## 4 Dynamical Analysis and Simulation Results
### Commensurate Order Case  
An investigation is conducted into the dynamic behavior variations of the proposed system (12) by constructing bifurcation diagrams; this approach allows visualization of the system's long-term behavior as the parameter changes. Initially, the state is \((S(0),I(0))=(40,30)\), the parameters used in the simulations are \((b,\lambda,d,e)=(20,0.04,0.2,0.13)\), and the commensurate order, denoted by \(\eta\), is treated as the main bifurcation and sensitivity value due to its intrinsic role in governing the system's memory and long-range interaction effects. By varying \(\eta\) while keeping other model parameters constant, we were able to observe rich dynamical behaviors, including transitions from steady-state to periodic and chaotic regimes. This sensitivity to \(\eta\) highlights its fundamental influence on the system's stability and complexity. The resulting bifurcation diagram and the corresponding MLEs are seen in Figure 2. The bifurcation charts in Figure 3a,b depict system behavior for varying \(\eta\) inside the interval \([0,1]\). These diagrams showcase various dynamical regimes, including stable fixed points, periodic oscillations, and chaotic dynamics. For small values of \(\eta\) (approximately \(\eta<0.2\)), the system shows stable behavior, marked by a single fixed point or periodic oscillations. As \(\eta\) increases beyond this range, a series of period-doubling bifurcations occurs, leading to the emergence of chaos around \(\eta=0.25\). The chaotic regime is evident in the bifurcation diagrams, where the state variable displays an irregular and dense structure, reflecting initial condition sensitivity. This chaotic behavior persists roughly in the range \(0.25\leq\eta\leq 0.7\), during which the system exhibits complex, non-periodic oscillations. The transition from order to chaos is further supported by the MLEs plot in Figure 2c. For \(\eta<0.2\), the MLE remains negative, confirming stability. As \(\eta\) approaches \(0.25\), the MLE crosses zero and becomes positive, signaling the onset of chaos. The chaotic regime is most pronounced in the interval \(0.25\leq\eta\leq 0.7\), where the MLE remains significantly positive, peaking around \(\eta=0.5\). This indicates a high degree of unpredictability and sensitivity to initial conditions. Beyond \(\eta=0.7\), the system gradually transitions back to stability, as indicated by decreasing MLE values. For \(\eta>0.8\), the MLE becomes negative again, suggesting a return to periodic or fixed-point behavior. This demonstrates that while moderate values of \(\eta\) induce chaotic dynamics, higher values restore stability. The commensurate fractional discrete computer virus System (12) exhibits a range of dynamical behaviors, as illustrated by the phase portraits in Figure 4. For different values of \(\eta\), the figure shows chaos at lower fractional orders \(\eta\) values. The choice of fractional orders in our simulations was guided by the goal of exploring transitions in system dynamics. Specific values were selected to capture the onset of chaotic behavior, as verified by positive Lyapunov exponents. These numerical simulations highlight the system's interesting dynamical properties. The oscillatory nature is due to nonlinearity and fractional memory, especially when chaos emerges. Fractional-order systems are known to exhibit oscillations even in the absence of external forcing. In summary, these findings emphasize the significant contribution of the fractional derivative parameter \(\eta\) in shaping the system's complexity. The occurrence of bifurcations and chaos within specific intervals highlights the rich dynamical behavior of fractional-order models, making them particularly valuable for studying complex systems. The capacity of fractional-order models to effectively simulate the delayed propagation and reinfection cycles characteristic of known computer worms, such as WannaCry or Conficker, positions them as a suitable tool for cyber-epidemiological forecasting.