## 6 Conclusions and Future Directions  
This study investigates the stability and chaotic dynamical behavior exhibited by a fractional discrete SI model designed for computer viruses, focusing on both commensurate and incommensurate fractional orders. By leveraging fractional calculus, long-term dependencies in virus transmission are captured, offering a more realistic depiction of digital epidemics. The stability analysis, based on the reproduction number \(R_{0}\), delineates the conditions under which the virus spreads is either contained or persists.  
Moreover, numerical investigations reveal that the system exhibits chaotic behavior in certain parameter regimes, with commensurate and incommensurate fractional orders introducing more intricate and unpredictable dynamics. To systematically examine the chaotic nature of the model, multiple analytical techniques were employed, including bifurcation diagrams, phase portraits, and the maximum Lyapunov exponent. Sophisticated complexity algorithms confirmed the system's nonlinear and chaotic nature, utilizing spectral entropy, approximate entropy, and the \(0-1\) test. These tools provided a comprehensive characterization of the system's nonlinear behavior, confirming that fractional order plays a crucial role in shaping the model's dynamical properties. The findings indicate that fractional orders contribute to heightened complexity, emphasizing the need for advanced cybersecurity strategies to address unpredictable virus propagation.  
While the current study initially focused on theoretical development and numerical analysis, a critical step towards demonstrating its practical utility involved bridging the gap to real-world applicability by applying the model to the well-documented case of the Stuxnet worm. Using empirical time-series data from [PERSON] et al. (2019), the parameters were calibrated, and the fractional-order model was shown to accurately reproduce the virus's diffusion trajectory. This real-world validation highlights the model's practical potential.  
Although retrospective modeling approaches, such as the one by [PERSON] [35], offer valuable statistical descriptions of cyberattacks, they often lack dynamic interpretability and flexibility. In contrast, the fractional-order framework proposed here offers a dynamic and tunable structure that can simulate complex virus behaviors, including chaos via memory effects and fractional parameters. Although tailored for computer virus propagation, the proposed model structure can be adapted to study biological epidemics by redefining the compartments and transition parameters, making it broadly applicable in epidemiological modeling.  
In future work, this validation will be extended to other real cyber threats, such as Conficker or WannaCry, and the generalizability of the model across different virus types will be explored. Plans also include integrating optimization-based data-fitting techniques to refine parameter estimation and collaborating with cybersecurity professionals to evaluate the model's integration into early detection systems, response mechanisms, and cryptographic schemes that leverage chaotic dynamics. By building on real-world data, the aim is to further enhance the impact of this model on both theoretical understanding and practical cybersecurity applications.  
Conceptualization, M.A.H. and G.G.; Methodology, I.Z.; Software, I.Z.; Validation, G.G.; Formal analysis, I.Z. and G.G.; Investigation, I.Z. and A.O.; Resources, M.A.H. and A.O.; Data curation, M.A.H. and A.O.; Writing--original draft, I.Z.; Writing--review & editing, M.A.H., A.O. and G.G.; Visualization, I.Z.; Supervision, A.O.; Project administration, A.O.; Funding acquisition, M.A.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The original contributions presented in this study are included in the article; further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.