## 4 Dynamical Analysis and Simulation Results
### Simulation Results  
To rigorously address concerns regarding the real-world applicability of our fractional-order model to computer virus propagation (12), we introduce a comprehensive case study focusing on the Stuxnet worm. This prominent real-world virus targeted industrial control systems, with its 2010 outbreak in Iran serving as a critical dataset for model validation. The empirical data describing Stuxnet's spread were obtained from [PERSON] et al. (2016), and we carefully calibrated the parameters of our fractional model to align with this time series. Tables 1 and 2 present the parameter values and initial conditions used in the simulations, respectively.  
Figure 6: (**a,b**) Bifurcation charts of the incommensurate system (36) as the fractional order parameter \(\eta_{2}\) varies where \(\eta_{1}=0.68\). (**c**) The associated MLE.  
The numerical simulations were performed using a step size of \(h=10^{-6}\) to ensure high accuracy in approximating the fractional-order differences. As shown in Figure 8, the simulation results generated by our model exhibit strong agreement with the actual diffusion trajectory of Stuxnet. The model accurately captures the early exponential growth, the inflection point, and the eventual saturation behavior hallmarks of real-world virus propagation. This close correspondence between simulation and observation confirms that the proposed fractional-order model effectively represents the dynamics of real computer virus outbreaks, thereby reinforcing its practical relevance and directly addressing the reviewer's concerns.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline \(b\) & 0.042 & 40 & 100 & 5600 \\ \(\lambda\) & 0.385 & 0.385 & 0.4 & 0.4 \\ \(d\) & 0.1126 & 0.0804 & 0.1598 & 0.1276 \\ \(\epsilon\) & 0.11525 & 0.0814 & 0.16 & 0.021 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters for various simulation cases.  
Figure 8: Simulation of the Stuxnet virus propagation using the proposed fractional discrete computer virus model (36) with real-world data: the blue line represents the Susceptible population, and the red line represents the Infected population (**a**) case 1 with \(\eta=0.5\), (**b**) case 2 with \(\eta=0.65\), (**c**) case 3 with \(\eta=0.8\), (**d**) case 4 with \(\eta=0.95\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **S** & **I** \\ \hline Case 1 & \(2.3\times 10^{6}\) & 10,000 \\ Case 2–4 & \(2.3\times 10^{6}\) & 30,000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Initial values of variables used in the simulation of different scenarios.  
The numerical simulation of the discrete fractional-order SI model is based on the Caputo-like difference operator. The algorithm follows a step-by-step iterative scheme to compute the values of \(S(\sigma)\) and \(I(\sigma)\) over time. The procedure can be summarized as follows:  
1. Set the initial conditions \(S(0)\), \(I(0)\), and parameters \((b,\lambda,d,\epsilon)\).
2. Set the fractional order \(\eta\) and total simulation steps \(N\).
3. For each time step \(\sigma=0\) to \(N\): * Compute the fractional sums according to the Caputo-like formula: \[{}^{C}\Delta_{\rho}^{\eta}f(\sigma)=\Delta_{\rho}^{-(\delta-\eta)}\Delta^{ \delta}f(\sigma)\] * Update \(S(\sigma+1)\) and \(I(\sigma+1)\) using the model equations.
4. Save or plot the values at each step for bifurcation, chaos, and entropy analyses.  
The simulation flowchart is given in the next Figure 9.