## 3 System Properties for Constant Concentrations
### Reduction to the Uncontrolled System and Basic Dynamical System Properties  
We carry out the dynamical systems analysis for constant controls, i.e., concentrations. We do not explicitly include pharmacokinetics (fluctuations in concentrations that depend on doses). The treatments considered are either administered daily or have long half-lives, and such pharmacokinetics are not expected to be significant for the treatment periods we consider here (five years or longer). We also mention the 2009 paper by [PERSON] et al. [16] that supports this assumption in the setting of hepatitis C.  
Keeping the \"\(C_{50}\)\" parameters in their original formulation in the controls, we define new drug-dependent parameters as  
\[\hat{k}_{P} =\left(1-\frac{U1_{\max,1}u_{1}}{U1C_{50}+u_{1}}\right)\left(1- \frac{U2_{\max,2}u_{2}}{U2C_{50}+u_{2}}\right)k_{P},\] \[\hat{r}_{P} =\left(1-\frac{U1_{\max,1}u_{1}}{U1C_{50}+u_{1}}\right)\left(1- \frac{U2_{\max,2}u_{2}}{U2C_{50}+u_{2}}\right)r_{P},\] \[\hat{\delta}_{P} =\left(1+\frac{U1_{\max,2}u_{1}}{U1C_{50}+u_{1}}\right)\left(1+ \frac{U2_{\max,3}u_{2}}{U2C_{50}+u_{2}}\right)\delta_{P},\] \[\hat{E}_{\max,1} =\left(1+\frac{U2_{\max,1}u_{2}}{U2C_{50}+u_{2}}\right)\left(1+ \frac{U3_{\max,1}u_{3}}{U3C_{50}+u_{3}}\right)E_{\max,1},\] \[\hat{E}_{\max,2} =\frac{\left(1+\frac{U2_{\max,1}u_{2}}{U2C_{50}+u_{2}}\right) \left(1+\frac{U3_{\max,1}u_{3}}{U3C_{50}+u_{3}}\right)}{\left(1+\frac{U1_{\max,1}u_{1}}{U1C_{50}+u_{1}}\right)\left(1+\frac{U2_{\max,3}u_{2}}{U2C_{50}+u_{2} }\right)}E_{\max,2},\] \[\hat{P}_{\max,1} =\left(1+\frac{U2_{\max,4}u_{2}}{U2C_{50}+u_{2}}\right)\left(1+ \frac{U3_{\max,2}u_{3}}{U3C_{50}+u_{3}}\right)P_{\max,1},\] \[\hat{P}_{\max,2} =\left(1-\frac{U2_{\max,5}u_{2}}{U2C_{50}+u_{2}}\right)\left(1- \frac{U3_{\max,3}u_{3}}{U3C_{50}+u_{3}}\right)P_{\max,2}.\]  
With these identifications, the dynamical system with constant controls is identical with the uncontrolled system and therefore, without loss of generality, the analysis can be done on the uncontrolled system. Returning to the original notation without the carets, we thus consider the following equations:  
\[\frac{dQ}{dt} =\left[r_{Q}-\delta_{Q}\left(1+\frac{E_{\max,1}E}{1+E}\right) \right]Q, \tag{6}\] \[\frac{dP}{dt} =k_{P}Q+\left[r_{P}\ln\left(\frac{P_{\mathrm{ss}}}{P}\right)- \delta_{P}\left(1+\frac{E_{\max,2}E}{1+E}\right)\right]P,\] (7) \[\frac{dE}{dt} =\left[s_{E}\left(1+\frac{P_{\max,1}P}{1+P}\right)\ln\left(\frac {E_{\mathrm{ss}}}{E}\right)-\delta_{E}\left(1+\frac{P_{\max,2}P}{1+P}\right) \right]E. \tag{8}\]  
The model with an exponential growth term on \(Q\) has various long-term behaviors. These include the extremes in which \(Q\) decays exponentially to zero or grows exponentially beyond limits, but there also is the possibility that nontrivial equilibrium points \((Q_{*},P_{*},E_{*})\) exist for which all three populations are positive. The first case corresponds to a scenario in which the patient goes into a stable deep molecular response. For the uncontrolled system, this may not seem to be of interest, but sincethe model includes the case with controls, this gives us information about which combinations of constant concentrations of the drugs would lead to an eradication of \(Q\). The case of exponential growth may characterize the _accelerated_ or _blast phase_ as these phases have short doubling times [17]. The conditions under which this is the long-term behavior of the system give information about what controls are needed for successful treatment. An asymptotically stable equilibrium point \((Q_{*},P_{*},E_{*})\) with positive values could be interpreted as describing a subset of the _chronic_ phase where net growth rate is zero, controlled by therapy or immune effects. Depending on the values of the parameters, this equilibrium point may be stable or unstable. Since in real life parameters may not be constant, bifurcation phenomena would be a mathematical description of the transition from chronic to the accelerated or blast phases. Knowing the parameter values when this may occur would be of interest.