## 3 Evaluation of the Model
One of the most important aspects of epidemic modeling is to accurately capture the dynamics of the spread of the disease over time. Fractional derivatives allow us to incorporate memory and non-local effects that are not accounted for in classical models. For example, the fractional derivative can model the non-exponential decay of infectiousness of a disease over time, which is a common characteristic of many epidemics. Moreover, fractional derivatives have the capability to account for heterogeneity within populations, encompassing variations in susceptibility and contact rates among individuals. This becomes especially crucial when modeling the spread of diseases in populations characterized by diverse demographics. When it comes to data fitting and analysis, the fractional framework offers an additional parameter that enhances the flexibility of the epidemic system. For our study on HBV infection dynamics, we have opted to employ fractional derivatives to depict the intricacies of the system. Our objective is to investigate the impact of different input factors of HBV on the system and to examine whether the fractional derivative order can be utilized as a viable control parameter or not. Therefore, we represent our model through Caputo FD as  
\[\left\{ \begin{array}{rcl}\frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{S}&=& \Pi(1-v)(1-q\mathbb{C})-(f_{1}\mathrm{A}+f_{2}\mathbb{C}+f_{3}\mathrm{H}) \frac{\mathbb{S}}{\mathrm{N}}-(g_{1}+d)\mathbb{S}+g_{2}\mathrm{V},\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathrm{V}&=&\Pi v+g_{1}\mathbb{S}-(d+g_{ 2})\mathrm{V},\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{E}&=&(f_{1}\mathrm{A}+f_{2} \mathbb{C}+f_{3}\mathrm{H})\frac{\mathbb{S}}{\mathrm{N}}-(d+\epsilon)\mathbb{E },\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{A}&=&\epsilon\mathrm{E}-(d+h_{1}+ \delta_{1})\mathrm{A},\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{C}&=&h_{1}\mathrm{a}\mathrm{A}+ \Pi(1-v)q\mathbb{C}-(d+h_{2}+\delta_{2})\mathbb{C}+\sigma\mathrm{H},\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{H}&=&\delta_{1}\mathrm{A}+\delta _{2}\mathbb{C}-(d+\phi+\sigma)\mathrm{H},\\ \frac{1}{\mathbb{C}}D_{f}^{\mathbb{S}}\mathbb{R}&=&h_{1}(1-\alpha)\mathrm{A}+h_{ 2}\mathbb{C}+\phi\mathrm{H}-d\mathrm{R},\end{array} \right. \tag{6}\]  
where \(\beta\) is the order of the LC derivative and \(0<\beta\leq 1\), moreover, system (6) becomes system (5) for \(\beta=1\). The theory of fractional calculus offers a wide range of practical applications and provides a more precise representation of the dynamics of biological phenomena compared to integer-order derivatives. In the recent literature, several new fractional operators have been introduced. Our future work aims to explore the dynamics of HBV infection with the help of these novel operators, and we will further investigate and  
Here, we determine the expressions for the basic reproduction number, symbolized by \(\mathcal{R}_{0}\), with the help of the next generation method [24]. In our system of HBV infection, we have four infected compartments E, A, C, and H, thus \(m=4\), then we have  
\[\mathcal{F}=\left[ \begin{array}{cccc}0&\frac{S^{0}f_{1}}{S+\mathcal{V}^{0}}& \frac{S^{0}f_{2}}{S^{0}+\mathcal{V}^{0}}&\frac{S^{0}f_{3}}{S^{0}+\mathcal{V}^{0 }}\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right]\text{ \ and \ }\mathcal{V}=\left[ \begin{array}{cccc}q_{1}&0&0&0\\ -\epsilon&q_{2}&0&0\\ 0&-ah_{1}&q_{3}&-\sigma\\ 0&-\delta_{1}&-\delta_{2}&q_{4}\end{array} \right].\]  
The spectral radius of \(\mathcal{F}\mathcal{V}^{-1}\) is the reproduction number, given by  
\[\mathcal{R}_{0}=\frac{\epsilon\{d(1-v)+g_{2}\}f_{1}}{q_{1}q_{2}(d+g_{1}+g_{2 })}+\frac{\epsilon\{d(1-v)+g_{2}\}(ah_{1}q_{4}+\sigma\delta_{1})f_{2}}{q_{1}q _{2}(d+g_{1}+g_{2})(q_{3}q_{4}-\sigma\delta_{2})}+\frac{\epsilon\{d(1-v)+g_{2 }\}(ah_{1}\delta_{2}+q_{3}\delta_{1})f_{3}}{q_{1}q_{2}(d+g_{1}+g_{2})(q_{3}q_{4 }-\sigma\delta_{2})}.\]  
in which \(q_{1}=d+\epsilon,\ q_{2}=d+h_{1}+\delta_{1},\ q_{3}=d+h_{2}+\delta_{2}-\Pi(1-v )q\), and \(q_{4}=d+\phi+\sigma\).