Variable \(x\left(.\right)\) shows the population of immune cells, and\(y\left(.\right)\) shows the population of cancer cells. The immune cells grow with the rate \(\sigma\left(.\right)\). The natural death rate of these cells is \(\delta\left(.\right)\). These cells are also eliminated by interaction with cancer cells with the rate \(\mu\). The immune system is excited to grow due to the presence of cancer cells, which is shown by the term \(\frac{\text{ρx}(t)y(t)}{y(t)+n}\ \). The growth of cancer cells is as a logistic term. The second term in cancer cells dynamics shows death due to the interaction with immune cells.\(M\left(.\right)\) shows the chemotherapy drug concentration in the blood, and \(v_{M}(.)\geq 0\) is the chemo-drug dose injected in the body. Some chemo-drugs are effective during a part of the cell cycle, such as doxorubicin. The concentration of the chemo-drug into the body decreases over time exponentially. Chemotherapy kills a fraction of all cells.
The effect of vaccine therapy is regarded in equations (4)-(5). Antiviral vaccines and anticancer vaccines have a significant difference. The antiviral vaccines are preventive, while anticancer vaccines are therapeutic. Antiviral vaccines are used before disease, but anticancer vaccines are used after beginning cancer. Anticancer vaccines reinforce the immune system. This vaccine has an effect on the parameters of the system (\(\sigma\left(.\right)\) and\(\delta\left(.\right)\)), which is shown by\(v_{v}\left(.\right)\ \)[21, 22]. The rate of variation of these parameters is assumed to be proportional to the input\(v_{v}\left(.\right)\). The values of \(\mu_{\sigma}\) and\(\mu_{\delta}\) ​​depend on the dynamics of \(\sigma\left(.\right)\)and \(\delta\left(.\right)\), respectively. These coefficients are saturated to the final limits \(k_{\sigma}\) and \(\ k_{\delta}\) that are related to the biological limitations of body organs and the accumulation of external effects [23]. Changes in the parameters of the system due to vaccine therapy are also regarded in [22], but not considered in the model. This type of modeling was presented in [23-25].
The estimated values of the parameters for the system (1)-(2) are shown in Table 1.
Steady states of the free system
The primary aim of the treatment is to settle the tumor-immune system to its tumor-free equilibrium point. Hence, examining the dyanmics of the free system is essential. Generally, two types of equilibrium points exist in the system: the tumor-free equilibrium point and the cancerous equilibrium points. The tumor-free equilibrium point has zero cancer cells and is biologically important, and a cancerous equilibrium point has non-zero cancer cells. Suppose the desired equilibrium point of the system (1)-(2) is \({X^{*}=(x}^{*},y^{*})\). The linearized system around the desired equilibrium point \({(x}^{*},y^{*})\) is: