Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

1  Introduction

One of the most dreaded and feared diseases is cancer. According to statistics, there are 1300 cancer-related fatalities every day in India. The incidence of cancer has been rising steadily over the past few decades. Several numerical models were created in the meantime to describe how cancer sickness spreads, for instance [1]. To better understand the characteristics of the disease cancer models have been created [2–4].

The biological phenomena procedures that are best described by fractional differential equations have been simulated using fractional calculus. Common integer-order derivative mathematical systems, such as nonlinear models, frequently fail. Fractional calculus has grown in importance during the past several years in fields like electrical engineering, chemistry, economics, control theory, mechanics, and image and signal processing. Fractional order calculus is as old as classical calculus, however, it has only been recently used in mathematics [5,6]. A hybrid fractional operator is built in [7]. Compared to Caputo’s fractional derivative operator, this new operator is more general. It is a linear combination of the Riemann Liouville integral and Caputo fractional derivative. The generalized fractional mathematical models are supposed to be helpful for modeling purposes in epidemiology. Its key benefits are that it captures factuality, captures memory effects, is multistage, and offers superior data fitting [8]. When compared to integer-order models, which either overlook or make it difficult to account for the systems’ memory and hereditary properties, fractional epidemic models provide a powerful tool for doing so. Additionally, the fractional version offers one more degree of freedom than the integer model when it comes to data fitting. A fractional order system provides several advantages, such as the most accurate data fitting, having its memory, and its ability to choose the fractional ordered value that will most accurately characterize a model in the real world. Due to hereditary characteristics, the fractional derivatives models are additionally strengthened and therefore better suited for illustrating a real-life phenomenon [9]. It is more efficient than the standard arrangement and provides advantages for crossover behavior. In [10–12] and the references therein, various mathematical models with intriguing findings are proposed.

Additionally, it can be used to investigate complex phenomena, such as disease models in [13–15]. The hybrid fractional operator, which can be defined as a linear combination of the Riemann-Liouville fractional integral and the Caputo fractional derivative, is one of the most efficient and dependable operators. This operator is more general than the Caputo fractional operator, as shown in [7]. Such fractional-order differential equations are challenging to solve analytically. Therefore, it is crucial to create numerical methods for approximating the solutions to these models.

Some real-world problems have complicated behaviors, so in the last years, piecewise calculus has been developed to more accurately depict these issues [16–19]. Applications in several real-life problems display crossover behavior. Lately, the idea of crossover phases, both differential and integral operators, has been created and applied to many difficult real-world scenarios, like that of chaos and many epidemiologic concerns. The theory seems to perform well when predicting scenarios involving crossover behaviors; for more details see [20].

The cancer models’ fractional-order derivatives are examined in [21]. We are attempting to alter the hyperchaotic cancer model in this scenario using the concept of a piecewise differential model, where its initial portion is dependable on an integer, the second portion is a hybrid order of fractions, while the last part is a hybrid fractional of variable order to identify some hidden insights of the model which will be very helpful both from biological and mathematical perspectives. Therefore, the main aim of this study is to develop two crossover hybrid variable-order derivatives of the cancer model. Moreover, Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator (CPC-AB5SM) will be developed to study the proposed models. Comparative studies with the generalized fifth-order Runge-Kutta method (GRK5M) will be given. Grünwald-Letnikov approximation is used to approximate the hybrid variable-order fractional operator. We will discuss the existence, uniqueness, and stability of the proposed model. Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations will be presented.

The basic structure of the paper is the following: “Basic Notations” introduces key concepts of hybrid fractional-order derivatives. The two types of crossover cancer models are given. Additionally, the proposed models’ existence, uniqueness, and stability analysis are presented in “The Cancer Mathematical Model”. The proposed models are solved numerically through the creation of schemes in the “Numerical Method” and the stability of these schemes is shown. The article discusses the approximation of solutions for the proposed system using CPC-AB5SM and GRK5M in “Numerical Simulations” along with a discussion. “Conclusions” presents the conclusions at the end of the paper.

2  Basic Notations

Definition 2.1. The Caputo proportional constant (CPC) fractional hybrid operator is defined as [7,8]:

 0CPCDtαy(t)=1Γ(1−α)(∫0t(t−s)−α(y(s)K1(α)+y′(s)K0(α))ds)=K1(α)0RLIt1−αy(t)+K0(α)0CDtαy(t),(1)

where, K0(α)=αQ(1−α), K1(α)=(1−α)Qα, Q is constant, 0<α≤1.

Definition 2.2. The variable order fractional CPC integration is stated as follows [7,8]:

 0CPCItα(t)=1K0(α(t))[∫0texp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)y(s)ds].(2)

The generalized mean value theorem being applied, we drive [7]:

y(t)=y(t1)+1K0(α(t))[∫t1texp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)y(s)ds].(3)

2.1 Types of Crossover Derivatives

In this part, we propose two types of crossover derivatives. Let y∈[0,T] be a differentiable function, the types of crossover differential equations are given as follows:

Type 1: The classical, fractional order, and variable order piecewise derivatives are stated as follows:

f(t,y(t))={y′(t),0≤t≤t1, CPCDαy(t),t1<t≤t2, CPCDα(t)y(t),t2<t≤T.(4)

Type 2: The variable order, integer order, and fractional order piecewise derivatives are stated as follows:

f(t,y(t))={ CPCDα(t)y(t),0≤t≤t1,y′(t),t1<t≤t2, CPCDαy(t),t2<t≤T.(5)

3  The Cancer Mathematical Model

Cancer models are developed to help better understand the traits of the illness, for further details, [2]. In [21], the fractional-order derivatives of the cancer models are investigated. Conventional derivatives are unable to simulate the sudden shifts that result in multibehavior. In light of this, some academics have lately noted that the aforementioned process can be modeled by differential equations utilizing piecewise equations. Recent publications in this field that we think are important include [22].

We will introduce two types for the crossover cancer mathematical model:

Type 1: Consider x,y,z are differentiable and defined in Table 1. The first type of crossover cancer model is given as follows:

dx(t)dt=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t),dy(t)dt=C3y(t)(1−y(t))−C4x(t)y(t),dz(t)dt=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}0≤t≤t1,(6)

where, x(0)=x0, y(0)=y0, z(0)=z0.

 0CPCDtαx(t)=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t), 0CPCDtαy(t)=C3y(t)(1−y(t))−C4x(t)y(t), 0CPCDtαz(t)=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}t1<t≤t2,(7)

where, x(t1)=x1, y(t1)=y1, z(t1)=z1.

 0CPCDtα(t)x(t)=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t), 0CPCDtα(t)y(t)=C3y(t)(1−y(t))−C4x(t)y(t), 0CPCDtα(t)z(t)=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}t2<t≤T.(8)

where, x(t2)=x2, y(t2)=y2, z(t2)=z2.

Such that, C1,C2,C3,C4,C5,C6,C7 and C8 are constants.

The definition of all variables within the system in Table 1 and the details of the parameters for the cancer model, see [23]. Here dx(t)dt,dy(t)dt and dx(t)dt are the integer order for 0≤t≤t1,  t1CPCDt2α is the fractional CPC derivative of order 0<α≤1 for t1<t≤t2 and  t2CPCDTα(t) is the variable order fractional of CPC derivative for t2<t≤T.

Type 2: Consider x,y,z are differentiable, then the second type of crossover cancer model given as follows:

 0CPCDtα(t)x(t)=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t), 0CPCDtα(t)y(t)=C3y(t)(1−y(t))−C4x(t)y(t), 0CPCDtα(t)z(t)=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}0≤t≤t1,(9)

where, x(t0)=x0, y(t0)=y0, z(t0)=z0.

dx(t)dt=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t),dy(t)dt=C3y(t)(1−y(t))−C4x(t)y(t),dz(t)dt=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}t1<t≤t2,(10)

where, x(t1)=x1, y(t1)=y1, z(t1)=z1.

 0CPCDtαx(t)=x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t), 0CPCDtαy(t)=C3y(t)(1−y(t))−C4x(t)y(t), 0CPCDtαz(t)=C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t),}t2<t≤T.(11)

where, x(t2)=x2, y(t2)=y2, z(t2)=z2.

Such that, C1,C2,C3,C4,C5,C6,C7 and C8 are constants. The variable fractional order of CPC derivative for 0≤t≤t1, the integer derivative for t1<t≤t2 and fractional CPC derivative for t2<t≤T given as (1).

3.1 Existence and Uniqueness

To study the existence of the solution for the model (7), we will use the fixed-point theorem [24]. First, convert (7) into an integral formula as shown below:

x(t)−x(0)= 0CPCItα(x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t)),y(t)−y(0)= 0CPCItα(C3y(t)(1−y(t))−C4x(t)y(t)),z(t)−z(0)= 0CPCItα(C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t)).(12)

Using the fractional integral of order α provided by (3), we obtain

x(t)=x(0)+1K0(α(t))[∫0texp(K1(α(t))K0(α(t))(t−s))0RLDt1−α(t)(x(s)(1−x(s))−C1x(s)y(s)−C2x(s)z(s))ds],y(t)=y(0)+1K0(α(t))[∫0texp(K1(α(t))K0(α(t))(t−s))0RLDt1−α(t)(C3y(s)(1−y(s))−C4x(s)y(s))ds],z(t)=z(0)+1K0(α(t))[∫0texp(K1(α(t))K0(α(t))(t−s))0RLDt1−α(t)(C5x(s)z(s)x(s)+C6−C7x(s)z(s)−C8z(s))ds].(13)

Among our definitions of kernels are

τ(t,x(t))=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t)),ν(t,y(t))=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C3(−y+1)y)−C4xy),ϕ(t,z(t))=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C5x(t)z(t)x(t)+C6−C7x(t)z(t)−C8z(t)).(14)

Theorem 3.1. The Lipschitz condition can be satisfied by the kernels τ,ν, and ϕ.

Proof. For each proposed kernel, we show this condition. Let x and X be two functions for kernel 1, y and Y for kernel 2, and z and Z for kernel 3. Next, we consider the following:

||τ(t,x(t))−τ(t,X(t))||=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)||(x(t)−X(t))[1−(x(t)−X(t))]−C1(x(t)−X(t))y(t)−C2(x(t)−X(t))z(t)||,||ν(t,y(t))−ν(t,Y(t))||=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C3||(y(t)−Y(t))[1−(y(t)−Y(t))]−Dx(t)(y(t)−Y(t))||),||ϕ(t,z(t))−ϕ(t,Z(t))||=exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C5||x(t)(z(t)−Z(t))x(t)+C6−C7x(t)(z(t)−Z(t))−C8(z(t)−Z(t))||).(15)

Cauchy’s inequality gives us the following:

||τ(t,x(t))−τ(t,X(t))||≤exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)||(x(t)−X(t))[1−(x(t)−X(t))]−C1(x(t)−X(t))y(t)−C2(x(t)−X(t))z(t)||,||ν(t,y(t))−ν(t,Y(t))||≤exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C3||(y(t)−Y(t))[1−(y(t)−Y(t))]−C4x(t)(y(t)−Y(t))||),||ϕ(t,z(t))−ϕ(t,Z(t))||≤exp(K1(α(t))K0(α(t))(t−s)) 0RLDt1−α(t)(C5||x(t)(z(t)−Z(t))x(t)+C6−C7x(t)(z(t)−Z(t))−C8(z(t)−Z(t))||).(16)

Given the recursive formula below, we obtain

x(t)=1K0(α(t))[∫0tτ(s,xn−1(s))ds],y(t)=1K0(α(t))[∫0tν(s,yn−1(s))ds],z(t)=1K0(α(t))[∫0tϕ(s,zn−1(s))ds].(17)

When we apply the norm and the triangle inequality to the difference between both successive terms, we obtain

||Yn(t)||=||xn(t)−Xn−1(t)||≤1K0(α(t))‖∫0t(τ(s,xn−1(s))−τ(s,xn−2(s))ds‖,||Φn(t)||=||yn(t)−Yn−1(t)||≤1K0(α(t))‖∫0t(ν(s,xn−1(s))−ν(s,xn−2(s))ds‖,||Ψn(t)||=||zn(t)−Zn−1(t)||≤1K0(α(t))‖∫0t(ϕ(s,xn−1(s))−ϕ(s,xn−2(s))ds‖,(18)

such that,

xn(t)=∑m=0∞Ym(t);yn(t)=∑m=0∞Φm(t);zn(t)=∑m=0∞Ψm(t).(19)

The Lipschitz condition is satisfied by the kernels τ,ν, and ϕ, hence we obtain

||Yn(t)||=||xn(t)−Xn−1(t)||≤1K0(α(t))[Δ1∫0t||xn−1(t)−Xn−2(t)||ds],||Φn(t)||=||yn(t)−Yn−1(t)||≤1K0(α(t))[Δ2∫0t||yn−1(t)−Yn−2(t)||ds],||Ψn(t)||=||zn(t)−Zn−1(t)||≤1K0(α(t))[Δ3∫0t||zn−1(t)−Zn−2(t)||ds],(20)

where, K0(α(t))=α(t)Q(1−α(t)), and Q,Δ1,Δ2,Δ3 are constants. This concludes Theorem 3.1 Proof. ■

Theorem 3.2. Kernels τ, ν and ϕ satisfed Lipsctize condition by considering (20) is bounded.

Proof. The Lipschitz condition is satisfied by the kernels τ,ν, and ϕ when (20) is bounded, as we showed. With the use of the recursive method and the results from (20), we can derive the relationship shown below:

||Yn(t)||≤||x(0)||+{1K0(α(t))Δ1t}n,||Φn(t)||≤||y(0)||+{1K0(α(t))Δ2t}n,||Ψn(t)||≤||z(0)||+{1K0(α(t))Δ3t}n.(21)

As a result, Eq. (21) is smooth and exists. However, to demonstrate that the aforementioned functions represent a system of solutions, we suppose

x(t)=xn(t)−Θ1(n);y(t)=yn(t)−Θ2(n);z(t)=zn(t)−Θ2(n),(22)

where the reminder terms for the series solution are Θ1(n),Θ2(n) and Θ3(n). Thus,

||x(t)−Xn(t)||=1K0(α(t))∫0tτ(s,x−Θ1(n)),||y(t)−Yn(t)||=1K0(α(t))∫0tν(s,y−Θ2(n))ds‖,||z(t)−Zn(t)||=1K0(α(t))∫0tϕ(s,z−Θ3(n))ds‖.(23)

Using the Lipschitz condition and the norm on both sides, we obtain

‖x(t)−x(0)−1K0(α(t))∫0t(τ(s,x(s))ds‖≤‖Θ1(n)‖+1K0(α(t))Δ1t‖Θ1(n)‖,‖y(t)−y(0)−1K0(α(t))∫0t(ν(s,y(s))ds‖≤≤‖Θ2(n)‖+1K0(α(t))Δ2t‖Θ2(n)‖,‖z(t)−z(0)−1K0(α(t))∫0t(τ(s,z(s))ds‖≤‖Θ3(n)‖+1K0(α(t))Δ3t‖Θ3(n)‖.(24)

The result of taking the limit n→∞ is

x(t)=x(0)+1K0(α(t))∫0t(τ(s,x(s))ds,y(t)=y(0)+1K0(α(t))∫0t(ν(s,y(s))ds,z(t)=z(0)+1K0(α(t))∫0t(ϕ(s,z(s))ds.(25)

■

Theorem 3.3. The system described in (7) has a unique solution.

Proof. To show that, let us obtain other solutions, such as x(t), y(t), and z(t).

x(t)−X(t)=1K0(α(t))∫0t(τ(s,x(s))−τ(s,X(s))ds,y(t)−Y(t)=1K0(α(t))∫0t(ν(s,y(s))−ν(s,Y(s))ds,z(t)−Z(t)=1K0(α(t))∫0t(ϕ(s,z(s))−ϕ(s,Z(s))ds.(26)

When we apply the norm on the two sides, the result is

‖x(t)−X(t)‖=1K0(α(t))∫0t‖(τ(s,x(s))−τ(s,X(s))‖ds,‖y(t)−Y(t)‖=1K0(α(t))∫0t‖(ν(s,y(s))−ν(s,Y(s))‖ds,‖z(t)−Z(t)‖=1K0(α(t))∫0t‖(ϕ(s,z(s))−ϕ(s,Z(s))‖ds.(27)

Given the Lipschitz condition and the knowledge that a solution is bounded, we obtain

‖x(t)−X(t)‖≤{1K0(α(t))Δ1W1t}n,‖y(t)−Y(t)‖≤{1K0(α(t))Δ2W2t}n,‖z(t)−Z(t)‖≤{1K0(α(t))Δ2W3t}n.(28)

Since this holds for all n, x(t)=X(t);y(t)=Y(t);z(t)=Z(t). As a result, it proves the uniqueness of the proposed solution. ■

3.2 Existence and Uniqueness of the Solution Piecewisely

We validate the linear growth and Lipschitz condition characteristics to achieve this. Additionally, consider B to be a Banach space with the norm ||ψ||∞=supt∈Dψ|ψ(t)|. ∀t∈(0,t1), we assume that there are four positive constants such that ||X||∞<A1, ||Y||∞<A2 and ||Z||∞<A3.

dxdt=g1(x,y,z,t),dydt=g2(x,y,z,t),dzdt=g3(x,y,z,t),(29)

When the right-hand side of the equations in the (6) system is represented by gρ(x,y,z,t),ρ=1,2,3. First, we make sure that

|gρ(x,t)|2<kρ(|xρ|2+1),|gρ(x1,t)−|gρ(x2,t)|2<k¯ρ|x1−x2|2.(30)

To show the existence and uniqueness piecewise. We assume a function g1(x,y,z,t)

|g1(x,y,z,t)|2=|x(t)(1−x(t))−C1x(t)y(t)−C2x(t)z(t)|≤4(|x|2−|x2|2−C1|x|2|y|2−C2|x|2|z|2),≤4(|x2|−|x4|−C1|x2|||y2||∞−C2|x2|||z2||∞),≤4(1−|x2|−C1||y2||∞−C2||z2||∞)|x2|,≤4(−C1||y2||∞−c2||z2||∞)(1+1C1||y2||∞+C2||z2||∞|x2|)|x2|.(31)

Continuing the same technique g2(x,y,z,t)

|g2(x,y,z,t)|2=|C3y(t)−y2(t)−C4x(t)y(t)|≤5C3|y|2−|y2|2−C4|x|2|y|2,≤5C3|y2|−|y4|−C4|x2||y2|,≤5C3|y2|−|y4|−C4||x2||∞|y2|,≤5(C3−|y2|−C4||x2||∞)|y2|,≤5(C3−C4||x2||∞)(1+1C3−C4||x2|||y2|)|y2|.(32)

In the case of the function g3(x,y,z,t)

|g3(x,y,z,t)|2=C5|xzx+C6−C7xz−C8z|2≤4(C5|xzx+C6|2−C7|x|2|z|2−C8|z|2),≤4(C5||x2||∞||(x+C6)2||∞|z2|−C7||x2||∞|z2|−C8|z2|),≤4(C5||x2||∞||(x+C6)2||∞−C7||x2||∞−C8)|z2|,≤4(C5||x2||∞||(x+C6)2||∞−C7||x2||∞−C8)|z2|,≤−4H(−C5||x2||∞H||(x+C6)2||∞+C7C8||x2||∞+1)|z2|.(33)