FRACTIONAL MATHEMATICAL MODEL UNDERLYING MIXED TREATMENTS USING ENDOCRINE DIET THERAPY AND IMMUNOTHERAPY FOR BREAST CANCER

. To understand cancer as a biomathematical process, we establish our model to give an analytical and a numerical examination of the fractional derivative impact on developing breast cancer with endocrine diet therapy and immunotherapy. So, we try, in this paper, to formulate the cancer dynamics involving the normal cells, tumor cells, immune cells, estrogen (endocrine parameter) and immunotherapy. We show the wellposedness of the breast cancer model and we analyze the existence and the stability of the equilibria, then, we discuss the numerical results in order to conclude that the use of fractional derivatives provides more useful information about the stability of the breast cancer dynamics with mixed treatments model.


INTRODUCTION
Cancer is a significant factor of death, the second cause of mortality worldwide after cardiovascular diseases [1], it can be caused by a variety of factors [2,3], like smoking, poor eating, genetic factors that can be transmitted from parents and others under research. Cancer can attack in any organ of the body like lung, breast, prostate, colon and others, which can be brought on by genetic changes that make healthy cells grow and divide uncontrollably with abnormal manners and form a masses or tissues called tumors. The treatments of cancer are vulnerable include surgery, chemotherapy, radiation therapy, immunotherapy, virotherapy and many others new cures that doctors and health scientists continue to develop and optimize to achieve a good results. The most frequent cancer diagnosis among female is breast cancer [4], according to "Global Cancer Statistics 2020," a collaborative report from the American Cancer Society (ACS) and the International Agency for Research on Cancer (IARC), it is estimated about 2.3 million new cases per year in the world. The reseach is permanently actif about this subject to recognize the main elements involved in the behavior, dynamics and evolution of cancer in order to highlight the aspect of healing, it describes the cell cycle control in breast cancer as potential oncogenes or tumor suppressor genes [5,6]. To better understand the cancer behavior and how to overcome this pathology, many researcher use theoretical and empirical tools as main manner to study the phenomenon biomathematically [7]. The development of epidemiology methods for tracking dynamics diseases could reap advantages from the use of modeling and simulation, which are crucial decision-making tools. In order to handle actual conditions, the models must be customized for each individual case because every disease has unique biological properties. Several recent studies have produced some intriguing findings, such as those demonstrating theorically how appropriate treatment might limit cells proliferation [8], the estimation of cancer velocity as a hybrid PDE-ODE model [9], many mathematical and analytical techniques have been developed to examine the link between tumor cells and the immune system throughout the therapeutic phase, most of them are based on ordinery differential equations (ODEs) [10,11], stochastic processes in modeling [12][13][14][15] can measure time courses of cells cancer growth, using several mathematical models of diffusion imaging. In [16], prognostic factors and genotypes among patients with breast cancer are predicted to opting for an appropriate therapeutic techniques and treatments, the dataset of cellular images is also used in the diagnosis and treatment processes of various diseases as an artificial intelligence-based technologies [17].
The model named Normal-Tumor-Immune-UnHealthy Diet Model (NTIUNHDM) [10,11] is studied to show that boosting of the immune system can contribute to reduce the risk of cancer.
In a comparable direction, we have noticed that among the most recent models, the model investigating the effects of estrogen combined with immunotherapy describe cancer in a realistic way by the following equations system. where r is the immune response rate, and o is the immune threshold rate; p E IE j E +E saturated term the immune response of due to the estrogen; p M IM j M +M saturated term the immune response due to the immunotherapy. In the fourth equation, the s, this hormonal variable estrogen exist with p, where p is the source rate of estrogen, and θ E is linear fonction decreasing the estrogen, where θ is the decay rate from the body. In the fifth equation, the production of immunotherapy from activated immune cells is injected with the fonction v, dissipate lineary nM and be saturated by To study this research in our situation, we formulate the dynamic of breast cancer with fractional derivatives approach to investigate these studies. We cite as examples to illustrate how the application of fractional calculations can be used to study the behavior of various dynamics [18][19][20][21][22]. This approach is regarded as the generalization of the standard theory of calculus to derivatives and integrals, and its success stems from its demonstrated efficacy in accurately. (2) Where D α is the fractional differentiation operator and α is fractional derivative order.
Every characterisation of the equation's system of model (2) is translated using the schematization of a compartmental diagram at Fig. 1 of breast cancer dynamic as follow.

. Schematic diagram of breast cancer behavior of the formulation (2)
This paper is organized into sections: The next one will provide some mathematical resources regarding fractional derivative, section 3 will establish the solutions and analyze the outcomes of the fractional system model (2), section 4 will provide numerical interpretations, and section 5 will conclude all sets.

MATHEMATICAL TOOLS
The fractional order integral and derivative are briefly defined in this section. We provide some preliminary definitions as follow: Definition 1. The α-order fractional integral of the function f is defined by: Definition 2. The Caputo fractional derivative of the function f is given by: In general case. Remark 1. The Caputo fractional derivative can be defined as an inverse operation of fractional integration. It can be presented as follow by the next definition:

Definition 3. [23] The Mittag-Leffler function is defined by
The Mittag-Leffler function, which may be thought of as a generalization of the role of the exponential function for ordinary differential equations, plays a significant role in case of fractional differential equations.
Due to biological limitations, all initial conditions of the solutions are therefore positive and logically constrained. So, we deduce the result of non-negativity of the solusions. Proof. For the boundness, we have from (2): We assume that λ N and C N two positive constants, to the effect that: We have also, with a = Max(a 1 , a 2 ) and b = Min(b 1 , b 2 ) : Then, we pose λ T and C T two positive constants, to the effect that: We pose λ I and C I two positive constants, to the effect that: From the model (2) we get the next equality.
That give the bounded solution for E.
The last equation of model (2) give: We pose λ T and C T two positive constants, to the effect that: We deduce the boundedness of M.
As a final result, the solution of the problem (2) is bounded.

Breast cancer equilibria.
The steady states of the model (2), the equilibrium instance verify the following equation's system: With simple calculation, we notice the equilibrium points as follow: With: With: We pose: With: With: We pose: With: It exist if n ≥ p I I d j I +I d and With: We pose: With: 3.3. Local stability of the equilibria. In this part, we give the stability resultats as the following theorems.
Theorem 2. At free tumor state P f (N f , 0, I f , E f , M f ).
The equilibrium point P f is stable if |Arg(Roots(Q f ))| > απ 2 .
Proof. The Jacobian matrix at P f is: The characteristic polynomial of J P f is: With: j M +M f ). So, we get the stability, since |arg(Roots(Q f ))| > απ 2 , according to the theorem 1, .
Theorem 3. At death free tumor stages P d f (0, 0, The equilibrium point P d f is stable if l 1 E d f ≥ a 1 , m 1 + g 1 I d f ≥ a 2 and |arg(Roots(Q d f ))| > απ 2 .
Proof. The Jacobian matrix at P d f is: The characteristic polynomial of J P d f is: With: Theorem 4. At death with tumor stages P d (0, T d , The equilibrium point P d is stable if |Arg(Roots(Q d ))| > απ 2 .
Proof. The Jacobian matrix at P d is: The characteristic polynomial of J P d is: With: We get the stability, since |arg(Roots(Q d ))| > απ 2 , according to the theorem 1.
Theorem 5. At Co-existing stages P c (N c , T c , I c , E c , M c ).
The equilibrium point P c is stable if |Arg(Roots(Q c ))| > απ 2 .
Proof. The Jacobian matrix at P c is: The characteristic polynomial of J P c is: With: . According to the theorem 1, we get the stability since |arg(Roots(Q c ))| > απ 2 .

NUMERICAL SIMULATION
In order to validate theoretical findings, in this part, we attempt to numerically represent the solution to the problem (2). We employ the numerical method for fractional differential equations based on the approximation of Lagrange interpolation. The general guidelines of this method is given by the next equation: By discretizing the integral, we set h is the subdivision step, we pose t n = nh for n = 0, 1, 2, . . .
considering uniform subdivision of a time line.
The Lagrange interpolation approximation of F(s, X(s)) function as polynomial P is: P(s) s − t n−1 t n − t n−1 F(t n , X(t n )) + s − t n t n−1 − t n F(t n−1 , X(t n−1 )) In accordance with the Adams technique, we simulate numerically the fractional system (2) to obtain graphical observations of the outcomes. The parameters listed in the Table 1  In figures 3, 4, 5, 6 and 7, we obtain a numerical results of free tumor equilibrium stability for logistic growth rates of normal and tumor cells values a 1 = 1.5 and a 2 = 1.3 respectively.
We also show that the curves of the solutions quickly converge for an ordinary time variation (α = 1). We should notice that for higher values of α, we gain a substantial result and an interpretation that describes the long memory behavior and the solutions converge more quickly to the regular state. The fractional derivative order α impact is shown efficiently for high values.
In figures 8, 9, 10, 11 and 12, we get a numerical results of death free tumor equilibrium state stability for logistic growth rates of normal and tumor cells values respectively a 1 = 0.7 and a 2 = 0.3. We also notice the same impact of fractional derivative.

CONCLUSION
In this work, we realize the existence and well-posedness of the solutions for the proposed fractional system (2). Specifically, our mathematical study can explain and show the evolution of breast cancer. Then, we analyze the stability of the equilibrium points known as free tumor, death free tumor, death free tumor and co-existing equilibruim, respectively. In order to analyze our results numerically, we also demonstrate how the stability of the steady states is influenced by the α-order of the fractional derivative for α values near to unit, which represents the long memory behavior of breast cancer.