# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 6 Optimal Control Strategy Framework
### Comprehensive Performance Evaluation and Comparative Analysis  
The control effectiveness is evaluated through multiple performance metrics comparing the baseline uncontrolled scenario against the optimal control implementation. The peak infected population \(M_{i}\) is reduced from 2634 devices in the uncontrolled case to 1847 devices under optimal control, representing a 29.9% reduction in maximum infection levels. Similarly, the peak quarantined population \(M_{q}\) decreases from 1285 devices to 892 devices, achieving a 30.6% reduction. The total infection duration is shortened from 45.2 days to 32.1 days, representing a 29.0% reduction in outbreak persistence. The area under the \(M_{i}\) curve, which represents the cumulative infection burden, is reduced from 47,850 device-days to 28,420 device-days, achieving a 40.6% reduction in total infection impact. The time to reach peak infection is delayed from 12.8 days to 15.3 days, providing a 19.5% extension that allows for better preparation and response. The recovery time to reach 90% of peak reduction is accelerated from 38.5 days to 25.7 days, representing a 33.2% faster recovery process. Resource cost analysis reveals significant economic benefits of the optimal control strategy. The contact reduction control cost is calculated as \(\int_{0}^{T}\frac{B_{1}}{2}u_{1}(t)^{2}dt=1247.8\), while the quarantine enhancement control cost is \(\int_{0}^{T}\frac{B_{2}}{2}u_{2}(t)^{2}dt=2156.3\), yielding a total control implementation cost of 3404.1. The damage cost savings are substantial, with infection damage avoided calculated as \(A_{1}\times\Delta(M_{i}\text{ area})=10\times 19,430=194,300\) and quarantine cost savings as \(A_{2}\times\Delta(M_{q}\text{ area})=5\times 8765=43,825\), resulting in total savings of 238,125. This yields an impressive cost-benefit ratio of 69.9:1, demonstrating the economic viability of the control strategy. Comparative benchmark analysis against alternative strategies demonstrates the superiority of the optimal control approach. The uncontrolled baseline scenario results in a peak \(M_{i}\) of 2634 devices with a duration of 45.2 days and zero control cost but also zero efficiency. A constant control strategy with \(u_{1}=0.4\) achieves a peak \(M_{i}\) of 2156 devices with 38.7 days in duration and total cost of 4500, yielding an efficiency score of 3.2. Another constant strategy with \(u_{2}=0.25\) produces a peak \(M_{i}\) of 2089 devices with 36.2 days duration and cost of 5625, achieving an efficiency score of 4.1. A periodic control strategy results in a peak \(M_{i}\) of 1978 devices with 34.8 days duration and cost of 6120, providing an efficiency score of 4.7. In contrast, the optimal control strategy achieves the best performance with a peak \(M_{i}\) of 1847 devices, duration of 32.1 days, cost of 3404, and the highest efficiency score of 6.8, where efficiency is calculated as damage reduction divided by control cost multiplied by \(10^{-3}\). Sensitivity analysis confirms the robustness of the optimal control strategy under parameter variations. When the infection rate \(\Pi\) varies by \(\pm 20\%\), the optimal control maintains infection reduction between 25-35%, demonstrating stability across different malware virulence levels. Variation in the base quarantine rate \(\theta_{1}\) by \(\pm 15\%\) results in control effectiveness variations of 12-18%, indicating reasonable sensitivity to quarantine infrastructure capabilities. Cost weight variations of \(\pm 50\%\) show that the control strategy adapts while maintaining an efficiency ratio greater than 45:1, confirming economic robustness. The baseline reproduction number \(R_{0}=5.56\) with peak infected population of 2634 devices reveals significant threat levels under standard parameter settings, emphasizing the critical need for proactive response planning and validating the importance of the proposed control framework. The baseline reproduction number is \(R_{0}=5.56\), with a peak infected population of 2634 devices.