# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 5 Numerical Methods and Implementation
The recruitment rate \(\lambda\) has moderate positive correlation, indicating that adding new devices to the network raises infection risk. The quarantine rate \(\theta_{1}\) exhibits the strongest negative correlation, proving that rapid detection and isolation of infected devices is the most effective control strategy. Both malware removal rate \(\delta_{2}\) and natural device retirement \(\delta_{1}\) show moderate negative correlations, demonstrating that cybersecurity measures and device turnover reduce epidemic spread. This analysis confirms that investing in automated detection and quarantine systems provides the highest return for mobile network security, supporting the finding that proactive containment outperforms responses to increased threat virulence. Figure 4 presents four contour plots and gradient color visualizations that illustrate the basic reproduction number, \(R_{0}\), in relation to different parameters. Contour plots and gradient color visualizations like the ones above constitute essential channels for the provision of insights and better understanding of how different parameters influence the basic reproduction number, \(R_{0}\), and can help in designing effective control strategies for managing the spread of infections. As per Figure 4a, showing the relationship between \(\theta_{1}\) and \(\theta_{2}\) with \(R_{0}<1\), the contour lines and gradient colors are consistent with conditions wherein the malware infection rate is under control. Moving to Figure 4b, which maps \(\theta_{1}\) against \(\theta_{2}\), for \(R_{0}>1\), we can see that there are contour lines and gradient colors that are consistent with a higher rate of infection and possible loss of control against the malware. Figure 4c examines the parameters \(\delta_{2}\) (x-axis) and \(\Pi\) (y-axis) for \(R_{0}>1\). The contour lines and gradient colors are consistent with the indication that the malware infection is likely to proliferate. Finally, Figure 4d, like Figure 4c, maps \(\delta_{2}\) against \(\Pi\) with \(R_{0}<1\). The contour lines and gradient colors in this case indicate conditions under which the infection spread is contained. Figure 3: Sensitivity of the basic reproduction number \(R_{0}\) to changes in various parameters through PRCC analysis. The transmission dynamics of mobile malware are governed by the autonomous system (1). The system's structural symmetries emerge from three fundamental transformations: time translation invariance manifests through the generator \(X_{t}=\partial_{t}\), reflecting the system's autonomy. For any solution \(\mathbf{x}(t)=(M_{s}(t),M_{i}(t),M_{q}(t),M_{r}(t))\), the time-shifted trajectory \(\mathbf{x}(t+\tau)\) satisfies the same dynamics. This symmetry implies that malware propagation patterns are independent of absolute time measurements: an outbreak beginning today will follow the same progression as one starting tomorrow given identical conditions. Reflection symmetry appears about the disease-free equilibrium plane \(\mathcal{P}=\{(M_{s},0,0,0)|\)\(M_{s}=\lambda/\delta_{1}\}\). The transformation  
\[\mathcal{R}: \begin{cases}M_{s}\mapsto 2M_{s}^{*}-M_{s}\\ M_{i}\mapsto M_{i}\\ M_{q}\mapsto M_{q}\\ M_{r}\mapsto M_{r}\end{cases} ,\quad M_{s}^{*}=\lambda/\delta_{1} \tag{20}\]  
yields approximately antisymmetric dynamics \(\mathbf{f}(\mathcal{R}\mathbf{x})\approx-\mathcal{R}\mathbf{f}(\mathbf{x})\) in the vicinity of \(\mathcal{P}\). Epidemiologically, this reveals a critical threshold behavior: populations with susceptible counts equidistant from \(M_{s}^{*}\) but on opposite sides exhibit mirrored infection trajectories. Figure 4: Contour plots illustrating regions of varying basic reproduction numbers. (**a**) \(R_{0}<1\) for varying \(\theta_{1}\) and \(\theta_{2}\), indicating controlled infection. (**b**) \(R_{0}>1\) for the same parameters, showing high infection risk. (**c**) \(R_{0}>1\) for \(\Pi\) and \(\delta_{2}\), indicating potential malware spread. (**d**) \(R_{0}<1\) for \(\Pi\) and \(\delta_{2}\), showing conditions for containment.