## 2 Mathematical Model and Analysis
### Numerical Results  
We assume that in the absence of drug therapy, the virus will persist, and the patient will experience chronic infection. We therefore set all parameters values and initial conditions to the values for the chronically-infected Patient 7 in [14] (see Table 1). The dynamics of the five variables in the absence of drug therapy, \(\epsilon=\eta=0\), throughout acute infection and transition to chronic disease is presented in Figure 1.  
We next investigate the change in the model dynamics in the presence of treatment. Initially, we set time-constant drug efficacy, \(\eta=0.5\) and \(\epsilon=0.9\) as in [17]; assume scalars \(a_{i}\) and \(b_{i}\) to be either 1 or 0, representing an effect or lack of effect in therapy; and set \(f=0.5\). We define HBV DNA clearance as the presence of less than one HBV DNA in the host serum. Since we assume that HBV DNA can distribute throughout the 3 liters of serum in an average 70-kg person, the viral extinction becomes \(V\leq V_{ext}=3\times 10^{-4}\) copies per mL. We notice that the speed of viral extinction is dependent on both the timing of therapy initiation and on the type of effects considered. Indeed, when the therapy is started at \(t_{d}=T_{peak}=96\) days post-infection (corresponding to the peak HBV DNA), our model predicts HBV DNA clearance 145 days after the start of therapy when \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\) and \(f=0.5\) (see Figure 2a, dashed line) and 2226 days (6.1 years) after the start of therapy when \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(a_{3}=0\) and \(f=0.5\) (see Figure 2a, dotted line). This suggests that the immune modulation effect of interferon is important in clearing the HBV DNA in a short amount of time. Similar immune-mediated effects of interferon are observed when drug therapy is started at the chronic state of Model (1), \(t_{d}=T_{\rm ss}=20\) years post-infection. Indeed, viremia clearance occurs faster when \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\) and \(f=0.5\) (see Figure 3a, dashed line) than when \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(a_{3}=0\) and \(f=0.5\) (see Figure 3a, dotted lines). However, viral clearance is delayed by 12 days compared to the case in which therapy is initiated at peak HBV DNA. The rapidity of clearance during peak therapy is due to the transient effects of immune cells, which are \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & \\ \hline \(T\) & Target cells & \(T_{0}=13.6\times 10^{6}\) per mL \\ \(I\) & Infected cells & \(I_{0}=0\) per mL \\ \(V\) & Free virus & \(V_{0}=0.33\) per mL \\ \(E\) & Effector cells & \(E_{0}=60\) per mL \\ \(R\) & Refractory cells & \(R_{0}=0\) per mL \\ \hline \multicolumn{3}{l}{Parameters} \\ \hline \(r\) & Hepatocyte maximum proliferation rate & \(1\) day\({}^{-1}\) \\ \(\beta\) & Infectivity rate constant & \(1.22\times 10^{-10}\) mL (virion \(\times\) day)\({}^{-1}\) \\ \(K\) & Hepatocyte carrying capacity & \(13.6\times 10^{6}\) cells per mL \\ \(\mu\) & Infected cell killing rate & \(1.2\times 10^{-4}\) mL (cell \(\times\) day)\({}^{-1}\) \\ \(\
u\) & Refractory cell killing rate & \(1.27\times 10^{-5}\) mL (cell \(\times\) day)\({}^{-1}\) \\ \(\rho\) & Cure rate & \(3.38\times 10^{-4}\) mL (cell \(\times\) day)\({}^{-1}\) \\ \(\alpha\) & Effector cell expansion rate & \(2\times 10^{-7}\) mL (cell \(\times\) day)\({}^{-1}\) \\ \(\tau\) & Delay & \(33.4\) days \\ \(\pi\) & Virus production rate & \(164\) virion (cell \(\times\) day)\({}^{-1}\) \\ \(c\) & Virus clearance rate & \(0.67\) day\({}^{-1}\) \\ \(s\) & Effector cell production & \(10\) day\({}^{-1}\) \\ \(d\) & Effector cell clearance rate & \(0.5\) day\({}^{-1}\) \\ \(q\) & Waning of refractory cell immunity & \(2\times 10^{-5}\) day\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables, parameters and values used in simulations with milliter (mL) and day (d). activated and expanding at the peak HBV DNA (see Figure 2b, solid lines) and tolerant at the chronic HBV infection (see Figure 3b, solid lines).