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ARTICLE

A Deterministic and Stochastic Fractional-Order Model for Computer Virus Propagation with Caputo-Fabrizio Derivative: Analysis, Numerics, and Dynamics

Najat Almutairi¹, Mohammed Messaoudi², Faisal Muteb K. Almalki³, Sayed Saber^3,4,*

1 Department of Mathematics, College of Science, Qassim University, Buraidah, Saudi Arabia
2 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3 Department of Mathematics, Faculty of Science, Al-Baha University, Alaqiq, Saudi Arabia
4 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

* Corresponding Author: Sayed Saber. Email:

Computer Modeling in Engineering & Sciences 2026, 146(3), 29 https://doi.org/10.32604/cmes.2026.076371

Received 19 November 2025; Accepted 23 January 2026; Issue published 30 March 2026

Abstract

This paper introduces a novel fractional-order model based on the Caputo–Fabrizio (CF) derivative for analyzing computer virus propagation in networked environments. The model partitions the computer population into four compartments: susceptible, latently infected, breaking-out, and antivirus-capable systems. By employing the CF derivative—which uses a nonsingular exponential kernel—the framework effectively captures memory-dependent and nonlocal characteristics intrinsic to cyber systems, aspects inadequately represented by traditional integer-order models. Under Lipschitz continuity and boundedness assumptions, the existence and uniqueness of solutions are rigorously established via fixed-point theory. We develop a tailored two-step Adams–Bashforth numerical scheme for the CF framework and prove its second-order accuracy. Extensive numerical simulations across various fractional orders reveal that memory effects significantly influence virus transmission and control dynamics; smaller fractional orders produce more pronounced memory effects, delaying both infection spread and antivirus activation. Further theoretical analysis, including Hyers–Ulam stability and sensitivity assessments, reinforces the model’s robustness and identifies key parameters governing virus dynamics. The study also extends the framework to incorporate stochastic effects through a stochastic CF formulation. These results underscore fractional-order modeling as a powerful analytical tool for developing robust and effective cybersecurity strategies.

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Keywords

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