## 3 System Properties for Constant Concentrations
### Dynamic Behavior for Positive \(Q\)-Values  
For the behavior of the overall system, the \(Q\) dynamics are essential. If one considers the above equilibria in the plane \(Q=0\) now in the full three-dimensional space, then the first row of the Jacobian matrix at \((0,P_{*},E_{*})\) takes the form  
\[\left(r_{Q}-\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right),0,0\right)\]  
and thus \((0,P_{*},E_{*})\) is unstable if \(r_{Q}>\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right)\) while the local stability properties for the overall system are the same as in the \((P,E)\)-plane if \(r_{Q}<\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right)\). If \(r_{Q}=\delta_{Q}\left(1+\frac{E_{\max,1}E_{*}}{1+E_{*}}\right)\), then there exists a 1-dimensional center manifold (corresponding to the 0 eigenvalue). In this case we have \(Q=0\) and there exists a curve of equilibria emerging from \((0,P_{*},E_{*})\) parameterized by \(Q\) or \(P\) (also see below). Generally, (1)-(3) is a time-varying linear system dominated by exponential growth and decay, depending on the parameter values. If  
\[r_{Q}\geq\delta_{Q}\left(1+E_{\max,1}\frac{E_{\rm ss}}{1+E_{\rm ss}}\right),\]  
then \(Q\) grows exponentially and no steady state exists. In this case, the influx \(k_{P}Q\) eventually becomes the dominant term in Equation (2) and \(P\) also grows beyond limits (Lemma 3). This represents the _malignant_ scenario in the model which corresponds to a highly-aggressive form of the disease or the _accelerated_ or _blast_ phase. The other extreme arises if \(r_{Q}<\delta_{Q}\). In this case \(Q\) exponentially decays to 0 for the uncontrolled system and overall trajectories converge to one of the equilibria \((0,P_{*},E_{*})\) in the plane \(Q=0\). If there exist multiple such equilibria, there exists a stable manifold for the unstable one that separates the regions of attraction for the stable equilibria. This would reflect a scenario when \(Q\) initiates the disease, but eventually dies off and the remaining \(P\) population determines the outcome of the disease. This could be benign if \(P_{*}\) is small (a form of successful _immune surveillance_) or malignant if this value is larger. In such a case, however, one only needs to deal with the proliferating cells as far as treatment is concerned. This appears less likely (unless it could be induced by the drugs) and in the uncontrolled case of the disease we would have \(r_{Q}>\delta_{Q}\). The interesting and most difficult case arises when the uncontrolled system has a chronic steady state or undergoes exponential growth without treatment, but has a negative net growth rate for \(Q\) with treatment. This is the case if the parameters satisfy the following condition (A):  
\[E_{\max,1}\frac{E_{\rm ss}}{1+E_{\rm ss}}>\frac{r_{Q}}{\delta_{Q}}-1>0, \tag{15}\]or, with the controls in the original form,  
\[E_{\max,1}\frac{E_{\rm ss}}{1+E_{\rm ss}}\left(1+\frac{U2_{\max,1}u_{2}}{U2C_{50} +u_{2}}\right)\left(1+\frac{U3_{\max,1}u_{3}}{U3C_{50}+u_{3}}\right)>\frac{r_{Q} }{\delta_{Q}}-1>0.\]  
Thus the replication rate constant \(r_{Q}\) needs to be greater than the death rate constant \(\delta_{Q}\) (this naturally will be satisfied for parameters in a disease state), but at the same time, the drugs need to be able to raise the maximum effectiveness \(\hat{E}_{\max,1}=E_{\max,1}\left(1+\frac{U2_{\max,1}u_{2}}{U2C_{50}+u_{2}} \right)\left(1+\frac{U3_{\max,1}u_{3}}{U3C_{50}+u_{3}}\right)\) high enough that the magnitude of the immune system effect can overcome the difference. These appear to be natural conditions. Assuming that (15) holds, there exists a unique value \(E_{*}\in(0,E_{\rm ss})\) for which \(\hat{Q}=0\), namely  
\[r_{Q}=\delta_{Q}\left(1+E_{\max,1}\frac{E_{*}}{1+E_{*}}\right)\quad \Longleftrightarrow\quad E_{*}=\frac{\frac{r_{Q}}{\delta_{Q}}-1}{E_{\max,1}- \left(\frac{r_{Q}}{\delta_{Q}}-1\right)} \tag{16}\]  
with \(Q\) increasing for \(E<E_{*}\) and decreasing for \(E>E_{*}\).