Dynamical Analysis of Fractional Order Model for Computer Virus Propagation with Kill Signals

References

[1] X. Han and Q. Tan, Dynamical behavior of computer virus on Internet, Appl. Math. Comput. 217 (2010), 2520–2526.10.1016/j.amc.2010.07.064Search in Google Scholar

[2] J. Kim, S. Radhakrishana and J. Jang, Cost optimization in SIS model of worm infection, ETRI J. 28 (2006), 692–695.10.4218/etrij.06.0206.0026Search in Google Scholar

[3] J. R. C. Piqueira and V. O. Araujo, A modified epidemiological model for computer viruses, Appl. Math. Comput. 213 (2009), 355–360.10.1016/j.amc.2009.03.023Search in Google Scholar

[4] L. Billings, W. M. Spears and I. B. Schwartz, A unified prediction of computer virus spread in connected networks, Phys. Lett. A 297 (2002), 261–266.10.1016/S0375-9601(02)00152-4Search in Google Scholar

[5] C. Gan, X. Yang, Q. Zhu, J. Jin and L. He, The spread of computer virus under the effect of external computers, Nonlinear Dyn. 73 (2013), 1615–1620.10.1007/s11071-013-0889-5Search in Google Scholar

[6] C. Gan, X. Yang, W. Liu and Q. Zhu, A propagation model of computer virus with nonlinear vaccination probability, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 92–100.10.1016/j.cnsns.2013.06.018Search in Google Scholar

[7] Y. Muroya, Y. Enatsu and H. Li, Global stability of a delayed SIRS computer virus propagation model, Int. J. Comput. Math. 91 (2013), 347–367.10.1080/00207160.2013.790534Search in Google Scholar

[8] B. K. Mishra and S. K. Pandey, Dynamic model of worms with vertical transmission in computer network, Appl. Math. Comput. 217 (2011), 8438–8446.10.1016/j.amc.2011.03.041Search in Google Scholar

[9] L. X. Yang, X. Yang, L. Wen and J. Liu, A novel computer virus propagation model and its dynamics, Int. J. Comput. Math. 89 (2012), 2307–2314.10.1080/00207160.2012.715388Search in Google Scholar

[10] L. X. Yang and X. Yang, The spread of computer viruses under the influence of removable storage devices, Appl. Math. Comput. 219 (2012), 3914–3922.10.1016/j.amc.2012.10.027Search in Google Scholar

[11] L. Feng, X. Liao, H. Li and Q. Han, Hopf bifurcation analysis of a delayed viral infection model in computer networks, Math. Comput. Modell. 56 (2012), 167–179.10.1016/j.mcm.2011.12.010Search in Google Scholar

[12] J. Ren, X. Yang, L.-X. Yang, Y. Xu and F. Yang, A delayed computer virus propagation model and its dynamics, Chaos Solitons Fractals 45 (2012), 74-79.10.1016/j.chaos.2011.10.003Search in Google Scholar

[13] Q. Zhu, X. Yang, L. X. Yang and C. Zhang, Optimal control of computer virus under a delayed model, Appl. Math. Comput. 218 (2012), 11613–11619.10.1016/j.amc.2012.04.092Search in Google Scholar

[14] B. K. Mishra and N. Jha, Fix period of temporary immunity after run of anti-malicious software on computer nodes, Appl. Math. Comput. 190 (2007), 1207–1212.10.1016/j.amc.2007.02.004Search in Google Scholar

[15] L. X. Yang, M. Draief and X. F. Yang, The optimal dynamics immunization under a controlled heterogeneous node-based SIRS model, Physica A. 450 (2016), 403–415.10.1016/j.physa.2016.01.026Search in Google Scholar

[16] L. X. Yang and X. F. Yang, The pulse treatment of computer viruses: a modeling study, Nonlinear Dyn. 76 (2014), 1379–1393.10.1007/s11071-013-1216-xSearch in Google Scholar

[17] Q. Y. Zhu, X. F. Yang and L. X. Yang, A mixing propagation model of computer viruses and countermeasures, Nonlinear Dyn. 73 (2013), 1433–1441.10.1007/s11071-013-0874-zSearch in Google Scholar

[18] J. O. Kephart and S. R. White, Measure and modeling computer virus prevalence, IEEE Computer Society Symposium Research in Security and Privacy (1993).10.1109/RISP.1993.287647Search in Google Scholar

[19] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujill, Models and numerical methods, World Scientific, Singapore, 2012.Search in Google Scholar

[20] L. Changpin, W. Yujiang and Y. Ruisong, Recent advances in applied nonlinear dynamics with numerical analysis, World Scientific, Singapore, 2013.Search in Google Scholar

[21] R. Gorenflo and F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer, New York, 223–276, 1997.10.1007/978-3-7091-2664-6_5Search in Google Scholar

[22] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, Amsterdam, 2006.Search in Google Scholar

[23] A. Dabiri, B. P. Moghaddam and J. A. T. Machado, Optimal variable-order fractional PID controllers for dynamical systems, J. Compsut. Appl. Math. 339 (2018), 40–48.10.1016/j.cam.2018.02.029Search in Google Scholar

[24] F. Evirgen and N. Ozdemir, A fractional order dynamical trajectory approach for optimization problem with HPM, Fractional Dyn. Control. (2012), 145–155.10.1007/978-1-4614-0457-6_12Search in Google Scholar

[25] I. Koca and N. Ozalp, Analysis of a fractional-order couple model with acceleration in feelings, Sci. World J. 2013 (2013), 6pp.10.1155/2013/730736Search in Google Scholar PubMed PubMed Central

[26] P. Li, L. Chen, R. Wu, J. A. T. Machado, A. M. Lopes and L. Yuan, Robust asymptotic stability of interval fractional-order nonlinear systems with time-delay, J. Franklin Inst. 355 (2018), 7749–7763.10.1016/j.jfranklin.2018.08.017Search in Google Scholar

[27] J. T. Machado and A. M. Lopes, Artistic painting: A fractional calculus perspective, Appl. Math. Modell. 65 (2019), 614–626.10.1016/j.apm.2018.09.009Search in Google Scholar

[28] N. Ozalp and I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Differ. Equ. 2012 (2012), 7pp.10.1186/1687-1847-2012-189Search in Google Scholar

[29] N. Ozdemir, D. Avci and B. B. Iskender, The numerical solutions of a two-dimensional space-time Riesz–Caputo fractional diffusion equation, Int. J. Optim. Control Theor. Appl. 1 (2011), 17–26.10.11121/ijocta.01.2011.0028Search in Google Scholar

[30] N. Ozdemir and M. Yavuz, Numerical solution of fractional Black–Scholes equation by using the multivariate Padé approximation, Acta Phys. Pol. A 132 (2016), 1050–1053.10.12693/APhysPolA.132.1050Search in Google Scholar

[31] E. Ucar, N. Ozdemir and E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Modell. Nat. Phenom. 14 (2019), 12.10.1051/mmnp/2019002Search in Google Scholar

[32] J. Ren and Y. Xu, A compartmental model for computer virus propagation with kill signals, Physica A. 486 (2017), 446–454.10.1016/j.physa.2017.05.038Search in Google Scholar

[33] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives to methods of their solution and some of their applications, Academic Press, San Diego, 1999.Search in Google Scholar

[34] Z. M. Odibat and N. T. Shawagfeh, Generalized taylors formula, Appl. Math. Comput. 186(1) (2007), 286–293.10.1016/j.amc.2006.07.102Search in Google Scholar

[35] W. Lin, Global existence theory and chaos control of fractional differential equations, Math. Anal. Appl. 332 (2007), 709–726.10.1016/j.jmaa.2006.10.040Search in Google Scholar

[36] P. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.10.1016/S0025-5564(02)00108-6Search in Google Scholar

[37] E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl. 325 (2007), 542–553.10.1016/j.jmaa.2006.01.087Search in Google Scholar

[38] E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, On some Routh-Hurwitz conditions for fractional differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Phys. Lett. 358 (2006), 1–4.10.1016/j.physleta.2006.04.087Search in Google Scholar

[39] T. Mekkaoui, Z. Hammouch, F. B. M. Belgacem and A. El Abbassi, Fractional-order nonlinear systems: chaotic dynamics, numerical simulation and circuits design, Fractional Dyn. (2015), 343–356.10.1515/9783110472097-021Search in Google Scholar

[40] K. Diethelm, N. Ford, A. Freed and Y. Luchko, Algorithms for the fractional calculus: a selection of numerical method, Comput. Methods Appl. Mech. Eng. 94 (2005), 743–773.10.1016/j.cma.2004.06.006Search in Google Scholar