Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted.


Introduction
The mathematical modeling and analysis of tumor growth is a crucial point to understand different aspects of cancer and to discover potential treatments [1][2][3][4]. Prof. H.I. Freedman has made important contributions in tumor-immune competitive systems and their application to chemotherapy [5,6].
In this paper, we investigate a tumor model with strong Allee effect in the following form [7] (1) with initial size of the tumor population is T(0) = T 0 ≥ 0 . We assume that tumor cells grow logistically as tumor growth slows down due to insufficient nutrients. Here, represents the intrinsic growth rate and k is the maximum carrying capacity of tumor cells [8]. It can be observed that 0, k and c are constant solutions. The parameter c ≥ 0 represents the strong Allee threshold. The growth of population subject to Allee effects is reduced at low density and it is related to the existence of a threshold size for the visibility of that population and they may altered the long-term persistence of the population. Note that for T between 0 and There is a long history of fractional order derivative and it plays a vital role in nonlinear mathematical model. There are some limitations in integer derivatives of nonlinear models. Therefore the application of fractional order derivative is very important in the mathematical model. In fractional order differential equation, there are mainly two types of derivatives known as Caputo-Fabrizio derivative and Riemann-Liouville derivative. In our study, we used Caputo-Fabrizio fractional order derivative.
The following fractional derivative of the model (1) has been considered as with ∈ (0, 1) and D the Caputo fractional derivative of a C 1 function and T is defined as In this paper, we investigate the fractional version of the model (1) where is the Caputo-Fabrizio fractional derivative [9][10][11] and in the line of [12]. The paper is organized in the following way. We investigate the Caputo-Fabrizio fractional order derivative and its corresponding fractional integral in the Sect. 2. We implicitly solved the Caputo-Fabrizio fractional derivative with order ∈ (0, 1) of the proposed model (1) in the next Section. In the same Section, we plot the fractional Caputo-Fabrizio logistic equation with strong Allee effect (7) for different values of by using MATH-EMATICA. The paper ends with a brief conclusion.

Fractional Calculus with Non-singular Kernel
Let us assume that ∈ (0, 1) . The classical Riemann-Lioville fractional integral is given by We have [13],

Solution for the Fractional Differential Equation
Assume that T is a solution of (3). Integration leads to where Therefore by (6), we get where T(0) = T 0 is the initial value. Taking derivative both sides of the above equation leads to

T(t) = P(t),
We notice that if = 1 then we recover the Allee model (1). After some algebraic manipulation, for 0 < < 1 equation (7) , , We plot the solutions of the proposed model (7) (7) and above all the solution of the Caputo-fractional logistic differential equation.

Conclusions
In this paper, we proposed a mathematical model of tumor cell population with strong Allee effect, which is constructed by Sardar et al. [7] . Then we introduce fractional order derivative in our model. To solve the fractional logistic differential equation with Allee effect, we use fractional calculus with non-singular kernel. We calculate the analytical solutions in implicit form of the tumor cell population with strong Allee effect. Finally, we plot our solutions for different values of . , . Fig. 1 Solution of the ordinary logistic differential equation with strong Allee effect for the initial value T 0 = 0.5. Classical logistic equation with strong Allee effect (blue colour), fractional Caputo-Fabrizio logistic equation with strong Allee effect (7) for = 1∕2 (black) and for = 1∕4 (green)