On Fractional Discrete-Time Computer Virus Model: Stability, Bifurcation, Chaos and Complexity Analysis

Omar Kahouli (), Imane Zouak (), Adel Ouannas, Lilia El Amraoui and Mohamed Ayari
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Omar Kahouli: Department of Electronics Engineering, Applied College, University of Ha’il, Ha’il 2440, Saudi Arabia
Imane Zouak: System Dynamics and Control Laboratory, Department of Mathematics and Computer Sciences, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
Adel Ouannas: Department of Mathematics and Computer Sciences, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
Lilia El Amraoui: Department of Electrical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Mohamed Ayari: Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Arar 91431, Saudi Arabia

Mathematics, 2025, vol. 13, issue 20, 1-18

Abstract: Computer viruses continue to threaten the security of digital networks, and their complex propagation dynamics require refined modelling tools. Most existing models rely on integer-order dynamics or assume uniform memory effects, which limit their ability to capture heterogeneous behaviours observed in practice. To address this gap, we propose a discrete incommensurate fractional-order virus model based on Caputo-like delta differences, where each compartment is assigned a distinct fractional order to represent mismatched time scales. The model’s dynamics are analysed in terms of stability, bifurcation, and chaos. Numerical results reveal the emergence of rich chaotic attractors, emphasizing the impact of fractional memory on system evolution. To quantify complexity, we employ Approximate Entropy and Spectral Entropy and relate these indicators to the maximum Lyapunov exponent, confirming the system’s sensitivity and unpredictability. All numerical simulations and visualizations were performed using MATLAB (R2015a). The findings highlight the importance of heterogeneous memory in computer-virus modeling and offer new insights for developing theoretical foundations of robust cybersecurity strategies.

Keywords: computer virus; chaos; incommensurate fractional-order; stability; complexity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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