## 2 Mathematical Model and Analysis  
### Model of Hepatitis B Virus (HBV) Drug Therapy  
To understand the various modes of action of antiviral therapy, we modify the model of acute infection published in [14] by considering the combined effects of NAs and interferon-\(\alpha\) drugs. As in [14], we consider five state variables, corresponding to uninfected hepatocytes (\(T\)), productively-infected hepatocytes (\(I\)), free virus (\(V\)), immune effector cells (\(E\)) and a population of refractory hepatocytes (\(R\)). All hepatocyte populations, uninfected, infected and refractory to reinfection, are maintained by homeostasis described by a logistic equation, with carrying capacity \(K\) and maximal per capita growth rate \(r\). Virus infects target cells at rate \(\beta\). Infected cells are killed by the immune responses at rate \(\mu\). As in [14], we assume that infected cells can be lost due to the non-cytolytic response at rate \(\rho\), dependent on the effector cell population \(E\)[34], and move into a refractory class \(R\). Refractory cells will still be assayed as infected, since surface antigens persist for some time [35]. We assume that they have lost their viral replicates and do not produce virus, which also makes them poor targets for cytotoxic T lymphocyte (CTL) responses. Therefore, the refractory population will be killed at a smaller rate, \(\
u<\mu\). The refractory state is not permanent, and the \(R\) population may eventually become susceptible to reinfection. We model this by allowing \(R\) cells to move into the uninfected population at rate \(q\). Free virus is produced at rate \(\pi\) and cleared at rate \(c\). In the absence of infection, we assume the immune effector cells \(E\) are at equilibrium value \(s/d\), where \(s\) corresponds to a source of effector cells specific for HBV and \(1/d\) is their average life-span. Upon encountering antigen (on the surface of infected liver cells), these cells expand at rate \(\alpha\), with a constant time delay \(\tau\) accounting for the lag between antigen encounter and effector cell expansion. We model drug therapy as interference with virus production and infection and as immune modulation, as follows. The contributions of interferon-\(\alpha\) are complex, with both direct antiviral and immunomodulatory effects. Here, we consider three effects of IFN-\(\alpha\): enhancement of effector cell killing rates, reduction of viral production rate and the delay of effector cell programmed death [36; 37]. We model this by changing rates \(\mu\), \(\pi\) and \(d\) to \(\mu_{1}=\mu(1+a_{1}\epsilon)\), \(\tau_{1}^{0}=\pi(1-a_{2}\epsilon)\) and \(d_{1}=d(1-a_{3}\epsilon)\), respectively. In each case, \(0\leq\epsilon\leq 1\) represents the efficacy of IFN-\(\alpha\), and \(a_{i}\geq 0\) are scalar parameters representing the strength of the corresponding effect of interferon. \(a_{1}\) can be any nonnegative number, while \(a_{2}\) and \(a_{3}\) are between \(0\) and \(1\). The nucleos(t)ide analogues, on the other hand, interfere with both the ability of virus to infect a cell and with the generation of HBV DNA by an infected cell [38]. We model this by assuming that the infectivity rate in the presence of NAs becomes \(\beta_{1}=\beta(1-b_{1}\eta)\), and the viral production rate becomes \(\pi_{1}=\pi(1-fa_{2}\epsilon-(1-f)b_{2}\eta)\). Here, \(0\leq\eta\leq 1\) represents NAs' efficacy; \(b_{i}\geq 0\) are scalar parameters representing the relative strength of the corresponding effect of NAs; and \(f\) and \(1-f\) represent the relative contribution of interferon-\(\alpha\) and NAs, respectively, to reducing viral production, for \(0\leq f\leq 1\). For \(t>t_{d}\), where \(t_{d}\) represents the time of therapy onset, the dynamics of these populations is governed by the following differential equations  
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