## 6 Numerical Results and Discussions
By analyzing the model, conducting simulations, and fitting the model to real-world data, researchers can gain insights into the key factors driving the epidemic's dynamics and make informed decisions to mitigate its impact. For simulation purposes, the state variables are assumed to be \(\mathrm{S}(0)=0.8,\mathrm{V}(0)=0.3,\mathrm{E}(0)=0.3,\mathrm{A}(0)=0.3, \mathrm{C}(0)=0.3,\mathrm{H}(0)=0.3\), and \(\mathrm{R}(0)=0.3\). In Figure 1, we illustrate the impact of the fractional parameter \(\beta\) on the solution pathways of the system of HBV infection to check whether this parameter of the system can be used as a preventive measure or not. The values of \(\beta\) are chosen to be 0.7, 0.8, 0.9, and 1.0 in the first simulation. It can be seen that the infection level can be decreased by decreasing the value of the input parameter \(\beta\). We noticed that this input factor contributes significantly and can be used as a control parameter, therefore, suggested this to the policymakers. In Figure 1, a comparative analysis of integer and non-integer derivatives are also illustrated. The curve for \(\beta=1\) represents the integer case, while curves with other values of \(\beta\) demonstrate non-integer cases. The effect of vaccination on the dynamics of HBV infection is shown in \[\mathrm{x}(t_{n+1})-\mathrm{x}(0)=\frac{1}{\Gamma(\beta)}\int_{0}^{t_{n+1}}(t_{ n+1}-t)^{\beta-1}\hbar(t,\mathrm{x}(t))dt, \tag{37}\]0 In this simulation, we assumed different values of the vaccination factor \(v\), i.e., \(v=0.45\), 0.55, 0.65, and 0.75. We observed that an increase in \(v\) decreases the exposed and chronically infected individuals of the system. We recommended increasing the efficiency of vaccination for better control of the disease. In Figure 3, the effect of the loss rate of immunity due to vaccination is illustrated on the vaccinated and exposed individuals of the system. It can be seen that this factor is dangerous and can increase the risk of infection in society. To visualize the impact of \(h_{1}\) and \(h_{2}\) on the system, we performed two different simulations, shown in Figures 4 and 5. In Figure 4, we take \(h_{1}=0.019\), 0.039, 0.059, and 0.079 while the values of \(h_{2}\) are assumed to be 0.025, 0.050, 0.075, and 0.100 in \[\mathrm{x}(t_{n+1})-\mathrm{x}(0)=\frac{1}{\Gamma(\beta)}\int_{0}^{t_{n+1}}(t_{ n+1}-t)^{\beta-1}\hbar(t,\mathrm{x}(t))dt, \tag{37}\]1 It is clear from the figures how the infected individuals behave with the variation of these parameters. In the final simulation, presented in Figure 6, the solution pathways of the system are highlighted with the variation in transmission rate \(f_{1}\). In Figure 6, we assumed the values of \(f_{1}\) to be 0.033, 0.063, 0.093, and 0.123 to identify the role of \(f_{1}\) on the dynamics of HBV infection. We noticed that this parameter is critical and highly increases the endemic level of the infection in society. This implies that the transmission rate \(f_{1}\) is sensitive and needs to be controlled. It is evident that HBV is a dangerous infectious disease, therefore, effective interventions are needed to eradicate and eliminate this viral infection from society. The control of the transmission rate and fractional parameter will reduce the intensity of the infection. Therefore, the control of these factors is recommended to officials. We also suggest increasing the efficiency of vaccination and medication through hospitalization for better prevention of the infection. \[\mathrm{x}(t_{n+1})-\mathrm{x}(0)=\frac{1}{\Gamma(\beta)}\int_{0}^{t_{n+1}}(t_{ n+1}-t)^{\beta-1}\hbar(t,\mathrm{x}(t))dt, \tag{37}\]2 Illustration of the solution pathways of **(a)** vaccinated individuals, **(b)** exposed individuals, **(c)** acutely infected, **(d)** chronically infected and **(e)** hospitalized individuals of the recommended system (6) of HBV infection with different values of \(\beta\), i.e., \(\beta=0.7,0.8,0.9,1.0\).