## 3 Stability Analysis of Equilibria
### The Incommensurate Case  
The focus of this section is the discrete computer virus model with incommensurate fractional orders. In such systems, each equation is governed by a distinct fractional order. The model is represented as follows:  
\[ \begin{cases}{}^{c}\Delta_{\rho}^{\eta_{1}}S(\sigma)=b-\lambda S(\sigma+\eta_{1} -1)I(\sigma+\eta_{1}-1)-dS(\sigma+\eta_{1}-1),\\ {}^{c}\Delta_{\rho}^{\eta_{2}}I(\sigma)=\lambda S(\sigma+\eta_{2}-1)I(\sigma+ \eta_{2}-1)-\epsilon I(\sigma+\eta_{2}-1)-dI(\sigma+\eta_{2}-1).\end{cases}  \tag{36}\]  
Theorem 1 yields the following numerical model for system (36):  
\[ \begin{cases}S(\sigma)=S(0)+\frac{1}{\Gamma(\eta_{1})}\sum_{j=1}^{\sigma}\frac {\Gamma(\sigma-j+\eta_{2})}{\Gamma(\sigma-j+1)}[b-\lambda S(j-1)I(j-1)-dS(j-1)],\\ I(\sigma)=I(0)+\frac{1}{\Gamma(\eta_{2})}\sum_{j=1}^{\sigma}\frac{\Gamma( \sigma-j+\eta_{2})}{\Gamma(\sigma-j+1)}[\lambda S(j-1)I(j-1)-\epsilon I(j-1)- dI(j-1)].\end{cases}  \tag{37}\]  
Here, the stability properties of the discrete computer virus model with incommensurate orders are explored. The theorem presented below serves as a basis for investigating the stability of discrete fractional nonlinear models with incommensurate orders. **Theorem 3** ([33]). : _For the following system where \(0<\eta_{\iota}<1\) for \(\iota=1,2,\ldots,n\)._  
\[ \begin{cases}{}^{c}\Delta_{0}^{\eta_{1}}\chi_{1}(\sigma)=\mathrm{Y}_{1}(\chi( \sigma+\eta_{1}-1)),\\ {}^{c}\Delta_{0}^{\eta_{2}}\chi_{2}(\sigma)=\mathrm{Y}_{2}(\chi(\sigma+\eta_{2 }-1)),\\ {}^{c}\Delta_{0}^{\eta_{n}}\chi_{n}(\sigma)=\mathrm{Y}_{n}(\chi(\sigma+\eta_{n }-1)),\end{cases}  \tag{38}\]  
_where \(\mathrm{Y}=(\mathrm{Y}_{1},\mathrm{Y}_{2},\ldots,\mathrm{Y}_{n}):\mathbb{R}^ {n}\to\mathbb{R}^{n}\) and \(\chi(\sigma)=(\chi_{1}(\sigma),\chi_{2}(\sigma),\ldots,\chi_{n}(\sigma))^{ \mathrm{T}}\in\mathbb{R}^{n}\). \(M\) is the LCM of the denominators \(\xi_{\iota}\) of \(\eta_{\iota}\), where \(\eta_{\iota}=\frac{\xi_{\iota}}{\xi_{\iota}}\), \((\epsilon_{\iota},\xi_{\iota})=1\) and \(\epsilon_{\iota},\xi_{\iota}\in\mathbb{Z}_{+}\) for \(\iota=1,2,\ldots,n\). Put \(i=\frac{1}{M}\)._  
\[\det\Bigl{(}\text{diag}(\gamma^{M\eta_{1}},\gamma^{M\eta_{2}},\ldots,\gamma^{M \eta_{n}})-(1-\gamma^{M})\mathcal{J}\Bigr{)}=0, \tag{39}\]  
_such that \(\mathcal{J}\) is the Jacobian matrix of (38). If all eigenvalues of Equation (39) remain within \(\mathbb{C}\setminus\mathbb{L}^{i}\), Then, system (38) has a locally asymptotically stable trivial solution, such that_  
\[\mathbb{L}^{i}=\Biggl{\{}l\in\mathbb{C}:|l|\leq\biggl{(}2\cos\frac{|\arg(l)|}{ i}\biggr{)}^{i},\text{ and }|\arg(l)|\leq\frac{i\pi}{2}\Biggr{\}}, \tag{40}\]  
Figure 1: Evolution of states of the commensurate discrete Model (12): **(a)** Example 1, **(b)** Example 2, **(c)** Example 3. **Example 4**. : _Consider system (36) with parameters \((b,\lambda,d,e)=(5,0.01,0.2,0.13)\), it follows that_  
\[\mathcal{J}_{0}= \begin{pmatrix}-0.2&-0.25\\ 0&-0.75\end{pmatrix} , \tag{41}\]  
_Let \((\eta_{1},\eta_{2})=(0.68,0.86)\), which implies that \(M=50\),_  
\[\det\Biggl{(} \begin{pmatrix}\gamma^{34}&0\\ 0&\gamma^{43}\end{pmatrix} -(1-\gamma^{50})\mathcal{J}_{0}\Biggr{)}=0, \tag{42}\]  
_equivalent to_  
\[\frac{1}{20}(-\gamma^{50}+5\gamma^{34}+1)(-3\gamma^{50}+4\gamma^{43}+3)=0. \tag{43}\]  
_Solving Equation (43) yields 100 solutions, \(\gamma_{j}(j=1,\ldots,100)\). All solutions remain within \(\mathbb{C}\setminus\mathbb{L}^{\frac{1}{50}}\). Theorem 3 proves the local asymptotic stability of \(E_{0}\)._  
**Example 5**. : _Consider System (36) with parameters \((b,\lambda,d,e)=(5,0.04,0.2,0.13)\), it follows that_  
\[\mathcal{J}^{*}= \begin{pmatrix}-0.606&-0.33\\ 0.406&0\end{pmatrix} , \tag{44}\]  
_Let \((\eta_{1},\eta_{2})=(0.68,0.86)\), which implies that \(M=50\),_  
\[\det\Biggl{(} \begin{pmatrix}\gamma^{34}&0\\ 0&\gamma^{43}\end{pmatrix}