# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 4 Equilibrium Analysis and Stability of the Malware Spread Model  
This involves proving that the total number of devices does not decrease below zero and remains within a certain bound. Equilibrium points are crucial as they represent the states where the system remains constant over time, meaning there are no changes in the number of devices in each compartment. Let the system (1) (right-hand side) equal to zero:  
\[(\dot{M}_{s},\,\dot{M}_{i},\,\dot{M}_{q},\,\dot{M}_{r})=\bar{\mathbf{0}} \tag{2}\]  
A trivial solution of (2) is the malware-free equilibrium point \(MFE_{0}=\left(\frac{\lambda}{\delta_{1}},\,0,\,0,\,0\right)\). The second equilibrium point, representing a successful mobile attack equilibrium (SME), can be derived by solving the system of equations. Using analytical methods, we obtain  
\[M_{s}^{*} =\frac{\lambda}{\delta_{1}}\frac{1}{\mathcal{R}_{0}} \tag{3}\] \[M_{i}^{*} =\frac{\delta_{1}}{\Pi}(\mathcal{R}_{0}-1)\] (4) \[M_{q}^{*} =\frac{\theta_{1}\delta_{1}}{(\delta_{1}+\delta_{3}+q)\Pi}( \mathcal{R}_{0}-1)\] (5) \[M_{r}^{*} =\frac{\theta_{2}M_{i}^{*}+qM_{q}^{*}}{\delta_{1}} \tag{6}\]  
where the basic reproduction number \(\mathcal{R}_{0}\) is given by  
\[\mathcal{R}_{0}=\frac{\lambda\Pi}{\delta_{1}(\delta_{1}+\delta_{2}+\theta_{1} +\theta_{2})} \tag{7}\]  
This expression for \(\mathcal{R}_{0}\) has a clear epidemiological interpretation: it represents the product of  
* \(\frac{\lambda}{\delta_{1}}\): the average number of susceptible mobile devices in the absence of malware;
* \(\frac{\Pi}{\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2}}\): the infection contribution of a single infected device during its infectious period. By substituting the expression for \(M_{i}^{*}\) into the formula for \(M_{r}^{*}\), we can obtain an explicit expression as follows:  
\[M_{r}^{*} =\frac{\theta_{2}M_{i}^{*}+qM_{q}^{*}}{\delta_{1}}=\frac{\theta_{ 2}\frac{\delta_{1}}{\Pi}(\mathcal{R}_{0}-1)+q\frac{\theta_{1}\delta_{1}}{( \delta_{1}+\delta_{3}+q)\Pi}(\mathcal{R}_{0}-1)}{\delta_{1}}\] \[=\frac{\delta_{1}}{\Pi\delta_{1}}(\mathcal{R}_{0}-1)\bigg{[} \theta_{2}+\frac{q\theta_{1}}{(\delta_{1}+\delta_{3}+q)}\bigg{]}=\frac{1}{ \Pi}(\mathcal{R}_{0}-1)\bigg{[}\theta_{2}+\frac{q\theta_{1}}{(\delta_{1}+ \delta_{3}+q)}\bigg{]}\]  
Importantly, the endemic equilibrium point \(SME\) only exists when \(\mathcal{R}_{0}>1\) as this ensures all compartments are non-negative. Introducing these equilibrium points helps us to understand the long-term behavior of the system and the conditions under which the malware infection can be eradicated or persist within the mobile network. Corollary 1: _The system (1) has exactly two equilibrium points:_  
* _Malware-free equilibrium:_ \(MFE_{0}=\left(\frac{\lambda}{\delta_{1}},\,0,\,0,\,0\right)\) _when_ \(\mathcal{R}_{0}\leq 1\)_._
* _Successful mobile attack equilibrium:_ \(SME^{*}=(M_{s}^{*},\,M_{i}^{*},\,M_{q}^{*},\,M_{r}^{*})\) _when_ \(\mathcal{R}_{0}>1\)_._  
Theorem 2: _For the malware-free equilibrium \(MFE_{0}=\left(\frac{\lambda}{\delta_{1}},\,0,\,0,\,0\right)\) of system (1), the following statements hold:_  
* \(MFE_{0}\) _is locally asymptotically stable when_ \(\mathcal{R}_{0}<1\)_._
* \(MFE_{0}\) _is unstable when_ \(\mathcal{R}_{0}>1\)_._
* \(MFE_{0}\) _is globally asymptotically stable when_ \(\mathcal{R}_{0}<1\)_._  
Proof: To determine the local stability, we evaluate the Jacobian matrix at \(MFE_{0}\):\[J(MFE_{0})= \begin{pmatrix}-\delta_{1}&-\Pi\frac{\lambda}{\delta_{1}}&0&0\\ 0&\Pi\frac{\lambda}{\delta_{1}}-(\delta_{1}+\delta_{2}+\theta_{1}+\theta_{2})&0 &0\\ 0&\theta_{1}&-(\delta_{1}+q+\delta_{3})&0\\ 0&\theta_{2}&q&-\delta_{1}\end{pmatrix}  \tag{8}\]  
The eigenvalues of \(J(MFE_{0})\) are shown in Table 2.  
When \(\mathcal{R}_{0}<1\), all eigenvalues listed in Table 2 have negative real parts, confirming that \(MFE_{0}\) is locally asymptotically stable. When \(\mathcal{R}_{0}>1\), \(\lambda_{2}>0\), making \(MFE_{0}\) unstable.