## 3 Optimal Control Problem
### Numerical Results with Optimal Controls
To further determine which of the interferon effects are the most influential, we remove the first two interferon effects \(a_{1}=a_{2}=0\), keep \(a_{3}=0.9\), \(b_{1}=b_{2}=1\) and set \(f=0\). Under these conditions, virus is cleared 91 days after the start of therapy, one and four days later than the previous two cases (see Figure 6b, third row) when: (i) NA dosage (\(u_{1}\)) is zero for \(0\leq t\leq 87\) and maximal (\(u_{1}=0.9\)) afterwards; and (ii) interferon dosage is maximal (\(u_{2}=0.9\)) for \(0\leq t\leq 91\) and zero afterwards (see Figure 6a, third row). This result implies, again, that interferon is needed for virus clearance. Lastly, when we consider monotherapy with NAs, i.e., \(b_{1}=b_{2}=1\), \(a_{1}=a_{2}=a_{3}=0\) and \(f=0\), the HBV DNA will not clear (see Figure 6b, bottom row) in spite of CTL levels being higher than the base of \(E=s/d_{1}=s/d\) (see Figure 6c, bottom row). Interestingly, our numerical results show that the optimal NA dosage for the first 42 days is zero (see Figure 6a, bottom row). Our study has used a delay of \(\tau=33.4\) days, since that represented the delay in the CTL activation in the only patient that developed chronic disease in [14]. To check whether the size of the delay affects the results, we have investigated the optimal control problem for a shorter delay of \(\tau=15.2\) days, which is the smallest delay among the patients in [14]. We found that the length of the delay does not influence the results when the therapy is started at the viral steady state (not shown). When the therapy is started at the peak viral load, we find, not surprisingly, that a shorter time lag speeds viral decay and CTL expansion (see Figure 7b,c). Moreover, the optimal control problem for the shorter delay of \(\tau=15.2\) days suggests no therapy (\(u_{1}=u_{2}=0\)) for the first three days post-peak, maximal interferon therapy for \(3\leq t\leq 75\) and \(t\geq 90\) and no interferon \(u_{2}=0\) otherwise. The role of NAs is transient, with maximal dosages \(u_{1}=0.9\) for \(20\leq t\leq 27\), \(38\leq t\leq 51\) and \(t>90\) days and \(u_{1}=0\) elsewhere (see Figure 7a, bottom row). Figure 6: (**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 peak virus concentration and: \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(a_{3}=0.9\), \(f=0.5\) (top row); \(a_{1}=a_{2}=1\), \(a_{3}=0.9\), \(b_{1}=b_{2}=0\), \(f=1\) (second row); \(a_{1}=a_{2}=0\), \(a_{3}=0.9\), \(b_{1}=b_{2}=1\), \(f=0\) (third row); \(a_{1}=a_{2}=a_{3}=0\), \(b_{1}=b_{2}=1\), \(f=0\) (bottom row). Figure 7: (**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 peak virus concentration for: \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\), \(f=0.5\) and \(\tau=33.4\) (top row); \(\tau=15.2\) (bottom row). Together, these results suggest that the immunomodulatory effects of interferon are needed for HBV DNA control and that immune modulation rather than suppression of virological factors is essential for inducing HBV clearance.