., Zhang, X.: An epidemic model of computer viruses with vaccination and generalized nonlinear incidence...
    • 22. Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6, 16 (2018)
    • 23. He, D., Xu, L.: Exponential stability of impulsive fractional switched systems with time delays. IEEE Trans. Circuits Syst. II Exp. Briefs...
    • 24. Hoang, M.T.: On the global asymptotic stability of a predator-prey model with Crowley–Martin function and stage structure for prey. J....
    • 25. Hoang, M.T., Nagy, A.M.: Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite...
    • 26. Hu, Z., Wang, H., Liao, F., Ma, W.: Stability analysis of a computer virus model in latent period. Chaos Solitons Fractals 75, 20–28 (2015)
    • 27. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Hoboken (2002)
    • 28. Kheiri, H., Jafari, M.: Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment. J. Computat....
    • 29. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull....
    • 30. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull....
    • 31. La Salle, J., Lefschetz, S.: Stability by Liapunov‘s direct method. Academic Press, New York (1961)
    • 32. Lyapunov, A. M.: The general problem of the stability of motion. Int. J. Control. Taylor & Francis (1992)
    • 33. Li, Y., Chen, Y., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)
    • 34. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler...
    • 35. Li, C., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71, 621–633 (2013)
    • 36. Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)
    • 37. Matignon, D.: Stability result on fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl....
    • 38. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor‘s formula. Appl. Math. Comput. 186, 286–293 (2007)
    • 39. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
    • 40. Parsaei, M.R., Javidan, R., Kargar, N.S., Nik, H.S.: On the global stability of an epidemic model of computer viruses. Theory Biosci....
    • 41. Piqueira, J.R.C., Araujo, V.O.: A modified epidemiological model for computer viruses. Appl. Math. Comput. 213, 355–360 (2009)
    • 42. Ren, J., Xu, Y.: A compartmental model for computer virus propagation with kill signals. Phys. A 486, 446–454 (2017)
    • 43. Singh, J., Kumar, D., Hammouch, Z., Atangana, A.: A fractional epidemiological model for computer viruses pertaining to a new fractional...
    • 44. Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Hoboken (1991)
    • 45. Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science...
    • 46. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission....
    • 47. Valerio, D., Trujillo, J.J., Rivero, M., Machado, J.A.T., Baleanu, D.: Fractional calculus: a survey of useful formulas. Eur. Phys. J....
    • 48. Vargas-De-Leon, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85...
    • 49. Wang, F., Yang, Y.: Fractional order Barbalat‘s lemma and its applications in the stability of fractional order nonlinear systems. Math....
    • 50. Xu, L., Chua, X., Hu, H.: Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses....
    • 51. Xu, L., Li, J., Ge, S.S.: Impulsive stabilization of fractional differential systems. ISA Trans. 70, 125–131 (2017)
    • 52. Xu, L., Chu, X., Hu, H.: Quasi-synchronization analysis for fractional-order delayed complex dynamical networks. Math. Comput. Simul....
    • 53. Yang, L.X., Yang, X., Zhu, Q., Wen, L.: A computer virus model with graded cure rates. Nonlinear Anal. Real World Appl. 144, 14–422 (2013)
    • 54. Yang, L.X., Yang, X., Wen, L., Liu, J.: A novel computer virus propagation model and its dynamics. Int. J. Comput. Math. 89, 2307–2314...
    • 55. Yang, L.X., Yang, X.: A new epidemic model of computer viruses. Commun. Nonlinear Sci. Numer. Simul. 19, 1935–1944 (2014)
    • 56. Yang, Y., Xu, L.: Stability of a fractional order SEIR model with general incidence. Appl. Math. Lett. 105, 106303 (2020)
    • 57. Yang, X.J., Gao, F., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular...
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