## 4 Discussion  
The management of chronic hepatitis B is based on guidelines regarding screening, diagnosis, duration of treatment, adherence and monitoring of immune and virological markers [12]. Combined antiviral therapy improves the survival rates of patients chronically infected with hepatitis B virus [42]. While the first goal of therapy is to reduce HBV replication, the ultimate goal is the removal of hepatitis B surface and e-antigens and of covalently-closed circular DNA (cccDNA). The removal of the hepatitis B e-antigen is associated with immunological factors, such as removal of the tolerant status of the hepatitis B-specific T cells [36]. Such effects have been demonstrated by interferon therapy, which leads to improved immune function, sustained response rates with combination therapy and improved overall prognosis [37]. These drugs have limitations, such as side effects, the use of injection and poor response in patients with compromised liver function. That makes them unsuitable for long-term administration [43]. To address the positive effects of interferon therapy and to account for its limitations due to adverse effects and limited time usage, we developed a control problem that accounts for: (i) three interferon functions: increased CTL life-span, reduced viremia and increased CTL killing of infected hepatocytes; (ii) two nucleos(t)ide analogue effects: blocking of viral infection and of viral production; (iii) hepatitis B DNA decay below the limits of detection; and (iv) minimal dosage administration over a short time period. The presence of significant side effects and the need for lengthy treatment make the question of the optimal therapy strategy relevant for the case of chronic hepatitis B. Little work has been previously done on the optimal control of HBV treatment, and the existing models do not include immune system involvement [30]. [PERSON] et al. demonstrated that consideration of delayed cytotoxic and non-cytotoxic immune reactions and the presence of cells refractory to infection was necessary to properly understand the dynamics of HBV acute infection and progression to chronic disease [14]. In this study, we derived an optimality system associated with this model, and we constructed the corresponding adjoint equations, which differed from the construction of adjoint equations for systems of ODE [44]. In particular, we found that for a delay \(\tau\) and a control period \([0,T]\), the adjoint equations are ODEs on \([T-\tau,T]\) and advance differential equations on \([0,T-\tau]\), meaning that there are terms with time argument \(t+\tau\). After deriving the proper form of the adjoint equations, we have investigated the optimal control system numerically using a forward-backward sweep method, as in [25]. The control problem indicates the need for high immunomodulatory effects of IFN-\(\alpha\) until HBV DNA is cleared. Most importantly, the immunomodulatory effects that increase the survival of effector cells are essential for timely reduction in viremia, which is needed to limit the IFN-\(\alpha\)-induced side effects [43]. We predict that starting interferon therapy at the peak viral load, rather than at viral equilibrium, shortens the time to HBV DNA removal. This is due to enhancement of an already activated T cell response. Most interestingly, monotherapy with interferon-\(\alpha\) is sufficient for virus control, while the effects of nucleos(t)ide analogues emerge only at the end stages of combination therapy. This result suggests that sequential single therapy (interferon followed by nucleos(t)ide analogues) may be the optimal course of action for both viral suppression and the limitation of drug effects. Our results imply that an increase in T cell response (in acute infections) and reversal of T cell inactivation (in chronic infections) are essential for fast control of viremia. However, experimental studies have shown that only a small percentage of patients on INF-\(\alpha\) therapy experience loss of surface-antigen, e-antigen loss and reduction in virus replication [45].