These models account for the cumulative impact of previous infections, providing valuable insights into how viruses persist, adapt, and evolve over time [19]. A fractional discrete SI model extends the classical SI model by incorporating fractional-order derivatives to capture long-term dependencies in virus transmission. While traditional SI models assume that the rate of infection relies solely on the information contained in the current system state, the fractional framework considers historical infection trends, leading to a more accurate representation of real-world virus behavior [20]. The utilization of discrete fractional models in cybersecurity makes it possible to identify hidden patterns, predict outbreaks, and develop proactive defense mechanisms [21]. By analyzing the system's dynamics, one can determine the conditions under which a virus persists, dies out, or exhibits chaotic behavior [22]. This insight is crucial for designing effective cybersecurity policies, intrusion detection systems, and digital defense strategies. Furthermore, nonlinear analysis, particularly bifurcation theory, is key to understanding the critical points where small changes in system parameters can lead to drastic shifts in behavior, such as the transition from stable infection dynamics to chaotic outbreaks. This allows for a more detailed exploration of how digital epidemics evolve and how they may be controlled or mitigated. This research investigates the stability and chaotic dynamics of a susceptible-infected (SI) model for computer virus propagation with fractional discrete dynamics, using the Caputo-like difference operator. The key motivation behind this work lies in the growing challenge of controlling sophisticated computer viruses whose spread exhibits unpredictable and highly sensitive dynamics. Understanding such chaotic behavior is essential for designing robust and adaptive cybersecurity strategies. In this context, fractional calculus offers a powerful framework for modeling long-term dependencies and capturing non-local behavior that classical models may overlook. When a virus exhibits chaotic behavior, even minor variations in initial conditions can lead to drastically different outcomes, making prediction and control significantly more difficult. This study provides an in-depth analysis of the relationship between fractional calculus and chaotic dynamics, contributing to the development of more resilient digital defense mechanisms. Unlike normalized continuous-time fractional SIR models [23], which use the Caputo derivative, our model employs a discrete-time Caputo-like fractional difference operator. This approach is more appropriate for modeling digital systems like computer virus spread, where interactions naturally occur at discrete time intervals. The novelty of our work lies in constructing a fractional discrete SI model with both commensurate and incommensurate fractional orders and uncovering its chaotic behavior using advanced nonlinear tools. To the best of our knowledge, our model is the first to explore chaotic dynamics in a discrete fractional SI framework with both commensurate and incommensurate orders. This analysis enables a deeper understanding of virus dynamics and offers insights for designing more robust cybersecurity mechanisms. The main contributions of this work are as follows:  
* We derive the stability conditions of the proposed fractional discrete SI model using the basic reproduction number \(R_{0}\), which helps predict whether a virus outbreak will persist or die out. * We examine the influence of both commensurate and incommensurate fractional orders on the system's dynamics, emphasizing their ability to capture memory effects in digital virus propagation. * We apply several chaos detection and complexity analysis tools, including bifurcation diagrams, phase portraits, maximum Lyapunov exponents, spectral entropy, approximate entropy, and the \(0-1\) test for chaos, to reveal nonlinear and chaotic behaviors in the model.