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).