Malware is a common word in modern era. Everyone using computer is aware of it. Some users have to face the problem known as Cyber crimes. Nobody can survive without use of modern technologies based on computer networking. To avoid threat of malware, different companies provide antivirus strategies on a high cost. To prevent the data and keep privacy, companies using computers have to buy these antivirus programs (software). Software varies due to types of malware and is developed on structure of malware with a deep insight on behavior of nodes. We selected a mathematical malware propagation model having variable infection rate. We were interested in examining the impact of memory effects in this dynamical system in the sense of fractal fractional (FF) derivatives. In this paper, theoretical analysis is performed by concepts of fixed point theory. Existence, uniqueness and stability conditions are investigated for FF model. Numerical algorithm based on Lagrange two points interpolation polynomial is formed and simulation is done using Matlab R2016a on the deterministic model. We see the impact of different FF orders using power law kernel. Sensitivity analysis of different parameters such as initial infection rate, variable adjustment to sensitivity of infected nodes, immune rate of antivirus strategies and loss rate of immunity of removed nodes is investigated under FF model and is compared with classical. On investigation, we find that FF model describes the effects of memory on nodes in detail. Antivirus software can be developed considering the effect of FF orders and parameters to reduce persistence and eradication of infection. Small changes cause significant perturbation in infected nodes and malware can be driven into passive mode by understanding its propagation by FF derivatives and may take necessary actions to prevent the disaster caused by cyber crimes.

Department of Computer Science and Mathematics Lebanese American University Byblos Lebanon

Department of Mathematics and Statistics Ripha International University Islamabad Pakistan

Department of Mathematics Quaid i Azam University Islamabad Pakistan

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

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Megira S, Pangesti AR, Wibowo FW. Malware analysis and detection using reverse engineering technique. InJournal of Physics: Conference Series 2018 Dec 1 (Vol. 1140, No. 1, p. 012042). IOP Publishing.

Tahir R. A study on malware and malware detection techniques. International Journal of Education and Management Engineering. 2018. Mar 1;8(2):20. doi: 10.5815/ijeme.2018.02.03 DOI

Gounder MP, Farik M. New ways to fight malware. Int. J. Sci. Technol. Res. 2017. Jun;6(06).

Raza A, Fatima U, Rafiq M, Ahmed N, Khan I, Nisar KS, et al.. Mathematical analysis and design of the nonstandard computational method for an epidemic model of computer virus with delay effect: application of mathematical biology in computer science. Results in Physics. 2021. Feb 1;21:103750. doi: 10.1016/j.rinp.2020.103750 DOI

Data S, Wang H. The effectiveness of vaccinations on the spread of email-borne computer viruses. InCanadian Conference on Electrical and Computer Engineering, 2005. 2005 May 1 (pp. 219–223). IEEE.

Feng L, Liao X, Han Q, Li H. Dynamical analysis and control strategies on malware propagation model. Applied Mathematical Modelling. 2013. Sep 1;37(16-17):8225–36. doi: 10.1016/j.apm.2013.03.051 DOI

Abbas S., Benchohra M., and Vityuk A. N. (2012). On fractional order derivatives and Darboux problem for implicit differential equations. Fractional Calculus and Applied Analysis, 15(2), 168–182. doi: 10.2478/s13540-012-0012-5 DOI

Oliveira EC, Machado JA. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering. 2014. Jun 10;2014:1–7. doi: 10.1155/2014/238459 DOI

Gerasimov AN. Generalization of linear deformation laws and their application to internal friction problems. Applied mathematics and mechanics. 1948;12:529–39.

Caputo M. Linear models of dissipation whose Q is almost frequency independent—II. Geophysical journal international. 1967. Nov 1;13(5):529–39. doi: 10.1111/j.1365-246X.1967.tb02303.x DOI

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. elsevier; 2006. Feb 16.

Tverdyi D, Parovik R. Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation. Fractal and Fractional. 2021. Dec 31;6(1):23. doi: 10.3390/fractalfract6010023 DOI

Sun H, Chang A, Zhang Y, Chen W. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fractional Calculus and Applied Analysis. 2019. Feb 25;22(1):27–59. doi: 10.1515/fca-2019-0003 DOI

Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications. 2015;1(2):73–85. 10.12785/pfda/010201 DOI

Losada J, Nieto JJ. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 2015. Apr;1(2):87–92. 10.12785/pfda/010202. DOI

Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408. 2016 Jan 20.

Hattaf K. A new generalized definition of fractional derivative with non-singular kernel. Computation. 2020. May 21;8(2):49. doi: 10.3390/computation8020049 DOI

Al-Refai M. On weighted Atangana–Baleanu fractional operators. Advances in Difference Equations. 2020. Jan 2;2020(1):3. doi: 10.1186/s13662-019-2471-z DOI

Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons and Fractals. 2016. Aug 1;89:447–54. doi: 10.1016/j.chaos.2016.02.012 DOI

Atangana A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, solitons and fractals. 2017. Sep 1;102:396–406. doi: 10.1016/j.chaos.2017.04.027 DOI

Alshabanat A, Jleli M, Kumar S, Samet B. Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits. Frontiers in Physics. 2020. Mar 20;8:64. doi: 10.3389/fphy.2020.00064 DOI

Borah M, Gayan A, Sharma JS, Chen Y, Wei Z, Pham VT. Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19?. Nonlinear dynamics. 2022. Jul;109(2):1187–215. doi: 10.1007/s11071-021-07196-3 PubMed DOI PMC

Partohaghighi M, Yusuf A, Karaca Y, Li YM, Ibrahim TF, Younis BA, et al.. A new fractal fractional modeling of the computer viruses system. Fractals. 2022. Aug 4;30(05):2240184. doi: 10.1142/S0218348X22401843 DOI

Samet B, Vetro C, Vetro P. Fixed point theorems for α–ψ-contractive type mappings. Nonlinear analysis: theory, methods and applications. 2012. Mar 1;75(4):2154–65. doi: 10.1016/j.na.2011.10.014 DOI

Mawhin J. Leray-Schauder degree: a half century of extensions and applications. 1999: 195–228.

Marsden JE. Elementary Classic Analysis. 1973.

Chen W, Sun H, Zhang X, Korošak D. Anomalous diffusion modeling by fractal and fractional derivatives. Computers and Mathematics with Applications. 2010. Mar 1;59(5):1754–8. doi: 10.1016/j.camwa.2009.08.020 DOI

Chen W. Time–space fabric underlying anomalous diffusion. Chaos, Solitons and Fractals. 2006. May 1;28(4):923–9. doi: 10.1016/j.chaos.2005.08.199 DOI

Avcı İ, Hussain A, Kanwal T. Investigating the impact of memory effects on computer virus population dynamics: A fractal–fractional approach with numerical analysis. Chaos, Solitons and Fractals. 2023. Sep 1;174:113845. doi: 10.1016/j.chaos.2023.113845 DOI

Rezapour S, Asamoah JK, Hussain A, Ahmad H, Banerjee R, Etemad S, et al.. A theoretical and numerical analysis of a fractal–fractional two-strain model of meningitis. Results in Physics. 2022. Aug 1;39:105775. doi: 10.1016/j.rinp.2022.105775 DOI

Omame A, Raezah AA, Okeke GA, Akram T, Iqbal A. Assessing the impact of intervention measures in a mathematical model for monkeypox and COVID-19 co-dynamics in a high-risk population. Modeling Earth Systems and Environment. 2024. Aug 30:1–5. doi: 10.1007/s40808-024-02132-x DOI

Adu IK, Wireko FA, Sebil C, Asamoah JK. A fractal–fractional model of Ebola with reinfection. Results in Physics. 2023. Sep 1;52:106893. doi: 10.1016/j.rinp.2023.106893 DOI

Nwajeri UK, Asamoah JK, Ugochukwu NR, Omame A, Jin Z. A mathematical model of corruption dynamics endowed with fractal–fractional derivative. Results in Physics. 2023. Sep 1;52:106894. doi: 10.1016/j.rinp.2023.106894 DOI

Poria S, Dhiman A. Existence and uniqueness theorem for ODE: an overview. arXiv preprint arXiv:1605.05317. 2016 May 17.

Hyers DH. On the stability of the linear functional equation. Proceedings of the National Academy of Sciences. 1941. Apr 15;27(4):222–4. doi: 10.1073/pnas.27.4.222 PubMed DOI PMC

Rassias TM. On the stability of the linear mapping in Banach spaces. Proceedings of the American mathematical society. 1978;72(2):297–300. doi: 10.1090/S0002-9939-1978-0507327-1 DOI

Solís-Pérez JE, Gómez-Aguilar JF, Atangana A. Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos, Solitons and Fractals. 2018. Sep 1;114:175–85. doi: 10.1016/j.chaos.2018.06.032 DOI