## 2 Mathematical Model and Analysis
### Numerical Results
Figure 1: Temporal evolution for the variables in Model (1) without drug therapy, i.e., \(\epsilon=\eta=0\), and the parameters in Table 1. The circles represent patient data. Figure 2: (a) Virus \(V\) per mL and (b) effector cells \(E\) per mL given by Model (1) and the parameters in Table 1 for: \(\eta=\epsilon=0\) (solid lines); \(\eta=0.5\), \(\epsilon=0.9\), \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\), \(f=0.5\) (dashed lines); and \(\eta=0.5\), \(\epsilon=0.9\), \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(a_{3}=0\), \(f=0.5\) (dotted lines). Here, time \(t=0\) corresponds to both the start of therapy and the peak viral load, i.e., the initial conditions are \(T(0)=1.32\times 10^{5}\) cells per mL, \(I(0)=1.23\times 10^{5}\) cells per mL, \(V(0)=3\times 10^{9}\) HBV DNA per mL, \(E(0)=20.23\) cells per mL and \(R(0)=1.23\times 10^{6}\) cells per mL. These results are supported by relative sensitivity curves. Briefly, sensitivity functions are numerical solutions of the following system:  
\[\frac{dx}{dt}=g(x(t,\xi),\;z(t,\xi),\;\xi) \tag{6}\] \[\frac{d}{dt}\frac{\partial x}{\partial\xi}=\frac{\partial g}{ \partial x}\frac{\partial x}{\partial\xi}+\frac{\partial g}{\partial z}\frac{ \partial z}{\partial\xi}+\frac{\partial g}{\partial\xi} \tag{7}\]  
where \(x(t,\xi)=(T(t,\xi),I(t,\xi),V(t,\xi),E(t,\xi),R(t,\xi))\), \(z(t,\xi)=x(t-\tau,\xi)\in\mathbb{R}^{5}\), the function \(g\) represents the right-hand side of Model (1) and:  
\[\xi=(r,\;K,\;\beta,\;q,\;\rho,\;\pi,\;c,\;s,\;\alpha,\;d,\;\eta,\;\
u,\;\mu,\; \epsilon,\;a_{1},\;a_{2},\;a_{3},\;b_{1},\;b_{2},\;f)\]  
The partial derivatives \(\partial x_{i}/\partial\xi_{j}\), for \(i=1,\ldots,5\) and \(j=1,\ldots,20\), are time functions denoting the rate of change in a state variable with respect to variations in model parameters (see [39] and the references therein). The functions \(\xi_{j}/x_{i}\times\partial x_{i}/\partial\xi_{j}\) are the relative sensitivity curves (similar to relative error), which allow for the comparison between the sensitivity of two variables \(x_{i}^{1}\) and \(x_{i}^{2}\) with respect to the same parameter \(\xi_{j}\). Here, we are interested in the sensitivity of \(V(t,\xi)\) and \(E(t,\xi)\) with respect to model parameters \(\xi=\{a_{2},a_{3},b_{2}\}\). As the numerical solutions displayed in Figure 4 show, the \(a_{3}\) effect is the most important drug effect in both reducing virus load and increasing immune response. Figure 3: (**a**) Virus \(V\) per mL and (**b**) Effector cells \(E\) per mL given by model (1) and parameters in Table 1 for \(\eta=\epsilon=0\) (solid lines); \(\eta=0.5\), \(\epsilon=0.9\), \(a_{1}=a_{2}=a_{3}=b_{1}=b_{2}=1\), \(f=0.5\) (dashed lines); and \(\eta=0.5\), \(\epsilon=0.9\), \(a_{1}=a_{2}=b_{1}=b_{2}=1\), \(a_{3}=0\), \(f=0.5\) (dotted lines). Here, \(t=0\) corresponds to both the start of therapy and the steady state of the viral load, i.e., the initial conditions are \(T(0)=3\times 10^{5}\) cells per mL, \(I(0)=3.8\times 10^{4}\) cells per mL, \(V(0)=9.33\times 10^{6}\) HBV DNA per mL, \(E(0)=20.3\) cells per mL and \(R(0)=1.33\times 10^{7}\) cells per mL. Our model suggests that the timing and the type of interferon effects are important in the success of the treatment. In the next sections, we will consider a time-dependent effect of both interferon and nucleo(s)tide analogues, and we will formulate an optimal control problem for our model to determine the connection between successful therapy and the optimal temporal efficacy of the effects considered.