## 3 Optimal Control Problem
### Numerical Results with Optimal Controls
In combination therapy, when \(u_{2}\) neither increases CTLkilling abilities nor reduces virus production, i.e., \(a_{1}=a_{2}=0\), and all other effects are maximal \(a_{3}=b_{1}=b_{2}=1\), \(f=0\) (so that NAs have maximal effect in blocking viral production), the HBV DNA will clear 341 days after the start of treatment (see Figure 5b, third row). The controls are: (i) \(u_{1}=0.9\) for \(0\leq t\leq 358\) and \(t\geq 396\), and zero otherwise; and (ii) \(u_{2}=0.9\) for \(0\leq t\leq 369.5\), and zero otherwise (see Figure 5a, third row). Figure 5: (a) Optimal evolution for controls \(u_{1}\) (dashed lines) and \(u_{2}\) (solid lines) as given by (15); (b) virus \(V\) per mL (solid lines) and infected cells \(I\) per mL (dashed lines) over time; (c) effector cells \(E\) per mL over time when drugs are introduced at equilibrium virus concentration and: \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\), \(f=0.5\) (top row); \(a_{1}=a_{2}=a_{3}=1\), \(b_{1}=b_{2}=0\), \(f=1\) (second row); \(a_{1}=a_{2}=0\), \(a_{3}=b_{1}=b_{2}=1\), \(f=0\) (third row); \(a_{1}=a_{2}=a_{3}=0\), \(b_{1}=b_{2}=1\), \(f=0\) (bottom row). Lastly, we consider monotherapy with NAs alone. We set \(a_{1}=a_{2}=a_{3}=0\), \(b_{1}=b_{2}=1\) and \(f=0\). The optimal control model does not result in viral clearance even when \(u_{1}=0.9\) throughout the duration of the treatment (see Figure 5a,b, bottom row). For the first three cases, we find that an increase in the effector cells' lifespan is needed for viral suppression. Indeed, when \(a_{3}=1\) and \(u_{2}>0\), the increase of CTL lifespan to \(1/d(1-a_{3}u_{2})\) leads to elevated CTL concentrations (see Figure 5c, top three rows) and subsequent HBV DNA removal (see Figure 5b, top three rows). By contrast, when \(a_{3}=0\), HBV DNA does not reach extinction during the \(T_{1}=400\) days of therapy (see Figure 5b, bottom row) due to low HBV-tolerant CTL concentrations, \(E=s/d_{1}=s/d\), which do not expand in the presence of HBV (see Figure 5c, bottom row). For optimal therapy during acute HBV infection, to further determine the relationship between CTL activation and the possible success of short-term anti-HBV therapy, we run the optimal control problem during acute HBV disease, where CTL activation has been reported [41]. We start by running the DDE system mainly to the peak virus concentration occurring at time \(T_{peak}\), record the values of all variables for the times \(-\tau+T_{peak}\leq t\leq T_{peak}\) and start the optimal control problem at \(t_{d}=T_{peak}\). The treatment will be given for \(T_{2}=95\) days and, as before, we include the following weights \(c_{1}=\frac{1}{x_{3}(T_{peak})}\), \(c_{2}=\frac{\pi}{cx_{3}(T_{peak})}\), \(c_{3}=c_{4}=1\), \(\epsilon_{1}=\epsilon_{2}=0.1\) and \(c_{5}=3\times 10^{3}\), so that all factors are normalized to one. As in the chronic HBV case, we consider four scenarios. If we include all effects of nucleos(t)ide analogues \(u_{1}\) and all effects of interferon \(u_{2}\), i.e., \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(f=0.5\) and \(a_{3}=0.9\), we predict that HBV DNA is cleared 90 days after the start of treatment when interferon dosage is maximal at each time step and NAs are zero for \(0\leq t\leq 90\) days and maximal from \(t\geq 90\) (see Figure 6a,b, top row). This result implies that NAs do not have a role in viral clearance. This is corroborated by the optimal control solution for interferon monotherapy. Indeed, when we set \(a_{1}=a_{2}=1\), \(a_{3}=0.9\), \(b_{1}=b_{2}=0\) and \(f=1\), virus is cleared even faster, 87 days after the start of therapy (see Figure 6b, second row) when \(u_{2}\) dosage is maximal \(u_{2}=0.9\) at each time step (see Figure 6a, second row). This is due to the higher efficacy of interferon-\(\alpha\) at blocking viral production assumed in this scenario. The results in both of these cases are due to fast expansion and transient maintenance of CTLs to high levels of 2500 cells per mL (see Figure 6c, top two rows).