## 1 Introduction  
Hepatitis B virus (HBV) is the leading viral cause of liver disease, affecting 250-350 million people worldwide. Chronic HBV leads to the development of liver cirrhosis and liver cancer [1]. Despite the availability of effective HBV vaccination [2], the prevalence of chronic HBV has only marginally declined [3, 4]. The natural course of chronic HBV includes an immune-tolerant phase, hepatitis B e-antigen positive immuno-active and -inactive phases and a hepatitis B e-antigen negative immuno-active phase [5]. Understanding the virological and immunological characteristics of each of these stages can provide a useful framework for the management of chronic HBV [6, 7]. Currently, seven drugs have been approved for treating chronic HBV disease: standard and PEGylated interferon (IFN-\(\alpha\)) and five nucleo(t)side analogs (NAs) [8]. These medications suppress HBV replication and liver inflammation, but do not lead to cure. The interferon-\(\alpha\) treatments modulate immune responses that may lower viral levels. It is given for a finite time (usually 12 months) due to its toxic side effects [8, 9]. The nucleo(t)side analogues are administrated for many years and sometimes for life and are responsible for viral suppression. However, life-long therapy is difficult due to costs, side effects, compliance and, most importantly, the development of antiviral drug resistance [10]. Combination therapy has not shown an increased effect on treatment response (but has reduced the rate of drug resistance) [11]. This has made it difficult to establish a universal guideline for treatment start, duration, which type or combinations of drugs to use [7] and how to define the success of therapy (virologically, serologically and/or immunologically) [7; 12]. To provide insight into the optimal combination therapy and, in particular, into the immune-mediated effects of interferon-\(\alpha\), we design an optimal control study for a mathematical model of hepatitis B infection. Mathematical models have been used to address transition from acute to chronic HBV infections [13; 14; 15; 16] and to study the effects of drug therapy [17; 18; 19]. Optimal control theory was developed by [PERSON] et al. for obtaining necessary conditions to characterize optimal controls for systems of ordinary differential equations (ODEs) [20]. Optimal control models have been used previously to design treatment strategies for disease models described by systems of ordinary differential equations with no delays [21; 22; 23; 24; 25] and systems of delay differential equations (DDEs) [26]. [PERSON] [27] developed the extension of the [PERSON]'s maximum principle for systems of DDEs with constant time delays. For approximation and numerical methods for such problems, see [28; 29]. In this paper, we modify an in-host DDE model of immune responses to hepatitis B infection introduced in [14] by adding the effects of combination drug therapy. In particular, we consider the drug effects in blocking viral production, reducing viral infection, enhancing the killing of infected cells by immune responses and removing immune cell exhaustion. We will use this as a starting model for designing an optimal control problem that advises what is the best combination therapy to ensure viral clearance, immune activation and the least amount of liver damage. Time-varying rates for therapies will be the controls in the system. Optimal control has been applied to the study of hepatitis B therapy using ODE [30; 31] and DDE [32] models that did not consider an immune system component and using an ODE model that considers the enhancement of immune responses following administration of traditional Chinese medicine [33]. Here, we introduce the DDE system with constant time delay from [14], derive its corresponding control formulation and use it to determine the temporal, quantitative and qualitative effects of the drugs that lead to hepatitis B virus control, balancing the goals of reducing viral load and minimizing the negative side effects of therapy.