UDC 517.9: 519.6

V.M. Bulavatsky
V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,  v_bulav@ukr.net

V.O. Bohaienko
V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine,
Kyiv, Ukraine,  sevab@ukr.net

MODELING OF FRACTIONAL-DIFFERENTIAL DYNAMICS
OF THE SPREAD OF COMPUTER VIRUS BASED ON
A DIFFUSIVE EPIDEMIOLOGICAL MATHEMATICAL MODEL

REFERENCES

• 1. Yang L.-X., Yang X., Liu J., Zhu Q., Gan C. Epidemics of computer viruses: A complex-network approach. Applied Mathem. and Comput. 2013. Vol. 219, Iss. 16. P. 8705–8717. https://doi.org/10.1016/j.amc.2013.02.031.
• 2. Cohen F. Computer viruses: theory and experiments. Comput. & Secur. 1987. Vol. 6, Iss. 1. P. 2235. https://doi.org/10.1016/0167-4048(87)90122-2.
• 3. Murray W.H. The application of epidemiology to computer viruses. Comput. & Secur. 1988. Vol. 7, Iss. 2. P. 139–145. https://doi.org/10.1016/0167-4048(88)90327-6.
• 4. Kephart J.O., White S.R. Directed-graph epidemiological models of computer viruses. Proc. 1991 IEEE Computer Society Symposium on Research in Security and Privacy (20–22 May 1991, Oakland, CA, USA). Oakland, 1991. P. 343–359. https://doi.org/10.1109/RISP.1991.130801.
• 5. Piqueira J.R.C., Navarro B.F., Monteiro L.H.A. Epidemiological models applied to viruses in computer networks. Journal of Computer Science. 2005. Vol. 1, N 1. P. 31–34. https://doi.org/10.3844/jcssp.2005.31.34.
• 6. Piqueira J.R.C., Araujo V.O. A modified epidemiological model of computer viruses. Appl. Math. Comp. 2009. Vol. 213, Iss. 2. P. 355–360. https://doi.org/10.1016/j.amc.2009.03.023.
• 7. Mishra B.K., Jha N. Fixed period of temporary immunity after run of anti-malicious software on computer nodes. Appl. Math. Comput. 2007. Vol. 190, Iss. 2. P. 1207–1212. https://doi.org/10.1016/j.amc.2007.02.004.
• 8. Mishra B.K., Saini D.K. SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 2007. Vol. 188, Iss. 2. P. 1476–1482. https://doi.org/10.1016/j.amc.2006.11.012.
• 9. Yuan H., Chen G. Network virus-epidemic model with the point-to-group information propagation. Appl. Math. Comput. 2008. Vol. 206, Iss. 1. P. 357–367. https://doi.org/10.1016/j.amc.2008.09.025.
• 10. Yang X., Yang L-X. Towards the epidemiological modeling of computer viruses. Discrete Dyn. Nat. Soc. 2012. Article number 259671. https://doi.org/10.1155/2012/259671.
• 11. Gan C., Yang X., Zhu Q. Propagation of computer virus under the influences of infected external computers and removable storage media. Nonlinear Dynamics. 2014. Vol. 78, Iss. 2. P. 1349–1356. https://doi.org/10.1007/s11071-014-1521-z.
• 12. Gan C., Yang X., Liu W., Zhu Q. A propagation model of computer virus with nonlinear vaccination probability. Commun. Nonlinear Sci. Numer. Simulat. 2014. Vol. 19, Iss. 1. P. 92–100. https://doi.org/10.1016/j.cnsns.2013.06.018.
• 13. Zhu Q., Yang X., Yang L., Zhang X. A mixing propagation model of computer viruses and countermeasures. Nonlinear Dynamics. 2013. Vol. 73, Iss. 3. P. 1433–1441. https://doi.org/10.1007/s11071-013-0874-z.
• 14. Ren J., Yang X., Zhu Q., Yang L., Zhang C. A novel computer virus model and its dynamics. Nonlin. Analysis: Real World Appl. 2012. Vol. 13, Iss. 1. P. 376–384. https://doi.org/10.1016/j.nonrwa.2011.07.048.
• 15. Gan C., Yang X., Zhu Q.T, Jin J., He L. The spread of computer virus under the effect of external computers. Nonlinear Dynamics. 2013. Vol. 73, Iss. 3. P. 1615–1620. https://doi.org/10.1007/s11071-013-0889-5.
• 16. Pinto C.M.A., Tenreiro Machado J.A. Fractional dynamics of computer virus propagation. Mathematical Problems in Engineering. 2014. Vol. 2014. Article number 476502. https://doi.org/10.1155/2014/476502.
• 17. Yang L.-X., Yang X. The effect of infected external computers on the spread of viruses: A compartment modeling study. Physica A: Statistical Mechanics and its Applications. 2014. Vol. 392, Iss. 24. P. 6523–6535. https://doi.org/10.1016/j.physa.2013.08.024.
• 18. Ren J., Yang X., Yang L.-X., Xu Y., Yang F. A delayed computer virus propagation model and its dynamics. Chaos, Solitons & Fractals. 2012. Vol. 45, Iss. 1. P. 74–79. https://doi.org/10.1016/j.chaos.2011.10.003.
• 19. Han X., Tan Q. Dynamical behavior of computer virus on Internet. Applied Mathematics and Computation. 2010. Vol. 217, Iss. 6. P. 2520–2526. https://doi.org/10.1016/j.amc.2010.07.064.
• 20. Podlubny I. Fractional differential equations. New York: Academic Press, 1999. 341 p. URL: https://books.google.com.ua/books?id=K5FdXohLto0C.
• 21. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. Amsterdam: Elsevier, 2006. 523 p. URL: https://books.google.com.ua/books?id=uxANOU0H8IUC.
• 22. Bohaienko V.O., Bulavatsky V.M., Khimich O.M. Mathematical and computer modeling in hydrogeomigration dynamics problems. Kyiv: Nauk. Dumka, 2022. 249 p. URL: http://www.irbis-nbuv.gov.ua/publ/REF-0000816324.
• 23. Bohaienko V.O., Bulavatsky V.M. Numerical-analytical solution of one problem of modeling fractional-differential dynamics of computer viruses. International scientific and technical journal "Problems of control and computer science". 2022. Vol. 67, No. 1. P. 56–65. https://doi.org/10.34229/1028-0979-2022-1-6.
• 24. Yasin M.W., Ashfaq S.M.H., Ahmed N., Raza A., Rafid M., Akgul A. Numerical modeling of reaction-diffusion e-epidemic dynamics. Eur. Phys. J. Plus. 2024. Vol. 139. Article number 431. https://doi.org/10.1140/epjp/s13360-024-05209-9.
• 25. Fatima U., Baleanu D., Ahmed N., Azam S., Raza A., Rafiq A., Rehman M.A. Numerical study of computer virus reaction diffusion model. Computers, Materials & Continua. 2021. Vol. 66, N 3. P. 3183–3194. https://doi.org/10.32604/cmc.2021.012666.
• 26. Shahid N., Rehman M.A., Khalid A., Fatima U., Shaikh T.S., Ahmed N., Alotaibi H., Rafiq M., Khan I., Nisar S. Mathematical analysis and numerical investigation of advection-reaction-diffusion computer virus model. Results in Physics. 2021. Vol. 26. Article number 104294. https://doi.org/10.1016/j.rinp.2021.104294.
• 27. Hayes T., Ali F. Mobile wireless sensor networks: Applications and routing protocols. In: Handbook of research on next generation mobile communications systems. Panagopoulos A.D. (Ed.). IGI Global, 2016. P. 256–292. https://doi.org/10.4018/978-1-4666-8732-5.ch011.
• 28. Wang X., He Z., Zhao X., Lin Ch., Pan Y., Cai Zh.-P. Reaction-diffusion modeling of malware propagation in mobile wireless sensor networks. Sci. China Inf. Sci. 2013. Vol. 56. P. 1–18. https://doi.org/10.1007/s11432-013-4977-4.
• 29. Serazzi G., Zanero S. Computer virus propagation models. In: Performance tools and applications to networked systems. Calzarossa M.C., Gelenbe E. (Eds.). MASCOTS 2003. LNCS. Vol. 2965. Berlin; Heidelberg: Springer, 2004. P. 26–50. https://doi.org/10.1007/978-3-540-24663-3_2.
• 30. Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler functions, related topics and applications. Berlin: Springer Verlag, 2020. 540 p. https://doi.org/10.1007/978-3-662-61550-8.
• 31. Колмогоров А.Н., Фомин С.В. Элементы теории функций и функционального анализа. Москва: Наука, Главная редакция физико-математической литературы, 1976. 543 с.
• 32. Granas A., Dugudji J. Fixed point theory. Springer, New York, 2003. 690 p. https://doi.org/10.1007/978-0-387-21593-8.
• 33. Tuan N.H., Huynh L.N., Ngoc T.B., Zhou Y. On a backward problem for nonlinear fractional diffusion equations. Appl. Math. Letters. 2019. Vol. 92. P. 76–84. https://doi.org/10.1016/j.aml.2018.11.015.
• 34. Tran N., Van Au V., Zhou Y., Huy Tuan N. On a final value problem for fractional reaction-diffusion equation with Riemann–Liouville fractional derivative. Math. Meth. Appl. Sci. 2019. Vol. 43, Iss. 6. P. 3086–3098. https://doi.org/10.1002/mma.6103.
• 35. Ariba ., Glgeleyen I., Yildiz M. Investigation of well-posedness for a direct problem for a nonlinear fractional diffusion equation and an inverse problem. Fractal Fract. 2024. Vol. 8, Iss. 6. Article number 315. https://doi.org./10.3390/fractalfract8060315.
• 36. Almalahi M.A., Panchal S.K. Existence and stability results of relaxation fractional differential equations with Hilfer–Katugampola fractional derivative. Adv. Theory Nonlin. Anal. Appl. 2020. Vol. 4, Iss. 4. P. 299–315. https://doi.org/10.31197/atnaa.686693.
• 37. Harikrishnan S., Kanagarajan K., Vivek D. Some existence and stability results for integro-differential equations by Hilfer-Katugampola fractional derivative. Palestine Journal of Mathematics. 2020. Vol. 9, N 1. P. 254–262. URL: https://pjm.ppu.edu/sites/default/files/papers/PJM_October2019_254to262.pdf.
• 38. Li C., Zhao Z., Chen Y. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 2011. Vol. 62, Iss. 3. P. 855–875. https://doi.org/10.1016/j.camwa.2011.02.045.
• 39. Li C., Zeng F. Finite difference methods for fractional differential equations. Internat. J. Bifur. Chaos. 2012. Vol. 22, N 4. Article number 1230014. https://doi.org/10.1142/S0218127412300145.
• 40. Samarskii A.A. The theory of difference schemes. Boca Raton: CRC Press, 2001. 786 p. https://doi.org/10.1201/9780203908518.