By using the parameters presented in Table 1, the system has one stable
cancerous equilibrium point (E2) and one unstable tumor-free equilibrium
point (E1) (see Figure 1). The system has one cancerous equilibrium
point. E1 is the unstable tumor-free equilibrium point and E2 is the
stable cancerous equilibrium point. In this case, the patient will die
in the absence of the treatment because the trajectory of the system
goes to the stable canceorus equilibrium point from any feasible initial
condition. The tumor free equilibrium point is an unstable node.
Bifurcation analysis
An abrupt change in the dynamics of a system due to changes in the
parameters of the system is called bifurcation. The effect of change in
two important parameters is examined: the growth rate of the immune
cells (\(\sigma\)) and the natural death rate of the immune cells
(\(\delta\)). The bifurcation diagrams due to change in these parameters
are shown in Figure 2, where other parameters are given in Table 1.
Three regions exist, which in each region the number of equilibrium
points is different.
For each parameter three regions exist. The phase plane of the system in
the region III is shown in the Fig 3. In region III, there is no
cancerous equilibrium point in the system. In other words, the patient
is healthy, and the immune system is capable of removing all cancer
cells in the body without any external treatment. The main target of the
treatment is to change the dynamics of the system such that there is no
cancerous equilibrium point in the system. Changing the parameters of
the system by using immunotherapies such as vaccines is the way of
reaching this target.
In the region I, three cancerous equilibrium points exist (see Figure 4)
and in the region II one cancerous equilibrium point exists in the
system (see Figure 1).
In Figure 4, the tumor-free equilibrium point (E1) is an unstable node,
E3 is a saddle point and the equilibrium points E2 and E4 are stable. It
has to be noted that, the stable eigenvector of the saddle point E3 is a
good estimation for the domain of attraction of the stable points E2 and
E4.
Immunotherapy
The target of the immunotherapy is to reinforce the immune system. This
is done by using vaccine therapy [21, 22]. The vaccine therapy
affects on the parameters \(\sigma\) and \(\delta\). The dynamics of the
cancer during the immunotherapy is time variant. During the
immunotherapy the phase plane of the system should change such that a
stable equilibrium point with low cancer cell population enters to the
system. In other words, the phase plane of the system should change from
Figure 1 to Figure 3. It means that, these two parameters should enter
to region III (see Figure 2).