Existence Results for a Computer Virus Spreading Model with Atangana-Baleanu Derivative

Öz

A computer virus is actually a kind of computer program that changes the operation of the computer and tries to hide itself in other files without the user's consent or knowledge. In this paper we deal with a computer virus spreading model benefiting from Atangana-Baleanu derivative in Caputo sense with non- local and non- singular kernels. The solution properties of our fractional model are established benefiting from Arzelo-Ascoli theorem.

Anahtar Kelimeler

• Arzelo-Ascoli theorem,Atangana-Baleanu derivative,existence and uniqueness

Kaynakça

1. 1. Piqueira, JRC, Araujo, VO. 2009. A modified epidemiological model for computer viruses, Applied Mathematics and Computation; 213(2): 355–360.
2. 2. Wierman, JC, Marchette, DJ. 2004. Modeling computer virus prevalence with a susceptible-infected susceptible model with reintroduction, Computational Statistics Data Analysis; 45(1): 3–23.
3. 3. Li, XZ, Zhou, LL. 2009. Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate, Chaos Solitons & Fractals; 40(2): 874–884.
4. 4. Li, G, Zhen, J. 2005. Global stability of an SEI epidemic model with general contact rate, Chaos, Solitons Fractals; 23(3): 997–1004.
5. 5. Jin, Y, Wang, W, Xiao, S. 2007. An SIRS model with a nonlinear incidence rate, Chaos, Solitons Fractals; 34(5): 1482–1497.
6. 6. Yang, LX, Yang, XF, Zhu, QY, Wen, LS. 2013. A computer virus model with graded cure rates, Nonlinear Analysis: Real World Applications; 14(1): 414-442.
7. 7. Yang, LX, Yang, XF. 2012. The spread of computer viruses under the influence of removable storage devices, Applied Mathematics and Computation; 219(8): 3419-3422.
8. 8. Wang, FG, Zhang, YK, Wang, CG, Ma, JF, Moon, SJ. 2010. Stability analysis of a SEIQV epidemic model for rapid spreading worms, Computers & Security; 29(4): 410-418.

9. 9. Özdemir, N, Karadeniz, D, İskender, BB. 2009. Fractional optimal control problem of a distributed system in cylindrical coordinates, Physics Letters A; 373(2): 221-226.
10. 10. Oldham, KB, Spanier, J. The Fractional Calculus; New York, Academic Press, 1974.
11. 11. Kilbas, AA, Srivastava, HM, Trujillo, JJ. Theory and applications of fractional differential equations; Amsterdam, Elsevier, 2006.
12. 12. Özdemir, N, Agrawal, OP, İskender, BB, Karadeniz, D. 2009. Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. Nonlinear Dynamics; 55:251-260.
13. 13. Yavuz, M, Özdemir, N. 2018. European vanilla option pricing model of fractional order without singular kernel. Fractal Fractional; 2(1): 3.
14. 14. Evirgen, F. 2011. Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, Journal of Computational and Nonlinear Dynamics; 6.
15. 15. Evirgen, F. 2016. Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA); 6(2): 75-83.
16. 16. Atangana, A, Koca, İ. 2016. On the new fractional derivative and application to nonlinear Baggs andFreedman model, Journal of Nonlinear Sciences and Applications; 9(5): 2467-2480.
17. 17. Alkahtani, BSTA, Atangana A, Koca İ. 2016. A new nonlinear triadic model of predator prey based on derivative with non-local and non-singular kernel, Advances in Mechanical Engineering; 8(11).
18. 18. Mekkaoui, T, Atangana, A. 2017. New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, European Physical Journal Plus; 132(10). 19. Morales-Delgado VF, Gomez-Aguilar JF, Taneco-Hernandez MA, Escobar-Jimenez RF, Olivares-Peregrino VH. 2018. Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, Journal of Nonlinear Sciences Applications; 11(8): 994-1014.
19. 20. Özdemir, N, Yavuz M. 2017. Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Pade Approximation, Acta Physica Polonica A; 132: 1050-1053.
20. 21. Yavuz, M, Özdemir, N, Başkonuş, HM. 2017. Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus; 133(6).
21. 22. Koca, İ. 2018. Analysis of rubella disease model with non-local and non-singular fractional derivatives, An International Journal of Optimization and Control: Theories & Applications (IJOCTA); 8(1): 17-25.
22. 23. Uçar, S, Uçar E, Özdemir, N, Hammouch Z. 2019. Mathematical analysis and numerical simulation for a smoking model with Atangana Baleanu derivative, Chaos, Solitons Fractals; 118: 300-306. 24. Özdemir, N, Uçar E. 2020. Investigating of an immune system-cancer mathematical model with Mittag-Leffler kernel, AIMS Mathematics, 5(2):1519-1531.
23. 25. Baleanu, D, Fernandez, A. 2018. On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications Nonlinear Science and Numerical Simulation; 59: 444–462.
24. 26. Fernandez, A, Baleanu D, Srivastava, HM. 2019. Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Communications Nonlinear Science and Numerical Simulation; 67: 517-527.
25. 27. Avcı, D, Yetim A. 2018. Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain, Journal of Balıkesir University Institute Science and Technology; 20(2): 382–395.
26. 28. Yavuz, M, Bonyah E. 2019. New approaches to the fractional Dynamics of schistosomiasis disease model, Physica A: Statistical Mechanics and its Applications; 525: 373-393.
27. 29. Xu, Y, Ren J. 2016, Propagation Effect of a Virus Outbreak on a Network with Limited Anti-Virus Ability; Plos One, Article ID:e0164415
28. 30. Atangana, A, Baleanu, D. 2016. New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model, Thermal Science; 20(2): 763-769.
29. 31. Toufik, M, Atangana, A. 2017. New numeriical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus; 132.