## 3 Optimal Control Problem
### Numerical Results with Optimal Controls  
For optimal therapy during chronic HBV infection, we assume that at the time of treatment initiation, the patient has reached chronic steady state values \(\bar{x}_{1}=3\times 10^{5}\) cells per mL, \(\bar{x}_{2}=3.8\times 10^{4}\) cells per mL, \(\bar{x}_{3}=9.33\times 10^{6}\) HBV DNA per mL, \(\bar{x}_{4}=20.3\) cells per mL and \(\bar{x}_{5}=1.33\times 10^{7}\) cells per mL. The terms in the objective functional (9) have different scales. In order to weigh each term equally, we choose parameters \(c_{i}\), such that \(c_{i}x_{i}\) (for \(i=1,2\)) and \(c_{i}u_{i}\) (for \(i=3,4\)) are one at the start of the optimal control. When the optimal control is started at the viral steady state, we normalize variables \(x_{1}\) and \(x_{2}\) by \(c_{1}=\frac{1}{\bar{x}_{3}}\) and \(c_{2}=\frac{1}{\bar{x}_{2}}=\frac{\pi}{\bar{x}_{3}}\). As a result, the factors \(c_{1}x_{1}\) and \(c_{2}x_{2}\) are one at the beginning of treatment and between zero and one from there on. Since we assume \(0\leq u_{1}(t),u_{2}(t)\leq 0.9\) for all \(t\), their corresponding weights are \(c_{3}=c_{4}=1\). Lastly, we choose the effects for the convex terms to be \(\epsilon_{1}=\epsilon_{2}=\mathbf{0.1}\). The treatment will be given for \(T_{1}=400\) days, which is sufficient for virus clearance and is still within the timeline of the HBV therapy guidelines [7]. The ideal clinical outcome of HBV drug therapy is to achieve viral suppression, such that \(x_{3}\leq V_{ext}=3\times 10^{-4}\) HBV DNA per mL, corresponding to one HBV DNA in the body (as in Section 2.3). To account for this final condition, we set \(c_{5}=3\times 10^{3}\) so that the final time condition is normalized to one. Under these assumptions, we run the optimal control problem over \(T_{1}=400\) days with the aim of finding the best temporal drug usage at each time point that ensures viral clearance while minimizing the drug levels. We will be considering several scenarios by varying the levels of control effects, described by constants \(a_{i}\) and \(b_{i}\). Our simulations show that for high virus levels (above extinction), the optimal control will always be the maximal drug dosage for the scenario considered. For example, for a case with any \(a_{i}>0\) and \(b_{1}=b_{2}=0\), the optimal control is \(u_{1}(t)=0\) and \(u_{2}(t)=0.9\) for all time points. Similarly, if \(a_{i}>0\) and \(b_{j}>0\), then the optimal control is \(u_{1}(t)=u_{2}(t)=0.9\) for all time points. Therefore, for the first 325 days, we assume the maximal drug dosage for a given scenario and only run the optimal control with variable \(u_{1}(t)\) and \(u_{2}(t)\) for the final 75 of the 400 days of treatment. Initially, we include all effects of nucleos(t)ide analogues \(u_{1}\) and interferon \(u_{2}\), i.e., \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\) and \(f=0.5\). For this choice of parameters, optimal control requires: (i) maximal NA drug dosage \(u_{1}=0.9\) for \(0\leq t\leq 347\) days and zero drug dosage for \(t\geq 347\) days; and (ii) maximal \(u_{2}=0.9\) interferon dosage for \(0\leq t\leq 366.5\) days, zero dosage for \(366.5\leq t\leq 399\) days and maximal dosage after that (see Figure 5a, top row). Under these drug regimes, HBV DNA is cleared 331 days after the start of treatment (see Figure 5b, top row). We next assume monotherapy with interferon-\(\alpha\). We set \(b_{1}=b_{2}=0\), \(a_{1}=a_{2}=a_{3}=1\) and change the anti-viral production effect to \(f=1\). The optimal control problem predicts that the virus is cleared 338 days after the start of therapy, seven days later than under combination therapy (see Figure 5b, second row). Another downside of this monotherapy is the need for interferon dosage to be maximal (\(u_{2}=0.9\)) for \(0\leq t\leq 377\) and for \(t\geq 396.5\) in order to compensate for the lack of NAs (see Figure 5a, second row).