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

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Abstract

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.

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References

  1. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80, 11–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, Ravi P., O‘Regan, D., Hristova, S.: Stability of Caputo fractional differential equations by Lyapunov functions. Appl. Math. 60, 653–676 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aguila-Camacho, N., Duarte-Mermoud, A.M., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allen, L.J.S.: An Introduction to Mathematical Biology. Prentice Hall, Pearson (2007)

    Google Scholar 

  5. Almeida, R.: Analysis of a fractional SEIR model with treatment. Appl. Math. Lett. 84, 56–62 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atangana, A., Baleanu, D.: New fractional derivative with non-local and non-singular kernel. Thermal Sci. 20, 757–763 (2016)

    Article  Google Scholar 

  7. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2021)

    MATH  Google Scholar 

  8. Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  11. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Diff. Appl. 1, 73–85 (2015)

    Google Scholar 

  12. Chu, X., Xu, L., Hu, H.: Exponential quasi-synchronization of conformable fractional-order complex dynamical networks. Chaos Solitons Fractals 140, 110268 (2020)

    Article  MathSciNet  Google Scholar 

  13. Cross, G.W.: Three types of matrix stability. Linear Algebra Appl. 20, 253–263 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  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. 36, 295–304 (2020)

    Google Scholar 

  15. Dang, Q.A., Hoang, M.T.: Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model. Int. J. Dyn. Control 8, 772–778 (2020)

    Article  MathSciNet  Google Scholar 

  16. Dang, Q.A., Hoang, M.T.: Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses. J. Comput. Appl. Math. 374, 112753 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag, Berlin (2010)

    Book  MATH  Google Scholar 

  18. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, A.J., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fatima, U., Baleanu, D., Ahmed, N., Azam, S., Raza, A., Rafiq, M., Rehman, M.A.: Numerical study of computer virus reaction diffusion epidemic model. Comput. Mater. Continua 66, 3183–3194 (2021)

    Article  Google Scholar 

  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 rate. Appl. Math. Comput. 222, 265–274 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6, 16 (2018)

    Article  MATH  Google Scholar 

  23. He, D., Xu, L.: Exponential stability of impulsive fractional switched systems with time delays. IEEE Trans. Circuits Syst. II Exp. Briefs (2020). https://doi.org/10.1109/TCSII.2020.3037654

  24. Hoang, M.T.: On the global asymptotic stability of a predator-prey model with Crowley–Martin function and stage structure for prey. J. Appl. Math. Comput. 64, 765–780 (2020)

    Article  MathSciNet  Google Scholar 

  25. Hoang, M.T., Nagy, A.M.: Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes. Chaos Solitons Fractals 123, 24–34 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Hoboken (2002)

    MATH  Google Scholar 

  28. Kheiri, H., Jafari, M.: Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment. J. Computat. Appl. Math. 346, 323–339 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 30, 615–626 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. La Salle, J., Lefschetz, S.: Stability by Liapunov‘s direct method. Academic Press, New York (1961)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, C., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71, 621–633 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Matignon, D.: Stability result on fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl. 2, 963–968 (1996)

    Google Scholar 

  38. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor‘s formula. Appl. Math. Comput. 186, 286–293 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  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. 136, 169–178 (2017)

    Article  Google Scholar 

  41. 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 

  42. Ren, J., Xu, Y.: A compartmental model for computer virus propagation with kill signals. Phys. A 486, 446–454 (2017)

    Article  MathSciNet  Google Scholar 

  43. Singh, J., Kumar, D., Hammouch, Z., Atangana, A.: A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl. Math. Comput. 316, 504–515 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Hoboken (1991)

    MATH  Google Scholar 

  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 and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  MATH  Google Scholar 

  46. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Valerio, D., Trujillo, J.J., Rivero, M., Machado, J.A.T., Baleanu, D.: Fractional calculus: a survey of useful formulas. Eur. Phys. J. Special Top. 222, 1827–1846 (2013)

    Article  Google Scholar 

  48. Vargas-De-Leon, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang, F., Yang, Y.: Fractional order Barbalat‘s lemma and its applications in the stability of fractional order nonlinear systems. Math. Modell. Anal. 22, 503–513 (2017)

    Article  MathSciNet  Google Scholar 

  50. Xu, L., Chua, X., Hu, H.: Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses. Appl. Math. Lett. 99, 106000 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. Xu, L., Li, J., Ge, S.S.: Impulsive stabilization of fractional differential systems. ISA Trans. 70, 125–131 (2017)

    Article  Google Scholar 

  52. Xu, L., Chu, X., Hu, H.: Quasi-synchronization analysis for fractional-order delayed complex dynamical networks. Math. Comput. Simul. 185, 594–613 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yang, L.X., Yang, X.: A new epidemic model of computer viruses. Commun. Nonlinear Sci. Numer. Simul. 19, 1935–1944 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yang, Y., Xu, L.: Stability of a fractional order SEIR model with general incidence. Appl. Math. Lett. 105, 106303 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yang, X.J., Gao, F., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Special Top. 226, 3567–3575 (2017)

    Article  Google Scholar 

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Acknowledgements

We would like to thank the editor and anonymous referees for useful comments that led to a great improvement of the paper.

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  1. Department of Mathematics, FPT University, Hoa Lac Hi-Tech Park, Km29 Thang Long Blvd, Hanoi, Vietnam

    Manh Tuan Hoang

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Hoang, M.T. Lyapunov Functions for Investigating Stability Properties of a Fractional-Order Computer Virus Propagation Model. Qual. Theory Dyn. Syst. 20, 74 (2021). https://doi.org/10.1007/s12346-021-00516-3

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