## 3 Evaluation of the Model
We represent  
\[\mathcal{R}_{01}=\frac{\epsilon\{d(1-v)+g_{2}\}f_{1}}{q_{1}q_{2}(d+g_{1}+g_{2 })},\]  
\[\mathcal{R}_{02}=\frac{\epsilon\{d(1-v)+g_{2}\}(ah_{1}q_{4}+\sigma\delta_{1}) f_{2}}{q_{1}q_{2}(d+g_{1}+g_{2})(q_{3}q_{4}-\sigma\delta_{2})},\]  
and  
\[\mathcal{R}_{03}=\frac{\epsilon\{d(1-v)+g_{2}\}(ah_{1}\delta_{2}+q_{3}\delta_{ 1})f_{3}}{q_{1}q_{2}(d+g_{1}+g_{2})(q_{3}q_{4}-\sigma\delta_{2})}.\]  
The EE of (5) is indicated by  
\[\mathcal{D}^{\star}=(\mathrm{S}^{\star}(t),\mathrm{V}^{\star}(t),\mathrm{E}^ {\star}(t),\mathrm{A}^{\star}(t),\mathrm{C}^{\star}(t),\mathrm{H}^{\star}(t), \mathrm{R}^{\star}(t)),\]  
where \(\mathrm{S}^{\star}(t),\mathrm{V}^{\star}(t),\mathrm{E}^{\star}(t),\mathrm{A}^ {\star}(t),\mathrm{C}^{\star}(t),\mathrm{H}^{\star}(t)\), and \(\mathrm{R}^{\star}(t)\) represent the endemic states of the state variables and are calculated as  
\[\left\{ \begin{array}{ll}\mathrm{S}^{\star}=&\frac{(d(1-v)+g_{2})\mathrm{N}^{ \star}}{\mathcal{R}_{0}},\\ \mathrm{E}^{\star}=&\frac{d_{2}}{\epsilon}\mathrm{A}^{\star},\\ \mathrm{V}^{\star}=&\frac{(11+g_{1})(d(1-v)+g_{2})N^{\star}}{(d+g_{2})\mathrm{N }_{0}},\\ \mathrm{C}^{\star}=&\frac{(ah_{1}q_{4}+\sigma\delta_{1})\Delta^{\star}}{q_{3}q _{4}-\sigma\delta_{2}},\\ \mathrm{H}^{\star}=&\frac{(ah_{1}q_{2}+\sigma\delta_{1})\Delta^{\star}}{q_{3}q _{4}-\sigma\delta_{2}},\\ \mathrm{R}^{\star}=&\left\{\frac{(1-\delta)\mathrm{V}_{1}}{d}+\frac{(ah_{1}q_{4 }+\sigma\delta_{1})h_{2}}{d(q_{3}q_{4}-\sigma\delta_{2})}+\frac{(ah_{1}\delta_ {2}+q_{3}\delta_{1})\phi}{d(q_{3}q_{4}-\sigma\delta_{2})}\right\}\mathrm{A}^{ \star}.\end{array} \right.\]  
From system (6), we calculate the below  
\[\mathrm{A}^{\star}=\frac{\Pi(q_{3}q_{4}-\sigma\delta_{2})^{2}q_{1}q_{2}(d+g_{1 }+g_{2})(\mathcal{R}_{0}-1)\epsilon}{(\mathrm{L}_{1}+\mathrm{L}_{2}+\mathrm{ L}_{3}+\mathrm{L}_{4}+\mathrm{L}_{5}+\mathrm{L}_{6}+\mathrm{L}_{7})},\]  
in which  
\[\left\{ \begin{array}{ll}\mathrm{L}_{1}=&\epsilon^{2}\Pi(1-v)q(d+g_{2})\{(q_{ 3}q_{4}-\sigma\delta_{2})\sigma\delta_{1}h_{1}+a^{2}h_{1}^{2}q_{4}(q_{4}f_{2}+ \delta_{2}f_{3})+\sigma\delta_{1}^{2}(\sigma f_{2}+q_{3}f_{3})\},\\ \mathrm{L}_{2}=&ah_{1}\epsilon^{2}\Pi(1-v)q(d+g_{2})\{q_{4}((q_{3}q_{4}-\sigma \delta_{2})f_{1}+2\sigma\delta_{1}f_{2})+\delta_{1}(q_{3}q_{4}-\sigma\delta_{2 })f_{2}\},\\ \mathrm{L}_{3}=&h_{1}\epsilon q_{1}q_{2}q_{4}(q_{3}q_{4}-\sigma\delta_{2})\{( \epsilon(d+h_{2}-\Pi(1-v)q)+(1-\alpha)q_{3})(d+\tau_{1}+\tau_{2})+\alpha f_{2} (d+\tau_{2})\},\\ \mathrm{L}_{4}=&q_{1}q_{2}(q_{3}q_{4}-\sigma\delta_{2})\delta_{2}\{\alpha f_{3}( d+g_{2})+(d+g_{1}+g_{2})(\alpha q_{4}-\sigma)\},\\ \mathrm{L}_{5}=&q_{1}q_{2}(q_{3}q_{4}-\sigma\delta_{2})^{2}\{\epsilon(f_{1}(d+g_ {2})+(d+g_{1}+g_{2})(d(\epsilon+q_{2})-q_{1}q_{2})\},\\ \mathrm{L}_{6}=&q_{1}q_{2}(q_{3}q_{4}-\sigma\delta_{2})\epsilon\sigma\delta_{1}\{( d+h_{2}-\Pi(1-v)q)(d+g_{1}+g_{2})+(d+g_{2})f_{2}\},\\ \mathrm{L}_{7}=&q_{1}q_{2}(q_{3}q_{4}-\sigma\delta_{2})\epsilon q_{3}\delta_{1}\{( d+\phi)g_{1}+(d+g_{2})(d+\phi+f_{3})\}.\end{array} \right.\]  
In the next step, we determine the local stability of the DFE of our system of HBV infection. **Theorem 3**. : _If \(\mathcal{R}_{0}<1\), then \(\mathcal{E}_{0}\) of the recommended HBV model presented in (5) is locally asymptotically stable \((LAS)\)._  
Proof of Theorem 3.: The Jacobian matrix associated with the model, denoted by \(J_{\mathcal{E}_{0}}\), evaluated around the FDE of system (6) is as follows:  
\[J_{\mathcal{D}_{0}}=\left( \begin{array}{cccccc}-d-\gamma_{1}&0&-\frac{(d- \mathrm{rd}+\gamma_{2})f_{1}}{d_{1}}&-\frac{(d-\mathrm{rd}+g_{2})f_{2}}{d_{1}} -\Pi(q-q_{q})&-\frac{(d-\mathrm{rd}+g_{2})f_{2}}{d_{1}}&g_{2}&0\\ 0&-q_{1}&\frac{(d(1-\gamma)+g_{2})f_{1}}{d+g_{1}+g_{2}}&\frac{(d(1-\gamma)+g_{ 2})f_{2}}{d+g_{1}+g_{2}}&\frac{(d(1-\gamma)+g_{2})f_{2}}{d+g_{1}+g_{2}}&0&0\\ 0&\epsilon&-q_{2}&0&0&0&0\\ 0&0&\dot{a}h_{1}&-q_{3}&\sigma&0&0\\ 0&0&\delta_{1}&\delta_{2}&-q_{4}&0&0\\ g_{1}&0&0&0&-d-g_{2}&0\\ 0&0&(1-\alpha)h_{1}&h_{2}&\phi&0&-d\end{array}