# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 5 Numerical Methods and Implementation  
To complement the analytical results, we perform numerical simulations to visualize the malware spread dynamics under various parameter settings. The model is implemented in Python using numpy, matplotlib.pyplot, scipy.integrate.solve_ivp, scipy.optimize.minimize,pandas,mpl_toolkits.mplot3d.Axes3D, and sympy libraries for numerical integration and plotting. Key outputs include time-series plots of each compartment and sensitivity analysis results, allowing for the assessment of control measures and model robustness. Table 4 values were obtained from (1) malware campaign analyses (Ghost Push, HummingBad [36]) for \(\Pi\) and \(\delta_{2}\), (2) antivirus reports (Google Play Protect, Kaspersky [37]) for \(\theta_{1}\), and (3) network statistics (GSMA, Ericsson [38]) for \(\lambda\) and \(\delta_{1}\). Fitted parameters (\(\theta_{2}\) and \(q\)) were calibrated via Monte Carlo simulations constrained by outbreak case studies [39]. All time-dependent parameters use units of day\({}^{-1}\). Limitations include the following: (1) spatial homogeneity assumption (15-25% transmission overestimation), (2) static parameters ignoring temporal variations, and (3) uniform \begin{table}
\begin{tabular}{c c} \hline \hline
**From Matrix \(B\)** & **From Matrix \(A\)** \\ \hline \(\lambda_{1}=-(\delta_{1}+\delta_{3}+q)\) & \(\lambda_{3}=\frac{-\text{tr}(A)+\sqrt{(\text{tr}(A))^{2}-4\det(A)}}{2}\) \\ \(\lambda_{2}=-\delta_{1}\) & \(\lambda_{4}=\frac{-\text{tr}(A)-\sqrt{(\text{tr}(A))^{2}-4\det(A)}}{2}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Eigenvalues of the Jacobian matrix at \(SME^{*}\). susceptibility despite real-world device heterogeneity (30% OS patch lag). Values represent global averages from 2022-2023 datasets, with regional variations expected. The dynamics of malware propagation, both over time and also under varying conditions of the basic reproduction number \(R_{0}\), are well illustrated in Figure 2a,b, representing those scenarios where \(R_{0}<1\), meaning the malware will eventually be disabled. Figure 2a shows this scenario of a short-term span, and Figure 2b shows this over a long-term span; both are consistent in showing the eventual decline (and eradication) of the malware as a result of effective containment. By contrast, when \(R_{0}>1\), this indicates the malware propagation impact is increasing over time. Figure 2 demonstrates this increase in impact in the short-term and long-term in Figure 2c and Figure 2d, respectively. Here, the variables either increase or maintain levels, indicating the persistence and spread of the malware. This comparison highlights the critical threshold of \(R_{0}\) in determining the success or failure of malware containment efforts. The bars in Figure 3 corresponding to \(\lambda\) and \(\Pi\) are positive, indicating a direct correlation with \(R_{0}\); as these parameters increase, so does \(R_{0}\). Conversely, \(\delta_{1}\), \(\delta_{2}\), \(\theta_{1}\), and \(\theta_{2}\) have \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Parameter** & \(\Pi\) & \(\theta_{1}\) & \(\theta_{2}\) & \(q\) & \(\lambda\) & \(\delta_{1}\) & \(\delta_{2}\) & \(\delta_{3}\) \\ \hline
**Value** & 0.015–0.085 & 0.12–0.18 & 0.05–0.10 & 0.20–0.35 & 0.001–0.005 & 0.01–0.03 & 0.15–0.25 & 0.08–0.12 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters and values in the \(M_{\text{siqr}}\) model. Figure 2: Malware dynamics over time (**a**), (**c**) short-term scenario, and (**b,d**) long-term scenario. negative PRCC values, suggesting an inverse relationship; as these parameters increase, \(R_{0}\) decreases. Notably, \(\delta_{1}\) has the smallest negative value among the parameters. The PRCC analysis of the mobile malware model reveals key parameter sensitivities affecting \(R_{0}\). The contact rate \(\Pi\) shows the strongest positive correlation with \(R_{0}\), meaning higher device-to-device transmission rates increase epidemic spread through Bluetooth, WiFi, and shared networks.