Mathematical Modelling and Control

: This study addresses a modiﬁed mathematical model of tumor growth with targeted chemotherapy consisting of e ﬀ ector cells, tumor cells, and normal cells. To investigate the dynamics of the model, local and global stability analyses have been performed at the equilibrium points of the model. It is found that the tumor-free steady state is globally asymptotically stable under certain conditions, which suggests that the prescribed treatment can eradicate tumor cells from the body for a threshold value of tumor growth rate. The main result of this study is that if the tumor growth rate is tiny, it is possible to eradicate the tumor from the body using a smaller amount of targeted chemotherapy drugs with less harm to the other healthy cells. If not, it requires a high dose of targeted chemotherapy drugs, which can increase the side e ﬀ ects of the drugs. Numerical simulations have been performed to verify our analytical results.


Introduction
Cancer, the abnormal growth of tumor cells that invade other body parts, is now the second leading cause of death worldwide after cardiovascular disease. So, managing cancer disease by developing new treatment strategies is a new research area for researchers. However, it is essential to understand the dynamics of tumor cells' growth and their complex interactions with the immune system to develop new treatment strategies. To do this, researchers heavily relied on mathematical models.
Many scientists have studied mathematical modelling of tumor evolution, interaction with different cells, and tumor proliferation by developing various models over the last few decades [1][2][3][4][5][6][7][8]. Allison et al. [9] developed an experimentbased, mathematical model to measure the effect of tumor growth rate, carrying capacity, and cytolytic activity of the effector cells on the progression of a tumor. Li et al. [10] explored the effects of angiogenic growth factors secreted by the tumor associated with the angiogenic process on tumor growth using a nonlinear time-delay model. They observed that the time delay on angiogenic growth is harmless for the model's local and global dynamical properties. In [11], the authors observed the dynamical properties such as stable steady states, unstable steady states, and limit There are several common treatment modalities for cancer, such as chemotherapy, targeted chemotherapy, radiotherapy, immunotherapy, and surgery. Among these cancer treatments, targeted chemotherapy is a systematic therapy that fights and kills the cancer cells in the tumor site without any significant effect on the effector-normal cells so that the tumor cells can not migrate to other parts of the body [14]. Mathematical modeling of tumor growth and treatment has been approached by several researchers using a variety of models over the past few decades [15][16][17]. De Pillis and his collaborators investigated several mathematical models to measure the impact of chemotherapy [18][19][20], immunotherapy [17,21], chemoimmunotherapy [22], and monoclonal antibody (mAb) therapy [23] on tumor growth and other healthy tissue.
In [19], the authors observed "Jeff's phenomenon" for a tumor growth competition model. They obtained new optimal treatment protocols that lower tumor growth and stabilize the normal cell population. A phase-space analysis of a mathematical model of tumor growth with immune response and chemotherapy has been performed, and it has been observed that optimally administered chemotherapy drugs could drive the system into a desirable basin of attraction [18]. In [24], the importance of CD8+ T cell activation in cancer therapy was addressed. Moreover, Pillis et al. developed a cancer treatment model [21] in which they observed that combining chemotherapy and immunotherapy can eliminate the entire tumor rather than the therapies applied alone. An optimally controlled chemodrug administration was presented in [20], which discussed the role of a quadratic control, a linear control, and a state constraint. In [22], it is observed that CD8+T cells play an active role in chemo-immunotherapy to kill tumor cells.
The effectiveness of mAbs treatment on colorectal cancer has been discussed in [23]. An optimal feedback scheme was proposed based on [24] that aims to tumor regression under a better health indicator profile and to improve treatment strategies in the case of mixed immunotherapy and chemotherapy of tumor [25]. Ansarizadeh et al. and chemo-drug therapy models for cancer treatment was investigated in [28], and the results suggested that the combination treatment may cure cancer and improve the patient's life. Recently, Liu and Liu [14] proposed a targeted chemotherapy cancer treatment mathematical model that suggests that the effectiveness of targeted chemotherapy in killing tumor cells is better than regular chemotherapy.
The effect and efficacy of the targeted chemotherapeutic drug were investigated in [29], which shows that an adequate dosage of the targeted chemotherapeutic drug of low molecular weight is necessary for removing tumor cells from the infected tumor system.
In the present study, we will examine the effectiveness of targeted chemotherapy on tumor and normal cells by modifying the model of de Pillis et al. [18], which describes the interaction between tumor cells and healthy tissue or normal cells. In the very next section, we formulate our model. In Section 3, we verified the positive invariance and boundedness of the models' solutions. A steady-state analysis has been performed in Section 4. We have checked the global stability of the tumor-free state in Section 5. A numerical simulation and a concluding remark have been carried out in Sections 6 and 7.

Model formulation
We consider the model developed by de Pillis et al. [18], in which a set of ordinary differential equations is used to characterize the dynamic interplay between effector cells, tumor cells, and healthy cells. We employ a novel adaptation

Positive invariance and boundedness
In this section, we will investigate whether the model

Integration of the above leads to
Again, Proceeding as above, we have Similarly, we have Using the considered initial values, we assume that Consequently, the corresponding domain region for the system (2.1) is The domain region ∆ is positively invariant, which verifies that the model system (2.1) is biologically feasible.

Equilibrium and stability
In this section, we will study the stability of the system and the system's stability around the equilibrium. To do this, we compute dI dt = 0, dT dt = 0, dN dt = 0, and dC dt = 0 and get the following equilibrium: • Tumor-free equilibrium: E 1 (I 1 , T 1 , N 1 , C 1 ) where, The tumor-free equilibrium E 1 exists if , C * = u d 2 +kT * , and from the following quadratic equation, we will calculate the value of T * .
After substituting the values of I * and C * , we get where, The co-axial equilibrium E * exists if the roots of the equation (4.1) is positive i.e., T * > 0 and following inequalities must holds: As N = 0 implies that the patients will not be alive, we do not consider those cases where N = 0. To investigate the linear stability of the system around the two above stability, we must compute the Jacobian of the system, and the Jacobian is where • The eigenvalues of the Jacobian matrix (4.2) corresponding to the steady-state E 1 are: Therefore, E 1 will stable if λ 1 2 < 0 =⇒ r 1 < c 2 I 1 + c 3 N 1 + a 2 C 1 and λ 1 3 < 0 =⇒ r 2 < 2r 2 N 1 + a 3 (1 − η)C 1 ; otherwise E 1 becomes unstable.
• The Jacobian corresponding to the stability E * is where At E * , the eigen values of the corresponding Jacobian matrix (4.3) are the roots of the following equation where According to Routh-Hurwitz stability rule, the equilibrium point E * will be asymptotically stable if (4.5)

Global stability at tumor-free steady state E 1
In this section, we will analyze the global stability around E 1 in order to investigate the behavior of the system (2.1) far away from the equilibrium point E 1 . Let us define a Lyapunov function for the model (2.1) at E 1 as: Differentiating V(t) with respect to t, we get where, By noting the second component of the vector Q, we must have, −r 1 + c 2 I 1 + c 3 N 1 + a 2 C 1 > 0 =⇒ c 2 I 1 + c 3 N 1 + a 2 C 1 > r 1 (5.3) so that Q t P ≥ 0. Furthermore, by considering the values of parameters from Table1, if I = s d 1 , T = 1 b 1 and C = u d 2 , then all minors are positive in the matrix R (all eigenvalues of R are also positive), and so P t RP > 0. Hence, it is clear that dV dt < 0. Therefore, the tumor-free equilibrium E 1 satisfies local stability conditions, making the point globally stable. In biological terms, it means that targeted chemotherapy will kill the tumor cells if must hold.

Numerical simulation
Analytical studies can only be completed with numerical verification of the derived results. In this section, we verified our analytical results of the considered system (2.1) graphically using MATHEMATICA, which is very important from a practical point of view. All the simulations have been carried out using the parameter values of Table   1 [14,18]. We take the units of the parameter values to be arbitrary.  The above scenarios can be justified in Figure 1. We observe that the density of fast proliferation tumor cells gets suppressed quickly as the amount of drugs increases from u = 0.019, u = 0.020 to u = 0.021 (see Figure 1) and gets stable at zero for u = 0.021, suggesting targeted chemotherapy's success. So, the prescribed drugs quickly affected the tumor cells, which is clinically reliable. In Figure 2, we observe that the density of effector cells decreases slowly (compared to tumor cells) while the amount of applied drug doses increases and gets stable at a required level. In Figure 3, the density of normal cells is reduced more slowly (compared to tumor cells) while the amount of drugs increases. Also, the normal cells become stable at the desired level.   In Figure 6, it is seen that the trajectory converges to the tumor-free steady state E 1 with the basin of attraction in the treatment case, indicating that it is a globally stable point for the system. The steady-state E 1 is a stable node, implying that the cell population incorporated with the treatment can suppress the cancer growth to zero with time increase.
Biologically, this indicates that the body is recovering from the tumor regardless of the initial condition, which includes tumor growth.
Overall, we observe that if the size of the tumor is small, i.e., the growth rate of tumor r 1 is tiny, then it is quite possible to eradicate the tumor from the body using a smaller amount of drugs with less harm to the other healthy cells. If not, we require a high drug dose that can increase the side effects of the drugs.

Conclusions
We have investigated a modified ODE mathematical So, in this regard, we must need an optimum period and drug dose for which the tumor is eradicated [18,19]; and it will be carried out in our future study.