Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Kamal Shah; Shabir Ahmad; Aman Ullah; Thabet Abdeljawad

doi:10.1515/phys-2024-0032

Article Open Access

Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Kamal Shah , Shabir Ahmad , Aman Ullah and Thabet Abdeljawad

Published/Copyright: May 21, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

Keywords: fractal-fractional operators; myeloid leukemia; model; Adams–Basforth method

1 Introduction

Abel first studied the notable application of fractional calculus (FC) in physics. In the same way, Riemann, Liouville, Lagrange, Hadmard, etc. have greatly contributed in this field [1]. Latter, different definitions were introduced by numerous researchers including Caputo, Fabrizio, Atangana and Baleanu, etc. Therefore, the area has given much more attention and also been applied to study various real-world problems. For some historical contributions in this regard, we refer to studies by AlBaidani et al. [2] and Ferrari et al. [3]. Recently, researchers have conducted many results for the existence theory and numerical solutions for different problems using fractional differential operators. Here, some significant contribution has been referred to previous studies [4–8]. Moreover, the use of FC in investigating different real-world process is an important area of research in recent times. Here, we refer some previous studies [9–13].

The different concepts of FC have been used in applied mathematics and proved to be a good operator for modeling any physical phenomena. Mostly, three types of operators are used in FC, which are based on three kernels, i.e., power law, exponential, and the Mittag–Leffler-type kernels. Modeling and graphical visualization using these three types of kernels are heavily influenced by the study of fractional differential equations (here, we refer previous studies [14,15]). Mathematical models have been studied very well using fractional order derivatives [16–18]. For more details, see some contributions on the said area in previous studies [19–23]. Kumar et al. [24] studied the susceptible infected recovered susceptible malaria infection model under Caputo–Fabrizio operator. A non-integer-order biological model with the consideration carrying capacity is studied in by Srivastava et al. [25]. Furthermore, human immunodeficiency virus-1 infection model of CD4+ T-cells with effects of antiviral drug therapy in fractional operator sense was analyzed by Kumar et al. [26]. We include recent studies of FC in different applied sciences in Singh [27] and Ahamed et al. [28]. For further analysis, numerical results, and stability theory for various problems of FC, we refer to previous studies [29–33]. Also, researchers have derived some appropriate results devoted to computational analysis of partial differential equations with Mittag–Leffler kernel as well as of some biological models. Readers can read the said work in previous studies [34–39].

Non-local derivatives, in general, are appropriate for such circumstances because, depending on whether there is a power law, fading memory, or crossover effects, they can preserve non-localities and also some memory effects. However, if more intricate behaviors cannot be replicated using a fading memory, crossover behavior, or power law, the recently developed fractals fractional derivative (FFD) operators may be more effective mathematical tools to cope with such behaviors [40]. The FFD operators have many applications in real-world problems. A non-Newtonian generalization of the derivative is the fractal derivative, which is often known as the Hausdorff derivative. This type of derivative was used to study a fluid flow problems in many studies. The FFD operators have recently attracted the attention of researchers very well. As the concept is still fresh, this area has to be studied. These new operators were introduced by Atangana: connecting the fractal calculus with FC [41]. Different complex behaviors were studied through these operators [42,43]. Also, recent contribution can be read in previous studies [44,45]. Compact in nature, fractal theory is a subfield of nonlinear science with important applications in turbulence, aquifers, porous media, and other media that typically display fractal qualities. When it comes to understanding concepts such as fractional-order integration and differentiation and their mutually inverse relationship, FC is crucial. Many branches of engineering and research, including electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signal processing, use FC [46].

The most powerful tools to investigate real-world problems from mathematical perspectives are devoted to modeling. The said area has given much more attention in the last many decades. With the help of mathematical formulations, we easily understand about the transmission dynamics of the process. Keeping in mind the applications, researchers have increasingly used the said area to model various chronic diseases. In this regard, significant work has been performed using classical and fractional order derivatives. Here, for reference, we refer few articles [47–49].

Chronic myelogenous leukemia (CML or chronic granulocytic leukemia) is a cancer of white blood cells (WBCs). In CML, a wide range of WBCs are produced by bone marrow, which also affects cell in blood circulation. This form of leukemia is uncommon. Adults are more likely to acquire CML than youngsters. CML occurs when the myeloid cells are transformed into immature cancer cells through a genetic alteration. Such cells then expand slowly and overtake the healthy cells in bone marrow and blood [50] called translocation. In translocation, a chromosome 9 (called the ABL gene) breaks and makes bond with section of the chromosome 22 (called the BCR gene), forming a Philadelphia chromosome (Ph chromosome). Therefore, Ph chromosome is a bond of two genes (ABL and BCR) that forms a fusion of single gene BCR-ABL [51,52]. It is only found in blood-forming cells. It causes myeloid cells that make an abnormal activated tyrosine kinase enzyme known as fusion protein, which allows us to grow WBCs out of control [53,54]. The number of WBCs is normally regulated by the body; however, more WBCs are generated in stress or during infections, but after the recovery the numbers become normal. In CML, the abnormal BCR-ABL enzyme behaves as a switch stuck-in the on position, and it proceeds to promote the growth and multiplication of WBCs. In addition to increasing WBCs, there is also an increase in the number of blood platelets that help in blood clotting, and amount of red blood cells carrying oxygen can be decreased. Therefore, an irregular or an unusual amount of platelets or even RBC are seen in CML. The symptoms of CML may include easy bleeding, pain in bones, feeling full by eating some food, feeling tired, weight loss with no exercise, fever, fullness or pain below ribs on left side, loss of appetite, and very high sweating in sleep. CML has three phases, which are accelerated, chronic, and blastic. The information from tests and procedures performed to diagnose CML is also used to plan treatment. Due to huge amount of blast-cells in blood and bone marrow, space reduces for healthy WBC, RBC, and platelets, which causes anemia, infections, bleeding, bone pain, or a sensation of fullness under the left ribs. Blast cells in the bone marrow and blood as well as the intensity of symptoms will determine the disease stage. During chronic CML process, blast cells are less than 10% of the cells in bone marrow and blood. During the accelerated phase of CML, the blast cells are 10 to 19% of the blood cells and the bone marrow. Blast cells are 20% or more of the cells in bone marrow or blood in blastic process CML.

Here, it should be worth to mention that researchers have used the concepts of mathematical model to understand and investigate the transmission dynamics of the aforementioned disease properly. For instance, Moore and Li [55] applied the traditional calculus to construct the following model as

(1) d T d t = s n − d n T − k n T C C + η , d ℰ d t = α n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , d C d t = r c C ln C max C − d c C − γ c C ℰ ,

along with initial conditions

(2) T ( 0 ) = T 0 , ℰ ( 0 ) = ℰ 0 , C ( 0 ) = C 0 ,

where T represents the native T-cells, ℰ represents the effector T-cells particular with CML, and C represents the CML cancer cells. Description of the nomenclatures is given in Table 1.

Table 1

Nomenclature and their description

Nomenclature	Description
s n	is the T source term
d n	represents the T death rate
d e	is the ℰ death rate
d c	represents the C death rate
k n	represents the T differentiation
η	is the Michaelis–Menten coefficient
α n	represents the ℰ proliferation
α e	represents the ℰ recruitment
C max	represents the maximum C
r c	is the C growth
γ e	represents the ℰ loss due to T
γ c	represents the C loss due to ℰ

Recently the significant applications of FFD operators have been found. Because the mentioned operators can describe many real-world process with irregular or complex geometry with more excellent ways. The said area has been found very fruitful in studying epidemiological as well as other chronic disease models in terms of mathematical concepts [56,57]. Therefore, our aim of this study is to establish sufficient conditions for the existence theory as well as numerical solutions for the proposed model given in Eq. (1) for three different FFD operators. The three different FFD operators contain power law, exponential and Mittag–Leffler-type kernels. Each and every operator has its own merits properties, which have been discussed in detail in the study by Khan and Atangana [58]. Therefore, we first consider the model given in Eq. (1) under the FFD operator with power-law kernel as

(3) D t ϱ , β 0 P ℱℱD ( T ( t ) ) = s n − d n T − k n T C C + η , D t ϱ , β 0 P ℱℱD ( ℰ ( t ) ) = ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , D t ϱ , β 0 P ℱℱD ( C ( t ) ) = r c C ln C max C − d c C − γ c C ℰ .

Furthermore, since the FFD operators have exponential and Mittag–Leffler kernels also attracted proper attentions from researchers in the past few years. These operators exhibit some more keen features to use in the analysis of disease models. Therefore, we also consider our proposed Model (1) under the FFD operator with exponential kernel as

(4) D t ϱ , β 0 E ℱℱD ( T ( t ) ) = s n − d n T − k n T C C + η , D t ϱ , β 0 E ℱℱD ( ℰ ( t ) ) = ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , D t ϱ , β 0 E ℱℱD ( C ( t ) ) = r c C ln C max C − d c C − γ c C ℰ .

The FFD operator with exponential kernel has been generalized to FFD operator with Mittag–Leffler kernel. Therefore, we also undertake our proposed Model (1) under the Mittag–Leffler-type kernel as

(5) D t ϱ , β 0 M ℱℱD ( T ( t ) ) = s n − d n T − k n T C C + η , D t ϱ , β 0 M ℱℱD ( ℰ ( t ) ) = ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , D t ϱ , β 0 M ℱℱD ( C ( t ) ) = r c C ln C max C − d c C − γ c C ℰ .

We use fixed point theory to establish the local existence and uniqueness for the aforementioned systems of FFD operators as (3)–(5), respectively. Furthermore, using the Adams–Bashforth numerical method [59], we establish numerical schemes for the three considered FFD operators systems. Various numerical results are presented graphically for the proposed model. Also, for the existence theory, we use a fixed point theory from Istratescu [60].

Our manuscript is arranged as follows: detailed introduction is given in Section 1. Preliminaries are given in Section 2. Our first part of main results is given in Section 3. Numerical schemes have developed in Section 4. In addition, numerical simulations are performed in Section 5. Finally, discussion and conclusion are given in Section 6.

2 Preliminaries

Consider that ℋ ( t ) be continuous and also fractal differentiable on the interval ( m , n ) . Let 0 < ϱ , β ≤ 1 , where ϱ , and β stand for the fractional orders and fractal dimension, respectively. Here, FFI represents the fractal-fractional integral.

Definition 2.1

[58] The Riemann–Liouville (R-L) FFD operator with power-law-type kernel of ℋ ( t ) is defined by

D t ϱ , β 0 P ℱℱD ( ℋ ( t ) ) = 1 Γ ( m − ϱ ) d d t β ∫ 0 t ( t − ξ ) m − ϱ − 1 ℋ ( ξ ) d ξ ,

where d d ξ β ℋ ( ξ ) = lim t → ξ ℋ ( t ) − ℋ ( ξ ) t β − ξ β .

Definition 2.2

[58] The FFD operator of ℋ ( t ) with exponential kernel is defined as

D t ϱ , β 0 E ℱℱD ( ℋ ( t ) ) = ℳ ( ϱ ) 1 − ϱ d d t β ∫ 0 t exp − ϱ 1 − ϱ ( t − ξ ) ℋ ( ξ ) d ξ ,

where ℳ ( ϱ ) = ( 1 − ϱ ) + ϱ Γ ( ϱ ) .

Definition 2.3

[58] The FFD operator with Mittag–Leffler kernel given by of ℋ ( t ) is presented by

D t ϱ , β 0 M ℱℱD ( ℋ ( t ) ) = ℳ ( ϱ ) 1 − ϱ d d t β ∫ 0 t E ϱ − ϱ 1 − ϱ ( t − ξ ) ϱ ℋ ( ξ ) d ξ ,

where ℳ ( ϱ ) = 1 − ϱ + ϱ Γ ( ϱ ) .

Definition 2.4

[58] The FFI of ℋ ( t ) in power-law case is given by

ℐ t ϱ , β 0 P ℱℱℐ ℋ ( t ) = β Γ ( ϱ ) ∫ 0 t ( t − ξ ) ϱ − 1 ξ β − 1 ℋ ( ξ ) d ξ .

Definition 2.5

[58] The FFI of ℋ ( t ) with exponential decay type kernel is given by

ℐ t ϱ , β 0 E ℱℱℐ ℋ ( t ) = β ( 1 − ϱ ) t β − 1 ℋ ( t ) ℳ ( ϱ ) + ϱ β ℳ ( ϱ ) ∫ 0 t ξ β − 1 ℋ ( ξ ) d ξ .

Definition 2.6

[58] The FFI of ℋ ( t ) with Mittag–Leffler-type kernel is

ℐ t ϱ , β 0 M ℱℱℐ ℋ ( t ) = ( 1 − ϱ ) β t β − 1 ℋ ( t ) ℳ ( ϱ ) + ϱ β ℳ ( ϱ ) ∫ 0 t ξ ϱ − 1 ( t − ξ ) β − 1 ℋ ( ξ ) d ξ .

3 Main work

We compute the equilibria points of Model (3). If we put left sides of Model (3) equal to zero, then one has T ˜ = s n d n , and from the second equation of Eq. (3), one has ℰ ˜ = 0 . In addition, there does not exist any other equilibria for which C ˜ = 0 . Hence, E 0 = ( T ˜ , ℰ ˜ , C ˜ ) = s n d n , 0 , 0 represents the healthy (trivial) equilibrium solution of System (3). Furthermore, it has been proved in the study by Moore and Li [55] that the equilibrium point E 0 is asymptotically stable.

Theorem 3.1

The trivial equilibrium point E 0 is locally asymptotically stable.

Proof

The Jacobian matrix of Model (3) at E 0 = s n d n , 0 , 0 is given by

J ( E 0 ) = − s n d n 0 k n γ C 0 − d e d n η d n 2 α n k n s n 0 0 − ( γ C + d C ) d n .

The eigen values of J ( E 0 ) are given by λ 1 = − s n d n , λ 2 = − d e d n , and λ 3 = − ( γ C + d C ) d n . We see that the real parts of all eigen values are negative. Therefore, the equilibrium point E 0 is locally asymptotically stable.□

Remark 3.2

Moreover, any other non-trivial equilibria are denoted by E * = ( T ˜ * , ℰ ˜ * , C ˜ * ) . In the third equation of the proposed model, the logarithmic function of C is involved. Therefore, third equation increases as C increases. Hence, for the term to be zero in third equation of the proposed model, C must be negative. Thus, the expression in the third equation decreases if C > 0 . Hence, there exists a unique value of C for which the third equation of Model (3) becomes zero. Thus, we conclude that there exists at most one equilibrium point E * , which occurs only if

d n > γ C ln γ C C max d n .

Hence, this equilibrium must have three nonnegative populations for it to have any physical significance. Keeping in mind this discussion, as all parameters of Model (3) are positive, hence from the first equation of Model (3), T 0 > 0 , ℰ 0 > 0 , C 0 > 0 , the solution T ( t ) > 0 , ℰ ( t ) > 0 , and C ( t ) > 0 , for every t ∈ [ 0 , b ] , b < ∞ . Hence, we conclude that if

d n < γ C ln γ C C max d n

holds, then both E 0 and E * are globally asymptotically stable.

We will explore the existence and the uniqueness of solution of the suggested model’s solution under the FFD with all the three different kernels using fixed point theory in this part. For other types of operators, one can easily derive the existence results. We will use the given theorem for the existence and uniqueness of solution. Here, X = C [ 0 , T ] is the Banach space, then Z = X × X × X is also Banach space with norm defined by

∥ Ψ ∥ ∞ = sup t ∈ [ 0 , T ] ∣ Ψ ( t ) ∣ .

Theorem 3.3

[60] Let O be a contraction operator on a Banach space say Z , then O has a unique fixed point.

Let us write System (3) as follows using Φ = ( T , ℰ , C ) :

(6) D t ϱ , β 0 P ℱℱD [ Φ ( t ) ] = F ( t , Φ ( t ) ) , Φ ( 0 ) = Φ 0

where

Φ ( t ) = T ( t ) , ℰ ( t ) , C ( t ) ,

and

F ( t , Φ ( t ) ) = ℋ 1 ( t , T , ℰ , C ) = s n − d n T − k n T C C + η , ℋ 2 ( t , T , ℰ , C ) = ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , ℋ 3 ( t , T , ℰ , C ) = r c C ln C max C − d c C − γ c C ℰ .

Replacing the R-L derivative of fractional order ϱ (6) by the Caputo to include initial condition, we obtain the equivalent integral form as follows:

(7) Φ ( t ) = Φ 0 + β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 F ( ξ , Φ ( ξ ) ) d ξ .

Since ( T , ℰ , C ) is bounded, we further described assumptions as:

F ( t , Φ ( t ) ) is continuous and bounded.
There is a constant M F > 0 , we have for Φ , Φ ¯ ∈ Z ,
∣ F ( t , Φ ) − F ( t , Φ ¯ ) ∣ ≤ M F ∣ Φ − Φ ¯ ∣ .

Theorem 3.4

The proposed Model (3) has a unique solution if K = β M F B ( β , ϱ ) Γ ( ϱ ) < 1 , where B ( β , ϱ ) is the beta function defined by

B ( β , ϱ ) = Γ ( β ) Γ ( ϱ ) Γ ( β + ϱ ) .

Proof

Let us define the operator O : Z → Z by

(8) O Φ ( t ) = Φ 0 + ϱ Γ ( β ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 [ F ( ξ , Φ ( ξ ) ) − F ( ξ , Φ ¯ ( ξ ) ) ] d ξ .

Let Φ , Φ ¯ ∈ Z , using the assumption A 2 , we have

‖ O Φ − O Φ ¯ ‖ ∞ = sup t ∈ [ 0 , T ] β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 F ( ξ , Φ ( ξ ) ) d ξ ≤ sup t ∈ [ 0 , T ] β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ∣ F ( ξ , Φ ( ξ ) ) − F ( ξ , Φ ¯ ( ξ ) ) ∣ d ξ ≤ sup t ∈ [ 0 , T ] β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 M F ∣ Φ ( ξ ) − Φ ¯ ( ξ ) ∣ d ξ ≤ β M F ‖ Φ − Φ ¯ ‖ ∞ Γ ( ϱ ) ∫ 0 T ξ β − 1 ( T − ξ ) ϱ − 1 d ξ = K ‖ Φ − Φ ¯ ‖ ∞ ,

where

∫ 0 T ξ β − 1 ( T − ξ ) ϱ − 1 d ξ = T ϱ + β − 1 B ( β , ϱ ) .

Hence, one has

‖ O Φ − O Φ ¯ ‖ ∞ ≤ K ‖ Φ − Φ ¯ ‖ ∞ ,

which shows that O satisfies the Banach contraction theorem. Thus, Problem (3) has a unique solution.□

By Definition 2.3, we have

D t ϱ , β 0 M ℱℱD [ Φ ( t ) ] = ℳ ( ϱ ) 1 − ϱ d d t β ∫ 0 t F ( ξ , Φ ( ξ ) ) E ϱ × − ϱ 1 − ϱ ( t − ξ ) ϱ d ξ .

Since the integral is differentiable, we achieve

D t ϱ 0 M ℱℱD [ Φ ( t ) ] = 1 β t β − 1 ℳ ( ϱ ) 1 − ϱ d d t × ∫ 0 t F ( ξ , Φ ( ξ ) ) E ϱ − ϱ 1 − ϱ ( t − ξ ) ϱ d ξ .

Eq. (7) becomes

(9) ℳ ( ϱ ) 1 − ϱ d d t ∫ 0 t F ( ξ , Φ ( ξ ) ) E ϱ × − ϱ 1 − ϱ ( t − ξ ) ϱ d ξ = β t β − 1 F ( t , Φ ( t ) ) .

Replacing the R-L derivative of fractional order ϱ (9) by the Caputo to include initial condition, we obtain the equivalent integral form as follows:

(10) Φ ( t ) = Φ ( 0 ) + 1 − ϱ ℳ ( ϱ ) β t β − 1 F ( t , Φ ( t ) ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ( t − ξ ) ϱ − 1 F ( ξ , Φ ( ξ ) ) ξ β − 1 d ξ .

Theorem 3.5

Model (4) has a unique solution if K 1 = β T β − 1 ( 1 − ϱ ) ℳ ( ϱ ) + ϱ β T ϱ + β − 1 B ( ϱ , β ) ℳ ( ϱ ) Γ ( ϱ ) M F < 1 holds.

Proof

Let us define the operator by

(11) O Φ ( t ) = Φ ( 0 ) + 1 − ϱ ℳ ( ϱ ) β t β − 1 F ( t , Φ ( t ) ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ( t − ξ ) ϱ − 1 F ( ξ , Φ ( ξ ) ) ξ β − 1 d ξ .

Let Ψ , Ψ ¯ ∈ Z , using assumption A 2 , we have

‖ O Φ − O Φ ¯ ‖ ∞ ≤ sup t ∈ [ 0 , T ] 1 − ϱ ℳ ( ϱ ) β t β − 1 F ( t , Φ ( t ) ) − 1 − ϱ ℳ ( ϱ ) β t β − 1 F ( t , Φ ¯ ( t ) ) + sup t ∈ [ 0 , T ] ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 F ( ξ , Φ ( ξ ) ) d ξ − ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 F ( ξ , Φ ¯ ( ξ ) ) d ξ ≤ M F ( 1 − ϱ ) ℳ ( ϱ ) β T β − 1 ‖ Φ − Φ ¯ ‖ ∞ + ϱ β ℳ ( ϱ ) Γ ( ϱ ) M F ‖ Φ − Φ ¯ ‖ ∞ ∫ 0 T ξ β − 1 ( T − ξ ) ϱ − 1 d ξ = β T β − 1 ( 1 − ϱ ) ℳ ( ϱ ) + ϱ β B ( ϱ , β ) T ϱ + β − 1 ℳ ( ϱ ) Γ ( ϱ ) M F ‖ Φ − Φ ¯ ‖ ∞ = K 1 ‖ Φ − Φ ¯ ‖ ∞ ,

which implies that

‖ O Φ − O Φ ¯ ‖ ∞ ≤ K 1 ‖ Φ − Φ ¯ ‖ ∞ ,

so O is a contraction and has a unique fixed point. Therefore, Model (5) has a unique solution.□

Remark 3.6

The same procedure of Theorem 3.5 can be repeated for Model (4) to show the existence of unique solution.

4 Numerical schemes

We will use the Lagrange piecewise interpolation to generate numerical schemes for the proposed models under three different kinds of FFD operators. In this section, we discuss the following three cases.

4.1 Numerical scheme for Model (3)

Now, we will derive the numerical scheme of Model (3) FFD operator with power-law kernel. From System (3), we have

(12) D 0 , t ϱ ℛℒ ( T ( t ) ) = β t β − 1 s n − d n T − k n T C C + η , D 0 , t ϱ ℛℒ ( ℰ ( t ) ) = β t β − 1 ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , D 0 , t ϱ ℛℒ ( C ( t ) ) = β t β − 1 r c C ln C max C − d c C − γ c C ℰ .

Replacing the R-L derivative of fractional order ϱ (12) by the Caputo to include initial condition, we obtain the equivalent integral system as follows:

(13) T ( t ) = T ( 0 ) + β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ ( t ) = ℰ ( 0 ) + β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 2 ( ξ , T , ℰ , C ) d ξ , C ( t ) = C ( 0 ) + β Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 3 ( ξ , T , ℰ , C ) d ξ ,

where

(14) ℋ 1 ( ξ , T , ℰ , C ) = s n − d n T − k n T C C + η , ℋ 2 ( ξ , T , ℰ , C ) = ϱ n k n T C C + η + α e ℰ C C + η − d e ℰ − γ e C ℰ , ℋ 3 ( ξ , T , ℰ , C ) = r c C ln C max C − d c C − γ c C ℰ .

Here, we derive the numerical algorithm of the aforementioned system at t = t n + 1 . So, Model (13) becomes

(15) T n + 1 = T 0 + β Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ 0 + β Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 2 ( ξ , T , ℰ , C ) d ξ , C n + 1 = C 0 + β Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 3 ( ξ , T , ℰ , C ) d ξ .

Then, the approximation of the aforementioned system is given as follows:

(16) T n + 1 = T 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 2 ( ξ , T , ℰ , C ) d ξ , C n + 1 = C 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 ℋ 3 ( ξ , T , ℰ , C ) d ξ .

We approximate the function ξ β − 1 ℋ 1 ( ξ , T , ℰ , C ) using the Lagrangian piece-wise interpolation (LPI) in [ t Ω , t Ω + 1 ] as follows:

(17) P Ω ( ξ ) = ξ − t Ω − 1 t Ω − t Ω − 1 t Ω β − 1 ℋ 1 ( t Ω , T Ω , ℰ Ω , C Ω ) − ξ − t Ω t Ω − t Ω − 1 t Ω − 1 β − 1 ℋ 1 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) , Q Ω ( ξ ) = ξ − t Ω − 1 t Ω − t Ω − 1 t Ω β − 1 ℋ 2 ( t Ω , T Ω , ℰ Ω , C Ω ) − ξ − t Ω t Ω − t Ω − 1 t Ω − 1 β − 1 ℋ 2 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) , R Ω ( ξ ) = ξ − t Ω − 1 t Ω − t Ω − 1 t Ω β − 1 ℋ 3 ( t Ω , T Ω , ℰ Ω , C Ω ) − ξ − t Ω t Ω − t Ω − 1 t Ω − 1 β − 1 ℋ 3 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) .

Thus, System (16) becomes

(18) T n + 1 = T 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 P Ω ( ξ ) d ξ , ℰ n + 1 = ℰ 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 Q Ω ( ξ ) d ξ , C n + 1 = C 0 + β Γ ( ϱ ) ∑ Ω = 0 n ∫ 0 t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 R Ω ( ξ ) d ξ .

Then, we reach

(19) T n + 1 = T 0 + β ( Δ t ) ϱ Γ ( ϱ + 2 ) ∑ Ω = 0 n [ t Ω − 1 β − 1 ℋ 1 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 1 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n + 1 − Ω ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] , ℰ n + 1 = ℰ 0 + β ( Δ t ) ϱ Γ ( ϱ + 2 ) ∑ Ω = 0 n [ t Ω − 1 β − 1 ℋ 2 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 2 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n + 1 − Ω ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] , C n + 1 = C 0 + β ( Δ t ) ϱ Γ ( ϱ + 2 ) ∑ Ω = 0 n [ t Ω − 1 β − 1 ℋ 3 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 3 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n + 1 − Ω ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] .

4.2 Numerical scheme for Model (4)

Here, we set a numerical scheme for the proposed model under the FFD operator with exponential kernel. Therefore, applying the Caputo–Fabrizio integral to Eq. (4) to obtain the following system of integral equations as follows:

(20) T ( t ) = T ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t , T , ℰ , C ) + ϱ β ℳ ( ϱ ) ∫ 0 t ξ β − 1 ℋ 1 ( t , T , ℰ , C ) , ℰ ( t ) = ℰ ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t , T , ℰ , C ) + ϱ β ℳ ( ϱ ) ∫ 0 t ξ β − 1 ℋ 2 ( t , T , ℰ , C ) , C ( t ) = C ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t , T , ℰ , C ) ϱ β ℳ ( ϱ ) ∫ 0 t ξ β − 1 ℋ 3 ( t , T , ℰ , C ) .

The derivation of the numerical scheme is presented at t = t n + 1 . Thus,

(21) T n + 1 = T 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ℋ 1 ( t , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ℋ 2 ( t , T , ℰ , C ) d ξ , C n + 1 = C 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t n , T n , ℰ n , C n ) ϱ β ℳ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ℋ 3 ( t , T , ℰ , C ) d ξ .

Taking the difference between the consecutive terms, one have

(22) T n + 1 = T n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) ∫ t n t n + 1 ξ β − 1 ℋ 1 ( t , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) ∫ t n t n + 1 ξ β − 1 ℋ 2 ( t , T , ℰ , C ) d ξ , C n + 1 = C n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) ∫ t n t n + 1 ξ β − 1 ℋ 3 ( t , T , ℰ , C ) d ξ .

Now, by LPI, we have

(23) T n + 1 = T n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) × 3 2 ( Δ t ) t n β − 1 ℋ 1 ( t n , T n , ℰ n , C n ) − Δ t 2 t n − 1 β − 1 ℋ 1 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) , ℰ n + 1 = ℰ n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) × 3 2 ( Δ t ) t n β − 1 ℋ 2 ( t n , T n , ℰ n , C n ) − Δ t 2 t n − 1 β − 1 ℋ 1 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) , C n + 1 = C n + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) − β t n − 1 β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) + ϱ β ℳ ( ϱ ) × 3 2 ( Δ t ) t n β − 1 ℋ 2 ( t n , T n , ℰ n , C n ) − Δ t 2 t n − 1 β − 1 ℋ 1 ( t n − 1 , T n − 1 , ℰ n − 1 , C n − 1 ) .

4.3 Numerical scheme for Model (5)

We apply FFI with Mittag–Leffler kernel to (5) to convert the proposed model into the equivalent integral system as

T ( t ) = T ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t , T , ℰ , C ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ ( t ) = ℰ ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t , T , ℰ , C ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 2 ( ξ , T , ℰ , C ) d ξ , C ( t ) = C ( 0 ) + β t β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t , T , ℰ , C ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t ξ β − 1 ( t − ξ ) ϱ − 1 ℋ 3 ( ξ , T , ℰ , C ) d ξ .

At t = t n + 1 , we have

(24) T n + 1 = T 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 2 ( ξ , T , ℰ , C ) d ξ , C n + 1 = C 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∫ 0 t n + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 2 ( ξ , T , ℰ , C ) d ξ .

Using the approximation of the integrals of System (24) gives

(25) T n + 1 = T 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∑ Ω = 0 n ∫ t Ω t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 1 ( ξ , T , ℰ , C ) d ξ , ℰ n + 1 = ℰ 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∑ Ω = 0 n ∫ t Ω t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 2 ( ξ , T , ℰ , C ) d ξ , C n + 1 = C 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + ϱ β ℳ ( ϱ ) Γ ( ϱ ) ∑ Ω = 0 n ∫ t Ω t Ω + 1 ξ β − 1 ( t n + 1 − ξ ) ϱ − 1 × ℋ 2 ( ξ , T , ℰ , C ) d ξ .

Now, using the Lagrangian polynomial with piece-wise interpolation, we obtain the following:

(26) T n + 1 = T 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 1 ( t n , T n , ℰ n , C n ) + β ( Δ t ) ϱ ℳ ( ϱ ) Γ ( ϱ + 2 ) × ∑ Ω = 0 n [ t Ω β − 1 ℋ 1 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 1 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n − Ω + 1 ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] , ℰ n + 1 = ℰ 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 2 ( t n , T n , ℰ n , C n ) + β ( Δ t ) ϱ ℳ ( ϱ ) Γ ( ϱ + 2 ) × ∑ Ω = 0 n [ t Ω β − 1 ℋ 2 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 2 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n − Ω + 1 ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] , C n + 1 = C 0 + β t n β − 1 ( 1 − ϱ ) ℳ ( ϱ ) ℋ 3 ( t n , T n , ℰ n , C n ) + β ( Δ t ) ϱ ℳ ( ϱ ) Γ ( ϱ + 2 ) × ∑ Ω = 0 n [ t Ω β − 1 ℋ 3 ( t Ω , T Ω , ℰ Ω , C Ω ) × ( ( n + 1 − Ω ) ϱ ( n − Ω + 2 + ϱ ) − ( n − Ω ) ϱ ( n − Ω + 2 + 2 ϱ ) ) − t Ω − 1 β − 1 ℋ 3 ( t Ω − 1 , T Ω − 1 , ℰ Ω − 1 , C Ω − 1 ) × ( ( n − Ω + 1 ) ϱ + 1 − ( n − Ω ) ϱ ( n − Ω + 1 + ϱ ) ) ] .

The presented numerical method for three cases has been proved that Adam–Bashforth method preserves accuracy in solving both linear and nonlinear problems involving fractional or fractal-fractional order derivatives (see details in the study by Zabidi et al. [59]).

5 Numerical simulations

For simulations, we consider the initial values for state variables along with parameter values from Table 2. The parameter values are also given in Table 2 and initial conditions are taken from the study of Moore and Li [55].

T ( 0 ) = 1,510 ; ℰ ( 0 ) = 20 ; C ( 0 ) = 10,000 ,

where all the initial values are considered in cells ∕ μ l .

Table 2

Values of the parameters

Parameters	Values	Units
s n	0.073	cell ∕ μ l day
d n	0.040	day ‒ 1
d e	0.06	day ‒ 1
d c	0.2	day ‒ 1
k n	0.001	day ‒ 1
η	100	cells/ μ l
ϱ n	0.41
α e	0.2	day ‒ 1
C max	3 × 1 0 5	cells/ μ l
r c	0.03	day ‒ 1
γ e	0.005	day ‒ 1 cells μ l ‒ 1
γ c	0.005	day ‒ 1 cells μ l ‒ 1

Here, first, we use the numerical scheme for power-law kernel given in Eq. (19) to plot the approximate results for the proposed model in Figures 1, 2, 3, respectively. For plotting, we assign the values to fractal order as β = 0.99 and simulate the results through power-law algorithm for different fractional-order ϱ = 0.75 , 0.85 , 0.95 , and 1. Using the given values of Table 2, we plot the graphs of approximate results obtained in Eqs. (23) and (26) for the considered model and obtain the graphs for Caputo Fabrizio and Atangana-Baleanu-Caputo case in Figures 4, 5, 6 and Figures 7, 8, 9, respectively. From Figures 1–3, we see that when naive T-cells are exposed to professional antigen-presenting cell (APC), then the number of naive T-cells are going to decreasing in approximately 1 month. As a result, the population of the effector T-cells specific to CML rapidly increases up to 5 days but later rebound, and the population of CML cancer cells rapidly decreases about in 1 day. From Figures 4–6, we note that the population of naive T-cells is steadily decreasing, resulting in a rise in the population of CML-specific effector T-cells up to 6 days but later rebounding and gradually decreasing in 4 months; the population of CML cancer is also decreasing. From Figures 7–9, we analyze that the population of naive T-cells decreases more steadily. As a result, the effector T-cell population increases and the CML cancer cell population gradually decreases. We see that the information given by the graphs of Mittag–Leffler kernel are more realistic than Caputo–Fabrizo type and power-law type for different fractional-order. By comparing the graphs obtained in the study by Moore and Li [55] with the plots for the fractal-fractional order model, it is clear that variation of the fractal-fractional order and control parameters can capture more complexities as compared to fractional differential operators.

$Figure 1 Graphical illustration for the naive T {\mathcal{T}} cells using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 1

Graphical illustration for the naive T cells using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 2 Graphical illustration for the effector ℰ {\mathcal{ {\mathcal E} }} cells specific to CML using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 2

Graphical illustration for the effector ℰ cells specific to CML using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 3 Graphical illustration for the CML cancer cells C {\mathcal{C}} using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 3

Graphical illustration for the CML cancer cells C using Model (3) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 4 Graphical illustration for the naive T {\mathcal{T}} cells using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 4

Graphical illustration for the naive T cells using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 5 Graphical illustration for the effector ℰ {\mathcal{ {\mathcal E} }} cells specific to CML using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 5

Graphical illustration for the effector ℰ cells specific to CML using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 6 Graphical illustration for the CML cancer cells C {\mathcal{C}} using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 6

Graphical illustration for the CML cancer cells C using Model (4) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 7 Graphical illustration for T {\mathcal{T}} cells using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 7

Graphical illustration for T cells using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 8 Graphical illustration for the effector ℰ {\mathcal{ {\mathcal E} }} cells specific to CML using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 8

Graphical illustration for the effector ℰ cells specific to CML using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

$Figure 9 Graphical illustration for the CML cancer cells C {\mathcal{C}} using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 \varrho =0.75,0.85,0.95,1 and β = 0.99 \beta =0.99 .$

Figure 9

Graphical illustration for the CML cancer cells C using Model (5) for ϱ = 0.75 , 0.85 , 0.95 , 1 and β = 0.99 .

Here, we present some plots for FFD operators with power-law kernel corresponding to different values of ϱ and β in Figures 10, 11, 12, respectively.

$Figure 10 Graphical illustration for the T {\mathcal{T}} cells using Model (3) for different values of ϱ \varrho and β \beta .$

Figure 10

Graphical illustration for the T cells using Model (3) for different values of ϱ and β .

$Figure 11 Graphical illustration for for the effector ℰ {\mathcal{ {\mathcal E} }} cells specific to CML using Model (3) for different values of ϱ \varrho and β \beta .$

Figure 11

Graphical illustration for for the effector ℰ cells specific to CML using Model (3) for different values of ϱ and β .

$Figure 12 Graphical illustration for the CML cancer cells C {\mathcal{C}} using Model (3) for different values of ϱ \varrho and β \beta .$

Figure 12

Graphical illustration for the CML cancer cells C using Model (3) for different values of ϱ and β .

6 Discussion and conclusion

In this article, under newly proposed differentials and integral operators, we have investigated the mathematical model of the relationship between CML and T-cells. We have derived the results for the existence and uniqueness of the solution of the proposed model. In general, we have presented three numerical schemes with the use of Lagrangian polynomial piece-wise interpolation for the solution of the model under Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractal-fractional operator. The first one is related to the power-law type kernel, the second one is concerned with the exponential-decay-type kernel, and the last one is related to the Mittag–Leffler-type kernel. We have presented the numerical results for fractional order ϱ = 0.75 , 0.8 , 0.95 , 1 and fractal order β = 0.99 . If ϱ = 1 and β = 1 , then we recover the results of the ordinary operators from these operators. The increase and decrease of the compartments of the proposed model are due to varying fractional order. As we see in simulations, the fractional order has a great impact on the fractal dimension. The researchers also studied the influence of the fractal dimension on the fractional orders in several papers. From the figures, we have concluded that modeling with the Atangana–Baleanu operator is better than the Caputo and Caputo–Fabrizio because the dynamics of the proposed model obtained through the Atangana–Baleanu operator is more realistic than the other two differential operators. The Atangana–Baleanu fractal-fractional operator is thus the better choice for any dynamic model that does not predict by ordinary differential operators to study complex behavior. Furthermore, one can extend the current work to study chaotic attractors and more complex behavior under different fractal and fractional operators of variable orders. In the future, young researchers can investigate different fractal-fractional differential operators in fuzzy sense, which is a more popular applicable area of research in the current time. Furthermore, the qualitative analysis of fractal-fractional differential operators can be investigated using newly established fixed point theorems. Also, spectral, collocations, and spline method of numerical analysis can be extended to investigate the said area.

kshah@psu.edu.sa

Acknowledgments

Prince Sultan University is appreciated for APC and support through TAS research lab.

Funding information: Prince Sultan University is appreciated for APC and support through TAS research lab.
Author contributions: Kamal Shah has edited revised the paper. Shabir Ahmad has written the initial dradt. Aman Ullah included literature. Thabet Abdeljawad edited the final version of the paper. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2023-12-16

Revised: 2024-04-11

Accepted: 2024-04-25

Published Online: 2024-05-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0032

Keywords for this article

fractal-fractional operators; myeloid leukemia; model; Adams–Basforth method

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