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Discontinuity, Nonlinearity, and Complexity 12(4) (2023) 837--848 | DOI:10.5890/DNC.2023.12.009

Ali Akg\"{u}l$^{1,2}$, Muhammad Farman$^{3,4}$, Muhammad Mannan Akram$^{5}$, Assad Sajjad$^{5}$, Muhammad Azeem$^{5}$

${^1}$ Art and Science Faculty, Department of Mathematics, Siirt University, 56100 Siirt Turkey

${^2}$ Department of Electronics and Communication Engineering, Saveetha School of Engineering, SIMATS,

Chennai, Tamilnadu, India

${^3}$ Institute of Mathematics, Khawaja Fareed University of Engineering and Information Technology, Rahim Yar

Khan, Pakistan

${^4}$ Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

${^5}$ Department of Mathematics and Statistics, University of Lahore, Lahore-54590, Pakistan

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Abstract

In this paper, we propose a novel technique for the computer virus epidemic which contains infected external computer effects and removable storage media on the computer viruses. The positivity and boundedness for validation of the model are also discussed. The existence and uniqueness of the system of solutions for the model are made by using fixed point theory and iterative method. Numerical simulation obtained with proposed scheme which shows the impacts of varying the fraction-al-order parameters and the support of the theoretical results.

References

1. [1]	Szor, P. (2005), The Art of Computer Virus Research and Defense, Addison-Wesley Educational Publishers Inc., Boston.

2. [2]	Cohen, F. (1987), Computer viruses: theory and experiments, Computers $\&$ security, 6(1), 22-35.

3. [3]	Murray, W.H. (1988), The application of epidemiology to computer viruses, Computers $\&$ Security, 7(2), 139-145.

4. [4]	Kephart, J.O. and White, S.R. (1992), Directed-graph epidemiological models of computer viruses, In Computation: the Micro and the Macro View, 71-102.

5. [5]	Yang, L.X., Yang, X., Wen, L., and Liu, J. (2012), A novel computer virus propagation model and its dynamics, International Journal of Computer Mathematics, 89(17), 2307-2314.

6. [6]	Caputo, M. and Fabrizio, M. (2015), A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation $\&$ Applications, 1(2), 73-85.

7. [7]	Losada, J. and Nieto, J.J. (2015), Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 87-92.

8. [8]	Atangana, A. and Alkahtani, B.S.T. (2015), Analysis of the Keller--Segel model with a fractional derivative without singular kernel, Entropy, 17(6), 4439-4453.

9. [9]	Atangana, A. and Baleanu, D. (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20, 763-769.

10. [10]	Farman, M., Saleem, M.U., Tabassum, M.F., Ahmad, A., and Ahmad, M.O. (2019), A linear control of composite model for glucose insulin glucagon pump, Ain Shams Engineering Journal, 10(4), 867-872.

11. [11]	Farman, M., Saleem, M.U., Ahmad, A., Imtiaz, S., Tabassum, M.F., Akram, S., and Ahmad, M.O. (2020), A control of glucose level in insulin therapies for the development of artificial pancreas by Atangana Baleanu derivative, Alexandria Engineering Journal, 59(4), 2639-2648.

12. [12]	Farman, M., Akg\"{u}l, A., Baleanu, D., Imtiaz, S., and Ahmad, A. (2020), Analysis of fractional order chaotic financial model with minimum interest rate impact. Fractal and Fractional, 4(3), 43.

13. [13]	Saleem, M.U., Farman, M., Ahmad, A., Haque, E.U., and Ahmad, M.O. (2020), A Caputo Fabrizio fractional order model for control of glucose in insulin therapies for diabetes, Ain Shams Engineering Journal, 11(4), 1309-1316.

14. [14]	Hussain, A. and Yaqoob, S. (2020), On a nonlinear fractional order model of novel coronavirus (nCoV-2019) under AB-fractional derivative, Authorea Preprints.

15. [15]	Khan, M.A. and Atangana, A. (2020), Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alexandria Engineering Journal, 59(4), 2379-2389.

16. [16]	Atangana, A., Bonyah, E., and Elsadany, A.A. (2020), A fractional order optimal 4D chaotic financial model with Mittag-Leffler law, Chinese Journal of Physics, 65, 38-53.

17. [17]	Toufik, M. and Atangana, A. (2017), New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, The European Physical Journal Plus, 132, 1-16.

18. [18]	Qureshi, S., Yusuf, A., and Aziz, S. (2021), Fractional numerical dynamics for the logistic population growth model under conformable Caputo: a case study with real observations, Physica Scripta, 96(11), 114002.

19. [19]	Qureshi, S. and Jan, R. (2021), Modeling of measles epidemic with optimized fractional order under Caputo differential operator, Chaos, Solitons $\&$ Fractals, 145, 110766.

20. [20]	Qureshi, S. (2021), Fox H-functions as exact solutions for Caputo type mass spring damper system under Sumudu transform, Journal of Applied Mathematics and Computational Mechanics, 20(1),

21. [21]	Musa, S.S., Qureshi, S., Zhao, S., Yusuf, A., Mustapha, U.T., and He, D. (2021), Mathematical modeling of COVID-19 epidemic with effect of awareness programs, Infectious Disease Modelling, 6, 448-460.

22. [22]	Jajarmi, A. and Baleanu, D. (2021), On the fractional optimal control problems with a general derivative operator, Asian Journal of Control, 23(2), 1062-1071.

23. [23]	Baleanu, D., Sajjadi, S.S., Jajarmi, A.M.I.N., Defterli, O.Z.L.E.M., Asad, J.H., and Tulkarm, P. (2021), The fractional dynamics of a linear triatomic molecule, Romanian Reports on Physics, 73(1), p.105.

24. [24]	Mohammadi, H., Rezapour, S., and Jajarmi, A. (2022), On the fractional SIRD mathematical model and control for the transmission of COVID-19: the first and the second waves of the disease in Iran and Japan, ISA transactions, 124, 103-114.

25. [25]	Mishra, L.N., Tiwari, S.K., Mishra, V.N., and Khan, I.A. (2015), Unique fixed point theorems for generalized contractive mappings in partial metric spaces, Journal of Function spaces, 2015.

26. [26]	Mishra, L.N., Tiwari, S.K., and Mishra, V.N. (2015), Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, Journal of applied Analysis and Computation, 5(4), 600-612.

27. [27]	Hu, X.Q., Zheng, M.X., Damjanovic, B., and Shao, X.F. (2013), Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces, Fixed Point Theory and Applications, 2013(1), 1-11.

28. [28]	Mishra, L.N., Dewangan, V., Mishra, V.N., and Karateke, S. (2021), Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, Journal of Mathematics and Computer Science, 22(2), pp.97-109.

29. [29]	Xu, C., Farman, M., Hasan, A., Akg\"{u}l, A., Zakarya, M., Albalawi, W., and Park, C. (2022), Lyapunov stability and wave analysis of Covid-19 omicron variant of real data with fractional operator, Alexandria Engineering Journal, 61(12), 11787-11802.

30. [30]	Abkar, A., Ghods, S., and Azizi, A. (2015), Coupled best proximity point theorems for proximally g-Meir-Keeler type mappings in partially ordered metric spaces, Fixed Point Theory and Applications, 2015(1), 1-16.

31. [31]	Sharma, N., Mishra, L.N., Mishra, V.N., and Pandey, S. (2020), Solution of delay differential equation via $N^{\nu}_1$ iteration algorithm, European Journal of Pure and Applied Mathematics, 13(5), pp.1110-1130.

32. [32]	Zhang, X. (2016), Modeling the spread of computer viruses under the effects of infected external computers and removable storage media, International Journal of Security and Its Applications, 10(3), 419-428.