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: