Optimal Control for Mathematical Models of Cancer Therapies by Heinz Schättler & Urszula Ledzewicz

Author:Heinz Schättler & Urszula Ledzewicz
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

5.5.1 The Dynamical System with Constant Controls

In order to see the effects that this modeling change has on the dynamics, it is useful to compare the resulting dynamical systems for constant control values. Figure 5.15 gives the phase portraits of the uncontrolled (left) and fully controlled (right) systems [I] and [I1] for the parameter values of Table 5.2 and . The uncontrolled dynamics very much exhibits the same behavior as for model [H]. For the fully controlled systems, the dynamic behavior of model [I] is much closer to the one for model [E] while the dynamic behavior for model [I1] qualitatively is very much identical with the one for model [H]. Note, however, that there is a significant quantitative difference in the values for the parameter values used. For system [I1], the stable equilibrium point of the uncontrolled system is in the unrealistically high range of 108. This obviously is caused by changing the term that multiplies q in the inhibition term from to which reduces the inhibitory effects of the tumor by one degree of magnitude. Thus the values are no longer realistic. However, as our analysis is completely independent of the specific values that these parameters actually have, we retained them for this illustration. If we counteract these effects by also scaling down the coefficient b that determines the stimulatory effect of the tumor by one order of magnitude, once more similar numerical values arise. As a comparison, in Figure 5.15 we also include as the fourth row the phase portraits for the system [I1] when the parameter b has been chosen as b = 0. 25.

Fig. 5.15Phase portraits for the uncontrolled (left) and fully controlled (right) systems [E] (top row), [I] (second row), [I1] for b = 5. 85 (third row), [I1] for b = 0. 25 (fourth row), and [H] (bottom row).

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