## 3 System Properties for Constant Concentrations
### Special Case: \(P_{\max,1}=P_{\max,2}\)
Furthermore  
\[a_{1}a_{2}-a_{0} =\left[\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+ \frac{P_{\max,1}P_{*}}{1+P_{*}}\right)\right]\times\] \[\qquad\times\left[\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)s_{E }\left(1+\frac{P_{\max,1}P_{*}}{1+P_{*}}\right)+\frac{E_{\max,2}\delta_{E}E_{* }}{\left(1+E_{*}\right)^{2}}\frac{\left(P_{\max,1}-P_{\max,2}\right)\delta_{P} P_{*}}{\left(1+P_{*}\right)\left(1+P_{*}+P_{\max,1}P_{*}\right)}\right]\] \[\qquad-\delta_{Q}Q_{*}\frac{E_{\max,1}\delta_{E}E_{*}}{\left(1+E_ {*}\right)^{2}}\frac{k_{P}\left(P_{\max,1}-P_{\max,2}\right)}{\left(1+P_{*} \right)\left(1+P_{*}+P_{\max,1}P_{*}\right)}\] \[\qquad>\frac{\delta_{E}E_{*}}{\left(1+E_{*}\right)^{2}}\frac{P_{ \max,1}-P_{\max,2}}{\left(1+P_{*}\right)\left(1+P_{*}+P_{\max,1}P_{*}\right)}k _{P}Q_{*}\left(\delta_{P}E_{\max,2}-\delta_{Q}E_{\max,1}\right).\]  
This expression is positive if we make the following assumption (B):  
\[\delta_{P}E_{\max,2}\geq\delta_{Q}E_{\max,1}. \tag{21}\]  
Note from Equations (2) and (3) that \(\delta_{P}E_{\max,2}\) represents the maximal size of the immune effect \(E\) on \(P\) while \(\delta_{Q}E_{\max,1}\) represents the maximal size of the immune effect \(E\) on \(Q\). This effect is assumed to be stronger on the proliferating class of cells than on the quiescent class of cells. Thus assumption (21) is a natural one to make. This assumption is invariant under the actions of the drugs:  
\[\hat{\delta}_{P}\hat{E}_{\max,2} =\left(1+\frac{U1_{\max,2}u_{1}}{U1C_{50}+u_{1}}\right)\left(1+ \frac{U2_{\max,3}u_{2}}{U2C_{50}+u_{2}}\right)\delta_{P}\cdot\frac{\left(1+ \frac{U2_{\max,1}u_{2}}{U2C_{50}+u_{2}}\right)\left(1+\frac{U3_{\max,1}u_{3}} {U3C_{50}+u_{3}}\right)}{\left(1+\frac{U1_{\max,2}u_{1}}{U1C_{50}+u_{1}} \right)\left(1+\frac{U2_{\max,4}u_{2}}{U2C_{50}+u_{2}}\right)}E_{\max,2}\] \[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]0  
while, letting \(\hat{\delta}_{Q}=\delta_{Q}\),  
\[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]1 \[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]2  
so that these terms are multiplied by the same coefficients. Hence we also have the following result:  
Theorem 4.2: _If \(P_{\max,1}>P_{\max,2}\) and \(\delta_{P}E_{\max,2}\geq\delta_{Q}E_{\max,1}\), then the positive equilibrium point \((Q_{*},P_{*},E_{*})\) is locally asymptotically stable._  
The limiting case \(P_{\max,1}=P_{\max,2}\) represents a degenerate scenario. In many cases no positive equilibrium exists. For example, if \(\frac{s_{E}}{\delta_{E}}\ln\left(\frac{E_{\max}}{E_{*}}\right)\
eq 1\), then it follows from (18) that  
\[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]3  
In such a case equilibria will cease to exist, as \(P_{\max,1}\searrow P_{\max,2}\), once the parameter values satisfy  
\[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]4Also, although the positive equilibrium point in Theorem 2 is stable, the value can be very high. In fact, \(P_{*}\) diverges to \(+\infty\) as these parameter relations are reached (c.f. (18)):  
\[a_{2} =\left(k_{P}\frac{Q_{*}}{P_{*}}+r_{P}\right)+s_{E}\left(1+\frac{P _{\max,1}P_{*}}{1+P_{*}}\right)>0,\]5  
For the equilibrium values to be relatively small ('chronic'), we see that \(P_{\max,1}\) must be significantly larger than \(P_{\max,2}\).