# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 4 Equilibrium Analysis and Stability of the Malware Spread Model
For global stability when \(\mathcal{R}_{0}<1\), consider the Lyapunov function:  
\[\mathcal{L}(t)=M_{i}(t) \tag{9}\]  
The time derivative of \(\mathcal{L}\) along the solutions of system (1) is  
\[\dot{\mathcal{L}} =\dot{M}_{i} \tag{10}\] \[=\Pi M_{\rm s}M_{i}-(\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2}) M_{i}\] (11) \[\leq\Pi\frac{\lambda}{\delta_{1}}M_{i}-(\delta_{1}+\delta_{2}+ \theta_{1}+\theta_{2})M_{i}\] (12) \[=(\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2})(\mathcal{R}_{0}-1 )M_{i} \tag{13}\]  
Since \(\mathcal{R}_{0}<1\), we have \(\dot{\mathcal{L}}\leq 0\) for all \(M_{i}>0\), with equality only when \(M_{i}=0\). By [PERSON]'s invariance principle, all solutions converge to the largest invariant set where \(\dot{\mathcal{L}}=0\), which is precisely \(MFE_{0}\). Therefore, \(MFE_{0}\) is globally asymptotically stable when \(\mathcal{R}_{0}<1\). **Theorem 3**. : _The endemic equilibrium point \(SME^{*}\) of system (1) is_  
* _locally asymptotically stable when_ \(\mathcal{R}_{0}>1\)_._
* _globally asymptotically stable when_ \(\mathcal{R}_{0}>1\)_._  
Proof. : For local stability analysis, we compute the Jacobian matrix at \(SME^{*}\):  
\[J(SME^{*})= \begin{pmatrix}-\delta_{1}-\[PERSON]_{i}^{*}&-\[PERSON]_{\rm s}^{*}&0&0\\ \[PERSON]_{i}^{*}&\[PERSON]_{\rm s}^{*}-(\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2})& 0&0\\ 0&\theta_{1}&-(\delta_{1}+\delta_{3}+q)&0\\ 0&\theta_{2}&q&-\delta_{1}\end{pmatrix} = \begin{pmatrix}A&\mathbf{0}\\ C&B\end{pmatrix}  \tag{14}\]  
where  
\[A = \begin{pmatrix}-\delta_{1}-\[PERSON]_{i}^{*}&-\[PERSON]_{\rm s}^{*}\\ \Pi M_{i}^{*}&\Pi M_{\rm s}^{*}-(\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2}) \end{pmatrix}  \tag{15}\] \[B = \begin{pmatrix}-(\delta_{1}+\delta_{3}+q)&0\\ q&-\delta_{1}\end{pmatrix} \] (16) \[C = \begin{pmatrix}0&\theta_{1}\\ 0&\theta_{2}\end{pmatrix}  \tag{17}\]  
The eigenvalues of \(J(SME^{*})\) are the eigenvalues of matrices \(A\) and \(B\). \begin{table}
\begin{tabular}{c c c c} \hline \(\lambda_{1}\) & \(\lambda_{2}\) & \(\lambda_{3}\) & \(\lambda_{4}\) \\ \hline \(-\delta_{1}\) & \((\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2})(\mathcal{R}_{0}-1)\) & \(-(\delta_{1}+\delta_{3}+q)\) & \(-\delta_{1}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Eigenvalues of the Jacobian matrix at \(MFE_{0}\). Since \(\mathcal{R}_{0}>1\), we can show that \(\text{tr}(A)<0\) and \(\det(A)>0\). This ensures that \(\lambda_{3}\) and \(\lambda_{4}\) have negative real parts. Combined with \(\lambda_{1},\lambda_{2}<0\), all eigenvalues have negative real parts, as shown in Table 3. By the Routh-Hurwitz criterion, \(SME^{*}\) is locally asymptotically stable. For global stability when \(\mathcal{R}_{0}>1\), consider the following Lyapunov function:  
\[\mathcal{L}=\frac{(M_{\text{s}}-M_{\text{s}}^{*})^{2}}{2}+M_{\text{s}}^{*}M_{ \text{i}}^{*}\left(\frac{M_{i}}{M_{i}^{*}}-\ln\frac{M_{i}}{M_{i}^{*}}-1\right) \tag{18}\]  
The time derivative of \(\mathcal{L}\) along the solutions of system (1) is  
\[\dot{\mathcal{L}}=-(\delta_{1}+\Pi M_{i})(M_{\text{s}}-M_{\text{s}}^{*})^{2} \tag{19}\]  
Since \(M_{i}\geq 0\) and \(\delta_{1}>0\), we have \(\dot{\mathcal{L}}\leq 0\) for all values in the feasible region, with equality only at \(SME^{*}\). By [PERSON]'s invariance principle, all solutions converge to \(SME^{*}\), proving that the endemic equilibrium is globally asymptotically stable when \(\mathcal{R}_{0}>1\). The stability analysis presented above provides a comprehensive understanding of the long-term behavior of the mobile malware propagation model. When \(\mathcal{R}_{0}<1\), the malware-free equilibrium is both locally and globally asymptotically stable, indicating that the malware will eventually be eradicated from the system. When \(\mathcal{R}_{0}>1\), the endemic equilibrium becomes both locally and globally asymptotically stable, meaning that malware will persist in the system at levels determined by the endemic equilibrium. These theoretical results have significant practical implications for mobile security strategies.