## 3 Evaluation of the Model  
To construct the dynamics of HBV, we indicate the total human population by N\((t)\), which is further distributed in seven different classes: susceptible, vaccinated, exposed, acutely infected, chronic HBV case, hospitalized, and recovered individuals, expressed by S\((t)\), V\((t)\), E\((t)\), A\((t)\), C\((t)\), H\((t)\), and R\((t)\), respectively. The birth rate of newborn babies (II) generates the susceptible population of humans, and successfully immunized newborns (\(\Pi v\), where \(0<v<1\)) transfer to V\((t)\) while the un-immunized newborn babies are expressed, with a factor \(q\), as carriers of the HBV infection. In addition to this, the fraction of newborns \((\Pi(q-qv)\mathrm{C})\) joins the chronic class, members of which are not fully immunized. This population is decreased due to the effective contact of the susceptible individuals with the population that are infected with HBV at a rate \(\frac{(f_{1}A+f_{2}C+f_{3}\mathrm{H})}{\mathrm{N}}\), where \(f_{1},f_{2}\), and \(f_{3}\) are the different rates of transmission due to A\((t)\), C\((t)\), and H\((t)\); moreover, the said portion moves to \((\mathrm{E}(t))\) to generate the exposed class. The vaccinated people leave this class and join the vaccinated class at the rate \(g_{1}\). In addition, those vaccinated individuals who lost their immunity join the S\((t)\) class at the rate \(g_{2}\). The rate of death which occurs naturally, is considered to be \(d\) in all epidemiological classes. The number of individuals in E\((t)\) decreases by the transfer of exposed people to the acutely infected class at the rate \(\epsilon\) and also the rate \(\mathrm{d}\). Similarly, the acutely infected A\((t)\) population is decreased at the rate \(\mathrm{d}\) and treatment rate \(\delta_{1}\). Furthermore, this class is also decreased by \(\alpha h_{1}\) to C\((t)\) and a portion \((1-\alpha)h_{1}\), where \(0<\alpha<1\), to the recovered class. The C\((t)\) class is also joined by the hospitalized at a rate \(\sigma\), which expresses medication failure in the hospitalized class. It is also reduced by the rate \(\mathrm{d}\) and the rate of recovery \(h_{2}\)as well as by medication at \(\delta_{2}\). The hospitalized class \(\mathrm{H}(t)\) is reduced by the rate \(\mathrm{d}\) and recovery of individuals at the rate \(\phi\), as well as the incomplete rate of medication \(\sigma\). Finally, the remaining two classes, of vaccination \(\mathrm{V}(t)\) and recovered \(\mathrm{R}(t)\), are decreased by the death rate \(d\). By utilizing all of this above-mentioned information about the dynamics of a life threatening infection, we have the following system of differential equations for the vaccinated and hospitalized classes. Consequently, all of the equations may be expressed as the nonlinear ODE system:  
\[\left\{ \begin{array}{rcl}\frac{d\mathbb{S}}{dt}&=&\Pi(1-v)(1-q\mathbb{C})-( f_{1}\mathrm{A}+f_{2}\mathbb{C}+f_{3}\mathrm{H})\frac{\mathbb{S}}{\mathrm{N}}-(g_{1}+d) \mathbb{S}+g_{2}\mathrm{V},\\ \frac{d\mathbb{S}}{dt}&=&\Pi v+g_{1}\mathbb{S}-(d+g_{2})\mathrm{V},\\ \frac{d\mathbb{S}}{dt}&=&(f_{1}\mathrm{A}+f_{2}\mathbb{C}+f_{3}\mathrm{H}) \frac{\mathbb{S}}{\mathrm{N}}-(d+\epsilon)\mathbb{E},\\ \frac{d\mathbb{A}}{dt}&=&\epsilon\mathrm{E}-(d+h_{1}+\delta_{1})\mathrm{A},\\ \frac{d\mathbb{S}}{dt}&=&h_{1}\mathrm{a}\mathrm{A}+\Pi(1-v)q\mathbb{C}-(d+h_{2 }+\delta_{2})\mathbb{C}+\sigma\mathrm{H},\\ \frac{d\mathbb{S}}{dt}&=&h_{1}(1-\alpha)\mathrm{A}+h_{2}\mathbb{C}+\phi \mathrm{H}-d\mathrm{R},\end{array} \right. \tag{5}\]  
with  
\[\mathbb{S}(0)=\mathbb{S}_{0}\geq 0,\ \ \mathrm{V}(0)=\mathbb{V}_{0}\geq 0,\ \ \mathrm{E}(0)=\mathbb{E}_{0}\geq 0,\ \ \mathrm{A}(0)=\mathbb{A}_{0}\geq 0,\]  
and  
\[\mathbb{C}(0)=\mathbb{C}_{0}\geq 0,\mathrm{H}(0)=\mathrm{H}_{0}\geq 0,\ \ \mathrm{R}(0)=\mathbb{R}_{0}\geq 0.\]  
A fractional derivative is a generalization of an ordinary derivative that can capture the memory effects and long-range correlations present in complex biological phenomena and epidemics.