## 3 Stability Analysis of Equilibria
### The Commensurate Case  
The next step is to establish sufficient conditions regarding the long-term stability of the two fixed points of System (12). Prior to examining stability, the following theorem is needed:  
**Theorem 2** ([32]). : _Take into account the fractional difference system_  
\[{}^{c}\Delta_{\rho}^{\eta}\chi(\sigma)=\mathcal{G}\chi(\sigma+\eta-1), \tag{18}\]  
_where \(\chi(\sigma)=(\chi_{1}(\sigma),\ldots,\chi_{n}(\sigma))^{\intercal},0<\eta<1\), and \(\gamma_{j}\), \((j=1,2,\ldots,n)\), are the eigenvalues of \(\mathcal{G}\in\mathbb{R}^{n\times n}\). If all \(\gamma_{j}\), fulfill this condition_  
\[\gamma_{j},\in\bigg{\{}\theta\in\mathbb{C}:|\theta|<\bigg{(}2\cos\frac{|\arg( \theta)|-\pi}{2-\eta}\bigg{)}^{\eta},\quad\text{and}\quad|\arg(\theta)|> \frac{\eta\pi}{2}\bigg{\}}, \tag{19}\]_then the asymptotic stability of the zero solution of (18) is achieved._  
The zero equilibrium point's stability in fractional discrete maps is able to be assessed following Theorem 2. However, for non-zero equilibrium points, a distinct approach is necessary. To this end, the following change of variables is considered. \[ \begin{cases}X_{0}(\sigma+\eta-1)=S(\sigma+\eta-1)-S_{0},\quad Y_{0}(\sigma+ \eta-1)=I(\sigma+\eta-1)-I_{0},\\ X_{*}(\sigma+\eta-1)=S(\sigma+\eta-1)-S_{*},\quad Y_{*}(\sigma+\eta-1)=I(\sigma+ \eta-1)-I_{*}.\end{cases}  \tag{20}\]  
The transformation yields two new maps, both of which have a zero equilibrium  
\[ \begin{cases}{}^{c}\Delta_{\rho}^{\eta}(X_{0}(\sigma)+S_{0})&=b-\lambda(X_{0} (\sigma+\eta-1)+\frac{b}{d})Y_{0}(\sigma+\eta-1)-d(X_{0}(\sigma+\eta-1)+\frac{ b}{d}),\\ {}^{c}\Delta_{\rho}^{\eta}(Y_{0}(\sigma)+I_{0})&=\lambda(X_{0}(\sigma+\eta-1) +\frac{b}{d})Y_{0}(\sigma+\eta-1)-eY_{0}(\sigma+\eta-1)-dY_{0}(\sigma+\eta-1),\end{cases}  \tag{21}\]  
and  
\[ \begin{cases}{}^{c}\Delta_{\rho}^{\eta}(X_{*}(\sigma)+S_{*})=b-\lambda(X_{*}( \sigma+\eta-1)+S_{*})(Y_{*}(\sigma+\eta-1)+I_{*})-d(X_{*}(\sigma+\eta-1)+S_{*}),\\ {}^{c}\Delta_{\rho}^{\eta}(Y_{*}(\sigma)+I_{*})=\lambda(X_{*}(\sigma+\eta-1)+ S_{*})(Y_{*}(\sigma+\eta-1)+I_{*})-\epsilon(Y_{*}(\sigma+\eta-1)+I_{*})-d(Y_{*}( \sigma+\eta-1)+I_{*}).\end{cases}  \tag{22}\]  
**Proposition 1**. : _The virus-free fixed point, denoted \(E_{0}\), of Model (12) is shown to be stable in a local asymptotic sense if the conditions outlined below are verified_  
\[d<2^{\eta},\quad R_{0}<1,\quad|(\epsilon+d)(R_{0}-1)|<2^{\eta}, \tag{23}\]  
Proof. : Determining the Jacobian matrix for model (21) at the zero equilibrium point yields  
\[\mathcal{J}_{0}= \begin{pmatrix}-d&-\frac{\lambda b}{d}\\ 0&\frac{\lambda b}{d}-\epsilon-d\end{pmatrix} , \tag{24}\]  
the corresponding characteristic equation:  
\[(-d-\gamma)\bigg{(}\frac{\lambda b}{d}-\epsilon-d-\gamma\bigg{)}=0, \tag{25}\]  
the eigenvalues of \(\mathcal{J}_{0}\) are \(\gamma_{1}=-d\) and \(\gamma_{2}=\frac{\lambda b}{d}-\epsilon-d=(\epsilon+d)(R_{0}-1)\). For \(\gamma_{1}=-d\)  
\[ \begin{cases}|\gamma_{1}|=d,\\ |\arg\gamma_{1}|=\pi>\frac{\kappa\pi}{2},\end{cases}  \tag{26}\]  
the condition \(|\gamma_{1}|<\left(2\cos\frac{|\arg\gamma_{1}|-\pi}{2-\eta}\right)^{\eta}=2^{\eta}\) is satisfied if \(d<2^{\eta}\). For \(\gamma_{2}=(\epsilon+d)(R_{0}-1)\)  
\[ \begin{cases}|\gamma_{2}|=|(\epsilon+d)(R_{0}-1)|,\\ |\arg\gamma_{2}|=\pi,\end{cases}  \tag{27}\]  
the condition \(|\gamma_{2}|<\left(2\cos\frac{|\arg\gamma_{2}|-\pi}{2-\eta}\right)^{\eta}=2^{\eta}\) is satisfied if \(|(\epsilon+d)(R_{0}-1)|<2^{\eta}\), the condition \(|\arg\gamma_{2}|>\frac{\kappa\pi}{2}\), is satisfied if \(\gamma_{2}<0\) i.e., \(R_{0}<1\). So the hypothesis of Theorem 13 is satisfied; the virus-free fixed point of (12) reveals local asymptotic stability. **Proposition 2**.