Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

Dynamical behavior of tumor-immune system with fractal-fractional operator

  • In this paper, the dynamical behavior of the fractional-order cancer model has been analyzed with the fractal-fractional operator, which discretized the conformable cancer model. The fractional-order model consists of the system of nonlinear fractional differential equations. Also, we discuss the fractional-order model to check the relationship between the immune system and cancer cells by mixing IL-12 cytokine and anti-PD-L1 inhibitor. The tumor-immune model has been studied qualitatively as well as quantitatively via Atangana-Baleanu fractal-fractional operator. The nonlinear analysis is used to check the Ulam-Hyres stability of the proposed model. Moreover, the dynamical behavior for the fractional-order model has been checked by using a fractal-fractional operator with a generalized Mittag-Leffler Kernel and verifying the effect of fractional parameters. Finally, the obtained solutions are interpreted biologically, and simulations are carried out to illustrate cancer disease and support theoretical results, which will be helpful for further analysis and to control the effect of cancer in the community.

    Citation: Muhammad Farman, Aqeel Ahmad, Ali Akgül, Muhammad Umer Saleem, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar. Dynamical behavior of tumor-immune system with fractal-fractional operator[J]. AIMS Mathematics, 2022, 7(5): 8751-8773. doi: 10.3934/math.2022489

    Related Papers:

    [1] Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
    [2] Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Nantapat Jarasthitikulchai, Marisa Kaewsuwan . A generalized Gronwall inequality via ψ-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system. AIMS Mathematics, 2024, 9(9): 24443-24479. doi: 10.3934/math.20241191
    [3] Mdi Begum Jeelani, Abeer S. Alnahdi, Mohammed A. Almalahi, Mohammed S. Abdo, Hanan A. Wahash, M. A. Abdelkawy . Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(2): 2001-2018. doi: 10.3934/math.2022115
    [4] Khaled M. Saad, Manal Alqhtani . Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225
    [5] Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
    [6] Aziz Khan, Thabet Abdeljawad, Manar A. Alqudah . Neural networking study of worms in a wireless sensor model in the sense of fractal fractional. AIMS Mathematics, 2023, 8(11): 26406-26424. doi: 10.3934/math.20231348
    [7] Muhammad Farman, Ali Akgül, Sameh Askar, Thongchai Botmart, Aqeel Ahmad, Hijaz Ahmad . Modeling and analysis of fractional order Zika model. AIMS Mathematics, 2022, 7(3): 3912-3938. doi: 10.3934/math.2022216
    [8] Yuehong Zhang, Zhiying Li, Wangdong Jiang, Wei Liu . The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays. AIMS Mathematics, 2023, 8(3): 6176-6190. doi: 10.3934/math.2023312
    [9] Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
    [10] Mohamed A. Barakat, Abd-Allah Hyder, Doaa Rizk . New fractional results for Langevin equations through extensive fractional operators. AIMS Mathematics, 2023, 8(3): 6119-6135. doi: 10.3934/math.2023309
  • In this paper, the dynamical behavior of the fractional-order cancer model has been analyzed with the fractal-fractional operator, which discretized the conformable cancer model. The fractional-order model consists of the system of nonlinear fractional differential equations. Also, we discuss the fractional-order model to check the relationship between the immune system and cancer cells by mixing IL-12 cytokine and anti-PD-L1 inhibitor. The tumor-immune model has been studied qualitatively as well as quantitatively via Atangana-Baleanu fractal-fractional operator. The nonlinear analysis is used to check the Ulam-Hyres stability of the proposed model. Moreover, the dynamical behavior for the fractional-order model has been checked by using a fractal-fractional operator with a generalized Mittag-Leffler Kernel and verifying the effect of fractional parameters. Finally, the obtained solutions are interpreted biologically, and simulations are carried out to illustrate cancer disease and support theoretical results, which will be helpful for further analysis and to control the effect of cancer in the community.



    The biological sciences have advanced at a breakneck pace during the last few decades, and it's reasonable to predict that this trend will continue, aided by massive technological advancements in a long time. Mathematics has always both contributed to and benefited society. There is huge progress in the natural sciences because of mathematics, and it can do the same for biological science [1]. Biology generates intriguing problems, and mathematics develops models to help us understand them, and biology then tests the mathematical models. The handling of complex mathematical systems has been made easier thanks to recent improvements in computer algebra systems. This has allowed scientists to concentrate on understanding mathematical biology rather than the mechanics of solving problems [2].

    Cancer has always been considered a complex system as it causes different diseases around two hundred with different characteristics. Therefore, many scientists are still trying to investigate the interactions between tumor cells and immune cells by employing different strategies to better understand the dynamics of cancer [3,4]. Research is devoted to understanding the interaction between the immune system and tumor cells [5]. Ordinary differential equations mathematical models have been proven valuable to understand the dynamics of the tumor-immune system how host immune cells and cancerous cells evolve interact; See e.g. [6,7,8]. However, fractional-order differential equations have more characteristics than classical derivatives in mathematical modeling.

    Accordingly, the subject of fractional calculus has gained popularity and importance due to its demonstrated applications in system biology [9], and different fields of sciences [10]. Derivatives and integrals of any non-negative order are allowed in fractional calculus. The benefit of fractional derivatives (integral as well) is that they are not a local (or point) attribute (or quantity) [11]. We noticed that in epidemiology, mathematical models are frequently employed to comprehend the complexity of infectious diseases. The stability theory of differential equations is used to study the dysentery modeling approach with controls [12]. The most often employed operators, such as Caputo-Fabrizio, and Atangana-Baleanu, involve a local derivative with an exponential function, power-law, and a Mittag-Leffler function, respectively. On the other hand, real-world processes display intricate fractal behavior, including volatility, aquifers, porous media, biomedicine, and Darcy's law. In this context, it is necessary to have operators involving nonlocal differentiation in the kernel, which Atangana provided in the form of fractional fractal operators [13]. In the field of mathematical biology, epidemiological models design has been examined using classical and certain fractional-order operators such as conformable derivatives, Caputo-Fabrizio, Atangana-Baleanu, beta derivatives, and a few more. Dengue fever, tuberculosis, measles, Ebola, and other diseases are among the epidemiological designs supported by fractional operators. Nevertheless, fresh real-life applications studied on fractal-fractional operators have shown that the operator's effectiveness is best suited to curves derived under Caputo type fractal operators using actual banking data [14]. Fractal-fractional derivative and integration analysis develop complicated behaviors of diarrhoeal sickness. These operators have created the best curve of the actual data of the diseased population, and the model of diarrhea proves that these novel operators have a unique solution [15]. Some application of fractional order model with the local and nonlocal, nonsingular kernel is also studied in [16,17,18,19,20,21] having memory results for a dynamical system.

    Fractional derivative in the Caputo sense is studied in [29,30,31]. A vitro model of HER 2 + breast cancer cells dynamics resulting from many dosages and timing of paclitaxel and trastuzumab combination regimens was proposed in [30]. Study the pattern and the trend of spread of this disease and prescribe a mathematical model which governs COVID-19 pandemic using Caputo type derivative [32]. Fractal-fractional operator in the Caputo sense, the condition for Ulam's type of stability of the solution to the models and its related work is in [33,34,35,36]. Distributed order time-fractional constitution model is put forward to analyze the unsteady natural convection boundary layer flow and heat transfer, in which the magnetic field effect is considered and its related work is in [37,38,39].

    This paper, in Sections 1 and 2, consists of an introduction and basic definitions for analysis. Section 3 is for the Ulam-Hyres stability and uniqueness of the proposed scheme of the fractional order cancer model with the fractal fractional operator by using generalized Mittag-Leffler Kernel. A numerical algorithm for simulation and results is developed in Section 4. Description and conclusion of results are described in Sections 5 and 6.

    We consider some basic refinement of fractional calculus [22,23,24,25] in this section which are helpful for analysis and simulation of the problem.

    Definition 2.1. Let 0ξ, η1, then U(t) in the Riemann-Liouville for fractal-fractional with power-law kernel and the fractal-fractional integral have been given as [26]:

    FFPDξ,η0,t(U(t))=1Γ(mξ)ddtξt0(tΨ)mξ1U(Ψ)dΨ,
    ddΨηU(Ψ)=limtΨU(t)U(Ψ)tηΨη

    and

    FFPIξ,η0,t(U(t))=1Γ(ξ)t0(tΨ)υ1Ψ1ηU(Ψ)dΨ.

    Definition 2.2. Let 0ξ, η1. Then U(t) in the Riemann-Liouville for fractal fractional operator having exponentially decaying kernel and fractal-fractional integral have been presented by [26,27]:

    FFEDξ,η0,t(U(t))=M(ξ)Γ(mξ)ddtηt0exp[ξ1ξ(tΨ)nξ1U(Ψ)dΨ,

    and

    FFEDξ,η0,t(U(t))=η(1ξ)tη1U(t)M(ξ)+ξηM(ξ)t0Ψξ1U(Ψ)dΨ.

    Definition 2.3. Let 0ξ, η1 then U(t). Then the Riemann-Liouville for fractal fractional operator with generalized Mittag-Leffler kernel and fractal-fractional integral have been presented by [26,27]:

    FFMDξ,η0,t(U(t))=AB(ξ)1ξddtηt0Eξ[ξ1ξ(tΨ)ξU(Ψ)dΨ,

    and

    FFMDξ,η0,t(U(t))=η(1ξ)tη1U(t)AB(ξ)+ξηAB(ξ)t0Ψξ1(tΨ)U(Ψ)dΨ.

    Recently, many researchers studies to examine the mathematical model of cancer-immune [27,28]. By this motivation, IL-12 cytokine is used to increase it ability by increasing the number of CD4+T, CD8+T lymphocytes, and the help of anti-PD-L1 inhibitor with effect of CD8+T cells are added in Castiglione-Piccoli model. So, [27,28] explains the new time-dependent ordinary differential system. Equation (1) tells about components of immune system against cancer cells, which is made by adding IL-12 and anti-PD-L1 to components of immune system. The main objective of adding new variables in (1) is to fight cancer cells for the immune system more effectively. The system of nonlinear fractional order with Fractal Fractional operator is given as

    {FFMDξ,η0,t(H(t))=a0+b0DH(1Hf0)+λ42I2K2+I2H(1Hf0)+λ412I12K12+I12H(1Hf0)c0H,FFMDξ,η0,t(M(t))=a1+b1I2K2+I2(M+D)C(1Cf1)+λ812I12K12+I12C(1Cf1)c1C,FFMDξ,η0,t(C(t))=b2M(1Mf2)d2FMC,FFMDξ,η0,t(D(t))=d3DC,FFMDξ,η0,t(I2(t))=b4DHe4I2Cc4I2,FFMDξ,η0,t(I12(t))=λDI12DdI12I12,FFMDξ,η0,t(Z(t))=γZ. (3.1)

    where

    F=cpd1π(tan1((Z1)kpd)+π2)
    H0=H(0),C0=C(0),M0=M(0),
    D0=D(0),I20=I2(0),I120=I12(0),Z0=Z(0)

    Here H,C,M,D,IL2,IL12 and Z symbolize CD+4T and CD+8T lymphocytes, cancer cells, dendritic cells, IL-2 and IL-12 cytokine, anti-PD-L1, respectively given in [27,28]

    Theorem 1. The solution of the given cancer fractal-fractional model (3.1) along initial conditions is unique and bounded in R7+.

    Proof. We obtain

    FFMDξ,η0,t(H(t))H=0=a00,FFMDξ,η0,t(M(t))M=0=a1+[b1I2K2+I2D+λ812I12K12+I12]C(1Cf1)c1C,0,FFMDξ,η0,t(C(t))C=0=b2M(1Mf2)0,FFMDξ,η0,t(D(t))|D=0=00,FFMDξ,η0,t(I2(t))|I2=0=b4DH0,FFMDξ,η0,t(I12(t))|I12=0=λDI12D0,FFMDξ,η0,t(Z(t))|Z=0=00. (3.2)

    If (S(0),U(0),C(0),Ca(0),R(0),B(0)R6+, then according to Eq (3.2) thee solution cannot escape from the hyperplane. Also on each hyperplane bounding the non-negative orthant, the vector field points into R7+, i.e., the domain R7+ is a positively invariant set.

    Stability and existences of proposed operator

    Here, we consider stability and existences with proposed operator,

    FFMDξ,η0,t(H(t))=a0+b0DH(1Hf0)+λ42I2K2+I2H(1Hf0)+λ412I12K12+I12H(1Hf0)c0H,FFMDξ,η0,t(M(t))=a1+b1I2K2+I2(M+D)C(1Cf1)+λ812I12K12+I12C(1Cf1)c1C,FFMDξ,η0,t(C(t))=b2M(1Mf2)d2FMC,FFMDξ,η0,t(D(t))=d3DC,FFMDξ,η0,t(I2(t))=b4DHe4I2Cc4I2,FFMDξ,η0,t(I12(t))=λDI12DdI12I12,FFMDξ,η0,t(Z(t))=γZ. (3.3)

    For existence, we have

    {ABR0Dξ,η0,t(H(t))=ηtη1Z1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(M(t))=ηtη1Y1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(C(t))=ηtη1X1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(D(t))=ηtη1W1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(I2(t))=ηtη1V1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(I12(t))=ηtη1U1(t,H,M,C,D,I2,I12,Z),ABR0Dξ,η0,t(Z(t))=ηtη1E1(t,H,M,C,D,I2,I12,Z). (3.4)

    where

    {Z(t,H,M,C,D,I2,I12,Z)=a0+b0DH(1Hf0)+λ42I2K2+I2H(1Hf0)+λ412I12K12+I12H(1Hf0)c0H,Y(t,H,M,C,D,I2,I12,Z)=a1+b1I2K2+I2(M+D)C(1Cf1)+λ812I12K12+I12C(1Cf1)c1C,X(t,H,M,C,D,I2,I12,Z)=b2M(1Mf2)d2FMC,W(t,H,M,C,D,I2,I12,Z)=d3DC,V(t,H,M,C,D,I2,I12,Z)=b4DHe4I2Cc4I2,U(t,H,M,C,D,I2,I12,Z)=λDI12DdI12I12,E(t,H,M,C,D,I2,I12,Z)=γZ.

    We can write system (3.4) as:

    {ABR0DξtΩ(t)=ηtη1Λ(t,Ω(t)),Ω(0)=Ω0.

    By replacing ABR0Dξ,η0 by ABC0Dξ,η0 and applying fractional integral, we get

    Ω(t)=Ω(0)+ηtη1(1ξ)CD(ξ)Λ(t,Ω(t))+ξηCD(ξ)Γ(ξ)t0ωη1(tω)η1Λ(t,Ω(t))dω,

    where

    Ω(t)={H(t)M(t)C(t)D(t)I2(t)I12(t)Z(t)Ω(0)={H(0)M(0)C(0)D(0)I2(0)I12(0)Z(0)Λ(t,Ω(t))={Z(t,H,M,C,D,I2,I12,Z),Y(t,H,M,C,D,I2,I12,Z),X(t,H,M,C,D,I2,I12,Z),W(t,H,M,C,D,I2,I12,Z),V(t,H,M,C,D,I2,I12,Z),U(t,H,M,C,D,I2,I12,Z)E(t,H,M,C,D,I2,I12,Z)

    For the existence theory, we define a Banach space B=C×C×C×C×C×C×C, where C=[0,T] under the norm

    Ω=maxtϵ[0,T]|H(t)+M(t)+C(t)+D(t)+I2(t)+I12(t),Z(t)|.

    Define an operator :BB as:

    (Ω)(t)=Ω(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Ω(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(t,Ω(t))dω. (3.5)

    Now, we suppose that Λ(t,(Ω(t)) is the non-linear function with growth and the Lipschitz conditions.

    ● For each ΩϵB, constants AΛ and PΛ such that

    |Λ(t,Ω(t))AΛ|Ω(t)|+PΛ. (3.6)

    ● For each Ω, ¯ΩϵB,a constant KΛ>0 such that

    |Λ(t,Ω(t))Λ(t,¯Ω(t)|KΛ|Ω(t)¯Ω(t)|. (3.7)

    Theorem 2. For the set of continuous function Λ=[0,T]×BR there exists at least single outcome for system (3.1) if the condition (3.6) holds.

    Proof.

    Let

    L= {ΩϵB:ΩR,R>0}.

    Now for any ΩϵB, we have

    |(Ω)|=maxtϵ[0,T]|Ω(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Ω(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(t,Ω(t))dω
    Ω(0)+ηTη1(1ξ)C1D1(ξ)(AΛΩ+PΛ)+maxtϵ[0,T]ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1|Λ(t,Ω(t))|dω
    Ω(0)+ηMη1(1ξ)C1D1(ξ)(AΛΩ+PΛ)+ξηC1D1(ξ)Γ(ξ)(AΛΩ+PΛ)Tξ+η1L(ξ,η)
    R.

    Hence, is uniformly bounded, for equicontinuity of , let us take t1<t2T. So, we have

    |(Ω)(t2)(Ω)(t1)|=|ηtη12(1ξ)C1D1(ξ)Λ(t2,Ω(t2))+ξηC1D1(ξ)Γ(ξ)t20ωη1(t2ω)η1Λ(ω,Ω(ω))dω
    ηtη11(1ξ)C1D1(ξ)Λ(t1,Ω(t1))+ξηC1D1(ξ)Γ(ξ)t10ωη1(t1ω)η1Λ(ω,Ω(ω))dω|
    |ηtη12(1ξ)C1D1(ξ)(AΛ|Ω(t)|,PΛ)+ξηC1D1(ξ)Γ(ξ)(AΛ|Ω(t)|,PΛ)tξ+η12L(ξ,η)
    |ηtη11(1ξ)C1D1(ξ)(AΛ|Ω(t)|,PΛ)+ξηC1D1(ξ)Γ(ξ)(AΛ|Ω(t)|,PΛ)tξ+η11L(ξ,η),

    when t1t2 then |(Ω)(t2(Ω)(t1|. Consequently, we can say that,

    (Ω)(t2(Ω)(t10,t1t2.

    Hence, is equicontinuous. So, theorem is proved. Thus, system has at least one solution.

    Theorem 3. Assume that (3.7) holds. If ϱ<1, where

    ϱ=(ηTη1(1ξ)C1D1(ξ)+ξηC1D1(ξ)Γ(ξ)Tξ+η1L(ξ,η))KΛ.

    Then the considered model has a unique solution.

    Proof. For Ω,¯ΩϵB, we have

    |(Ω)(¯Ω)|=maxtϵ[0,T]|ηtη1(1ξ)C1D1(ξ)(Λ(t,Ω(t))Λ(t,¯Ω(t)))+ξηC1D1(ξ)Γ(ξ)
    t0ωη1(tω)η1dω(Λ(ω,Ω(ω))Λ(ω,¯Ω(ω)))|
    [ηTη1(1ξ)C1D1(ξ)+ξηC1D1(ξ)Γ(ξ)Tξ+η1L(ξ,η)]Ω¯Ω
    ϱΩ¯Ω.

    Hence, is a contraction. So, system has unique solution by using Banach contraction principle.

    Definition 3.1. The proposed model is Ulam-Hyres stable if ξ,η0 such that for any ε>0 and for every Ω(C[0,T],R) satisfies the following inequality:

    |FFM0Dξ,ηtΩ(t)Λ(t,Ω(t))|ε,tϵ[0,T], (3.8)

    and there exists a unique solution Υ(C[0,T],R) such that

    |Ω(t)Υ(t)|ξ,ηε,tϵ[0,T]. (3.9)

    Consider a small perturbation ΨC[0,T] such that Ψ(0)=0. Let

    |ψ(t)|εforε>0.
    FFM0Dξ,ηtΩ(t)=Λ(t,Ω(t))+ψ(t).

    Theorem 4. The solution of the perturbed model

    FFM0Dξ,ηtΩ(t)=Λ(t,Ω(t))+ψ(t),Ω(0)=Ω0,

    fulfills the relation given below

    |(t)(Ω(0)+ηtη1(1ξ)C1D1ξ)Λ(t,Ω(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(ω,Ω(ω))dω,)|
    ξξ,ηε, (3.10)

    where

    ξξ,η=ηTη1(1ξ)C1D1(ξ)+ξηC1D1(ξ)Γ(ξ)Tξ+η1L(ξ,η).

    Proof. Under condition (3.7) along with Theorem 3.5, the solution of the proposed model is Ulam-Hyres stable if ϱ<1.

    Let ΥB be a unique solution and ΩB be any solution of the proposed model, then

    |Ω(t)Υ(t)|=|Ω(t)(Υ(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Υ(t))+ξηC1D1(ξ)Γ(ξ)
    t0ωη1(tω)η1Λ(ω,Υ(ω))dω)|
    |Ω(t)(Ω(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Ω(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(ω,Ω(ω))dω)|
    +|Ω(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Ω(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(ω,Ω(ω))dω)|
    |Υ(0)+ηtη1(1ξ)C1D1(ξ)Λ(t,Υ(t))+ξηC1D1(ξ)Γ(ξ)t0ωη1(tω)η1Λ(ω,Υ(ω))dω)|
    ξξ,ηε+(ηTη1(1ξ)C1D1(ξ)+ξηC1D1(ξ)Γ(ξ)Tξ+η1L(ξ,η))KΛ|Ω(t)Υ(t)|
    ξξ,ηε+ϱ|Ω(t)Υ(t)|.

    Consequently, one can write

    ΩΥξξ,ηε+ϱΩΥ.

    We can write the above relation is

    ΩΥξ,ηε,

    where ξ,η=ξξ,η1ϱ. Hence, proposed scheme is Ulam-Hyres stable.

    In this section, The numerical scheme is establish for developed fractional order model, we get

    {H(t)=H(0)+ηtη1(1ξ)C1D1(ξ)L1(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L1(ϕ,H,M,C,D,I2,I12,Z)dϕ,M(t)=M(0)+ηtη1(1ξ)C1D1(ξ)L2(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L2(ϕ,H,M,C,D,I2,I12,Z)dϕ,C(t)=C(0)+ηtη1(1ξ)C1D1(ξ)L3(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L3(ϕ,H,M,C,D,I2,I12,Z)dϕ,D(t)=D(0)+ηtη1(1ξ)C1D1(ξ)L4(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L4(ϕ,H,M,C,D,I2,I12,Z)dϕ,I2(t)=I2(0)+ηtη1(1ξ)C1D1(ξ)L5(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L5(ϕ,H,M,C,D,I2,I12,Z)dϕ,I12(t)=I12(0)+ηtη1(1ξ)C1D1(ξ)L6(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L6(ϕ,H,M,C,D,I2,I12,Z)dϕZ(t)=Z(0)+ηtη1(1ξ)C1D1(ξ)L7(t,H,M,C,D,I2,I12,Z)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)L7(ϕ,H,M,C,D,I2,I12,Z)dϕ. (4.1)

    Now, we derive the numerical scheme at t=tx+1, we have

    {Hx+1=H0+ηtη1(1ξ)C1D1(ξ)L1(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L1(ϕ,H,M,C,D,I2,I12,Z)dϕ,Mx+1=M0+ηtη1(1ξ)C1D1(ξ)L2(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L2(ϕ,H,M,C,D,I2,I12,Z)dϕ,Cx+1=C0+ηtη1(1ξ)C1D1(ξ)L3(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L3(ϕ,H,M,C,D,I2,I12,Z)dϕ,Dx+1=D0+ηtη1(1ξ)C1D1(ξ)L4(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L4(ϕ,H,M,C,D,I2,I12,Z)dϕ,Ix+12=I02+ηtη1(1ξ)C1D1(ξ)L5(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L5(ϕ,H,M,C,D,I2,I12,Z)dϕ,Ix+112=I012+ηtη1(1ξ)C1D1(ξ)L6(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L6(ϕ,H,M,C,D,I2,I12,Z)dϕZx+1=Z0+ηtη1(1ξ)C1D1(ξ)L7(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)t0ϕη1(tϕ)ξ1L7(ϕ,H,M,C,D,I2,I12,Z)dϕ. (4.2)

    On system (4.2), we have

    {Hx+1=H0+ηtη1x(1ξ)C1D1(ξ)L1(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L1(ϕ,H,M,C,D,I2,I12,Z)dϕ,Mx+1=M0+ηtη1x(1ξ)C1D1(ξ)L2(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L2(ϕ,H,M,C,D,I2,I12,Z)dϕ,Cx+1=C0+ηtη1x(1ξ)C1D1(ξ)L3(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L3(ϕ,H,M,C,D,I2,I12,Z)dϕ,Dx+1=D0+ηtη1x(1ξ)C1D1(ξ)L4(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L4(ϕ,H,M,C,D,I2,I12,Z)dϕ,Ix+12=I02+ηtη1x(1ξ)C1D1(ξ)L5(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L5(ϕ,H,M,C,D,I2,I12,Z)dϕ,Ix+112=I012+ηtη1x(1ξ)C1D1(ξ)L6(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L6(ϕ,H,M,C,D,I2,I12,Z)dϕZx+1=Z0+ηtη1x(1ξ)C1D1(ξ)L7(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+ξηC1D1(ξ)Γ(ξ)xf=0tf+1tfϕη1(tx+1ϕ)ξ1L7(ϕ,H,M,C,D,I2,I12,Z)dϕ. (4.3)

    Using Lagrangian piece-wise interpolation with in the finite interval [tf,tf+1], we get

    {Jf(ϕ)=ϕtf1tftf1tη1fL1(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L1(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Of(ϕ)=ϕtf1tftf1tη1fL2(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L2(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Gf(ϕ)=ϕtf1tftf1tη1fL3(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L3(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Nf(ϕ)=ϕtf1tftf1tη1fL4(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L4(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Ff(ϕ)=ϕtf1tftf1tη1fL5(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L5(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Qf(ϕ)=ϕtf1tftf1tη1fL6(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L6(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1),Rf(ϕ)=ϕtf1tftf1tη1fL7(tf,Hf,Mf,Cf,Df,If2,If12,Zf)ϕtftftf1tη1f1L7(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1). (4.4)

    We have

    {Hx+1=H0+ηtη1x(1ξ)C1D1(ξ)L1(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL1(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L1(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Mx+1=M0+ηtη1x(1ξ)C1D1(ξ)L2(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxff=0[tη1fL2(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L2(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Cx+1=C0+ηtη1x(1ξ)C1D1(ξ)L3(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL3(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L3(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Dx+1=D0+ηtη1x(1ξ)C1D1(ξ)L4(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL4(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L4(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Ix+12=I02+ηtη1x(1ξ)C1D1(ξ)L5(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL5(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L5(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Ix+112=I012+ηtη1x(1ξ)C1D1(ξ)L6(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL6(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)times((x+1f)ξ(xf+2+ξ)(x2)ξ(ξ2+2ξ+x2))tη1f1L6(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))],Zx+1=Z0+ηtη1x(1ξ)C1D1(ξ)L7(tx,Hx,Mx,Cx,Dx,Ix2,Ix12,Zx)+η(Θt)ξC1D1(ξ)Γ(ξ+2)Σxf=0[tη1fL7(tf,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((x+1f)ξ(xf+2+ξ)(x2)ξ(2+2ξ+x2))tη1f1L7(tf1,Hf1,Mf1,Cf1,Df1,If12,If112,Zf1)×((1+xf)ξ+1(xf)ξ(1+ξ+xf))]. (4.5)

    The mathematical analysis of the tumor-immune system consisting of a fractional differential equation has been presented. In order to conduct a credible study, several α values are evaluated. By using non-integer parameter values, surprising reactions are achieved from the compartments of the proposed fractional-order model. Solution for H(t),M(t),C(t), I2(t) and I12(t) start decreasing by decreasing the fractional values while D(t) and Z(t) start increasing by decreasing fractional values respectively and can be seen easily from Figures 17 which converges to steady state with dimension 0.7. Solution for H(t),M(t),D(t), I2(t), I12(t) and Z(t) start increasing by decreasing the fractional values while C(t) start decreasing by decreasing fractional values respectively and can be seen easily from Figures 814 which converges to steady state with dimension 0.9. It is easily observed from Figure 3 and Figure 9 that cancer cell start decreasing by decreasing the fractional with both dimensions. But Fractal-fractional technique with dimension 0.9 gives better results for all the compartments which converges rapidly to steady state as compare to dimension 0.7. It is easily observed from simulation that fractional order model check the relationship between cancer cells and immune system which further increase the immune system of CD4+T and CD8+T lymphocytes production and decrease the cancer cell by adding IL-12 cytokine and anti-PD-L1 inhibitor. The variable anti-PD-L1 inhibitor was introduced to catch cells escaping from the immune system and destroy these cells which can be seen in Figures 7 and 14 with both dimensions. Also the variable IL-12 cytokine is used to increase the number of CD4+T and CD8+T lymphocytes production which can be seen in Figures 1, 2 and 8, 9 with dimension 0.7 and 0.9 respectively. This kind of study helps to analyze the effect of cancer cell and its anti-PD-L1 inhibitor in human body. Numerical simulations are important in sense for developing algorithm of treatment cell and impact of vaccination to control the disease. This kind of study is helpful for physician for planing and decision making for treatment cells.

    Figure 1.  H(t) with fractal fractional derivative at ξ=0.7.
    Figure 2.  M(t) with fractal fractional derivative at ξ=0.7.
    Figure 3.  C(t) with fractal fractional derivative at ξ=0.7.
    Figure 4.  D(t) with fractal fractional derivative at ξ=0.7.
    Figure 5.  I2(t) with fractal fractional derivative at ξ=0.7.
    Figure 6.  I12(t) with fractal fractional derivative at ξ=0.7.
    Figure 7.  Z(t) with fractal fractional derivative at ξ=0.7.
    Figure 8.  H(t) with fractal fractional derivative at ξ=0.9.
    Figure 9.  M(t) with fractal fractional derivative at ξ=0.9.
    Figure 10.  C(t) with fractal fractional derivative at ξ=0.9.
    Figure 11.  D(t) with fractal fractional derivative at ξ=0.9.
    Figure 12.  I2(t) with fractal fractional derivative at ξ=0.9.
    Figure 13.  I12(t) with fractal fractional derivative at ξ=0.9.
    Figure 14.  Z(t) with fractal fractional derivative at ξ=0.9.

    We developed a scheme for the analytical solution of the fractional order cancer model by using the Fractal Fractional operator. The model represents population dynamics during the disease as a set of non-linear fractional-order ordinary differential equations. The Fractional-order system is analyzed qualitatively to verify the state of the disease as well as determine the unique positive solutions. Uniqueness and boundedness for solution are proved using the fixed point theory. The solution is obtained for the fractional-order cancer model using the Fractal Fractional operator. Numerical simulations are carried out to check the actual behavior of the cancer outbreak using a developed scheme for fractional-order model, which will be helpful in future understanding of this disease and control strategics. The simulation easily observed that the fractional-order model examines that IL-12 cytokine and anti-PD-L1 inhibitor are used to check the relationship between the immune system and cancer cells, which cause to increase the immune system and decrease the cancer cell is due to IL-12 cytokine and anti-PD-L1 inhibitor with the impact of fractional parameters. Results show significant changes using different fractional values with different dimensions. Numerical simulations are important for developing an algorithm of treatment cells and the impact of vaccination to control the disease. In the future, the other cytokineses are also added to the mathematical model, which can further check with different types of derivatives and different types of techniques.

    All authors declare no conflicts of interest in this paper.



    [1] C. S. Chou, A. Friedman, Introduction to mathematical biology, Springer undergraduate texts in mathematics and technology, Springer International Publishing, 1 (2016), 1–10.
    [2] E. K. Yeargers, R. W. Shonkwiler, J. V. Herod, An introduction to the mathematics of biology: With computer algebra models-Chapter 1: Biology, mathematics and a mathematical biology laboratory, Springer Science and Business Media, 2013, 1–8.
    [3] M. A. Medina, Mathematical modeling of cancer metabolism, Crit. Rev. Oncol. Hemat., 124 (2018), 37–40. https://doi.org/10.1016/j.critrevonc.2018.02.004 doi: 10.1016/j.critrevonc.2018.02.004
    [4] N. Bellomo, A. Bellouquid, M. Delitala, Mathematical topics on the modeling of multicellular systems in competition between tumor and immune cells, Math. Mod. Meth. Appl. S., 2004, 1683–1733.
    [5] T. Roose, S. J. Chapman, P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), 179–208. https://doi.org/10.1137/S0036144504446291 doi: 10.1137/S0036144504446291
    [6] N. Bellomo, L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interactions with the immune system, Math. Comput. Model., 32 (2000), 413–452. https://doi.org/10.1016/S0895-7177(00)00143-6 doi: 10.1016/S0895-7177(00)00143-6
    [7] H. M. Byrne, T. Alarcon, M. R. Owen, S. D. Webb, P. K. Maini, Modeling aspects of cancer dynamics: A review, Philos. T. R. Soc. A, 364 (2006), 1563–1578.
    [8] F. Castiglione, B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, J. Theor. Biol., 247 (2007), 723–732.
    [9] F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan, E. Ahmed, Fractional order delayed predator-prey systems with Holling type-Ⅱ functional response, Nonlinear Dynam., 1 (2015). https://doi.org/10.1007/s11071-015-1905-8 doi: 10.1007/s11071-015-1905-8
    [10] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 91 (2018), 403–420.
    [11] Y. Wang, J. Cao, X. Li, A. Alsaedi, Edge-based epidemic dynamics with multiple routes of transmission on random networks, Nonlinear Dynam., 91 (2018), 1683–1733. https://doi.org/10.1007/s11071-017-3877-3 doi: 10.1007/s11071-017-3877-3
    [12] H. W. Berhe, O. D. Makinde, D. M. Theuri, Modelling the dynamics of direct and pathogens induced dysentery diarrhea epidemic with controls, J. Biol. Dyn., 13 (2019), 192–217.
    [13] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
    [14] Z. Li, Z. Liu, M. A. Khan, Fractional investigation of bank data with fractal-fractional Caputo derivatives, Chaos Soliton. Fract., 131 (2019), 1–12. https://doi.org/10.1016/j.chaos.2019.109528 doi: 10.1016/j.chaos.2019.109528
    [15] S. Qureshi, A. Atangana, Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data, Chaos Soliton. Fract., 136 (2020), 1–14. https://doi.org/10.1016/j.chaos.2020.109812 doi: 10.1016/j.chaos.2020.109812
    [16] M. Farman, M. U. Saleem, A. Ahmad, S. Imtiaz, M. O. Ahmad, A control of glucose level in insulin therapies for the development of artificial pancreas by Atangana Baleanu fractional derivative, Alex. Eng. J., 59 (2020), 2639–2648.
    [17] M. Farman, A. Akgul, A. Ahmad, D. Baleanue, M. U. Saleem, Dynamical transmission of coronavirus model with analysis and simulation, CMES-Comp. Model. Eng., 127 (2021), 753–769. https://doi.org/10.32604/cmes.2021.014882 doi: 10.32604/cmes.2021.014882
    [18] M. U. Saleem, M. Farman, A. Ahmad, H. Ehsan, M. O. Ahmad, A Caputo Fabrizio fractional order model for control of glucose in insulintherapies for diabetes, Ain Shams Eng. J., 11 (2020), 1309–1316.
    [19] M. Farman, A. Ahmad, A. Akgul, M. U. Saleem, M. Naeem, D. Baleanue, Epidemiological analysis of the coronavirus disease outbreak with random effects, CMC-Comput. Mater. Con., 67 (2021), 3215–3227. https://doi.org/10.32604/cmc.2021.014006 doi: 10.32604/cmc.2021.014006
    [20] M. Aslam, M. Farman, A. Akgul, M. Sun, Modeling and simulation of fractional order COVID-19 model with quarantined-isolated people, Math. Meth. Appl. Sci., 2021. https://doi.org/10.1002/mma.7191 doi: 10.1002/mma.7191
    [21] A. Alshabanat, M. Jleli, S. Kumar, B. Samet, Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits, Front. Phys., 8 (2020), 1–20. https://doi.org/10.3389/fphy.2020.00064 doi: 10.3389/fphy.2020.00064
    [22] R. Kanno, Representation of random walk-in fractal space-time, Phys. A, 248 (1998), 165–175. https://doi.org/10.1016/S0378-4371(97)00422-6 doi: 10.1016/S0378-4371(97)00422-6
    [23] S. Ahmad, A. Ullah, T. Abdeljawad, A. Akgül, N. Mlaiki, Analysis of fractal-fractional model of tumor-immune interaction, Results Phys., 25 (2021), 104178. https://doi.org/10.1016/j.rinp.2021.104178 doi: 10.1016/j.rinp.2021.104178
    [24] J. F. Gómez, L. Torres, R. F. Escobar, Fractional derivatives with Mittag-Leffler kernel: Trends and applications in science and engineering, Springer Nature Switzerland, 2019.
    [25] T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107. https://doi.org/10.22436/jnsa.010.03.20 doi: 10.22436/jnsa.010.03.20
    [26] A. Atangana, D. Baleanu, New fractional derivative with non-local and non-singular kernel, Therm. Sci., 20 (2016), 757–763.
    [27] E. Ucara, N. Ozdemir, A fractional model of cancer-immune system with Caputo and Caputo-Fabrizio derivatives, Eur. Phys. J. Plus., 2021.
    [28] X. Lai, A. Friedman, Combination therapy for melanoma with BRAF/MEK inhibitor and immune checkpoint inhibitor: A mathematical model, BMC Syst. Biol., 11 (2017), 70. https://doi.org/10.1186/s12918-017-0446-9 doi: 10.1186/s12918-017-0446-9
    [29] P. A. Naik, J. Zu, M. Naik, Stability analysis of a fractional-order cancer model with chaotic dynamics, Int. J. Biomath., 14 (2021), 2150046. https://doi.org/10.1142/S1793524521500467 doi: 10.1142/S1793524521500467
    [30] K. Owolabi, A. Shikongo, Fractal fractional operator method on HER2+ breast cancer dynamics, Int. J. Appl. Comput. Math., 7 (2021), 85. https://doi.org/10.1007/s40819-021-01030-5 doi: 10.1007/s40819-021-01030-5
    [31] P. A. Naik, J. Zu, K. M. Owolabi, Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order, Physica A, 545 (2020), 123816. https://doi.org/10.1016/j.physa.2019.123816 doi: 10.1016/j.physa.2019.123816
    [32] P. A. Naik, K. M. Owolabi, J. Zu, M. U. Din, NaikModeling the transmission dynamics of COVID-19 pandemic in Caputo type fractional derivative, J. Multiscale Model., 12 (2021), 2150006.
    [33] Z. Ali, F. Rabiei, K. Shah, T. Khodadadi, Fractal-fractional order dynamical behavior of an HIV/AIDS epidemic mathematical model, Eur. Phys. J. Plus, 136 (2021), 36. https://doi.org/10.1140/epjp/s13360-020-00994-5 doi: 10.1140/epjp/s13360-020-00994-5
    [34] Z. Ali, F. Rabiei, K. Shah, Z. M. MAJID, Dynamics of SIR mathematical model for COVID-19 outbreak in Pakistan under fractal-fractional derivative, Fractals, 29 (2021), 2150120. https://doi.org/10.1142/S0218348X21501206 doi: 10.1142/S0218348X21501206
    [35] Z. Ali, F. Rabiei, K. Shah, T. Khodadadi, Modeling and analysis of novel COVID-19 under fractal-fractional derivative with case study of Malaysia, Fractals, 29 (2021), 2150020. https://doi.org/10.1142/S0218348X21500201 doi: 10.1142/S0218348X21500201
    [36] Z. Ali, F. Rabiei, K. Shah, T. Khodadadi, Qualitative analysis of fractal-fractional order COVID-19 mathematical model with case study of Wuhan, Alex. Eng. J., 60 (2021), 477–489. https://doi.org/10.1016/j.aej.2020.09.020 doi: 10.1016/j.aej.2020.09.020
    [37] L. L. Feng, L. B. Xu, Q. Zheng, L. C. Liu, Fawang flow and heat transfer of generalized Maxwell fluid over a moving plate with distributed order time fractional constitutive models, Int. Commun. Heat Mass, 116 (2020), 104679.
    [38] S. Yang, L. Liu, Z. Long, L. Feng, Unsteady natural convection boundary layer flow and heat transfer past a vertical flat plate with novel constitution models, Appl. Math. Lett., 120 (2021), 107335. https://doi.org/10.1016/j.aml.2021.107335 doi: 10.1016/j.aml.2021.107335
    [39] Z. Long, L. Liu, S. Yang, L. Feng, L. Zheng, Analysis of Marangoni boundary layer flow and heat transfer with novel constitution relationships, Int. Commun. Heat Mass, 127 (2021), 105523.
  • This article has been cited by:

    1. Sugandha Arora, Trilok Mathur, Kamlesh Tiwari, A fractional-order model to study the dynamics of the spread of crime, 2023, 426, 03770427, 115102, 10.1016/j.cam.2023.115102
    2. Shahram Rezapour, Chernet Tuge Deressa, Robert G. Mukharlyamov, Sina Etemad, Watcharaporn Cholamjiak, On a Mathematical Model of Tumor-Immune Interaction with a Piecewise Differential and Integral Operator, 2022, 2022, 2314-4785, 1, 10.1155/2022/5075613
    3. Kottakkaran Sooppy Nisar, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Rafia Tabassum, Ahmed Morsy, A novel design of evolutionally computing to study the quarantine effects on transmission model of Ebola Virus Disease, 2023, 22113797, 106408, 10.1016/j.rinp.2023.106408
    4. CHERNET TUGE DERESSA, ROBERT G. MUKHARLYAMOV, HOSSAM A. NABWEY, SINA ETEMAD, İBRAHIM AVCI, ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS, 2024, 32, 0218-348X, 10.1142/S0218348X2440053X
    5. Huda Alsaud, Muhammad Owais Kulachi, Aqeel Ahmad, Mustafa Inc, Muhammad Taimoor, Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals, 2024, 9, 2473-6988, 10208, 10.3934/math.2024499
    6. Kottakkaran Sooppy Nisar, Muhammad Farman, Mahmoud Abdel-Aty, Jinde Cao, A review on epidemic models in sight of fractional calculus, 2023, 75, 11100168, 81, 10.1016/j.aej.2023.05.071
    7. Anusmita Das, Kaushik Dehingia, Evren Hinçal, Fatma Özköse, Kamyar Hosseini, A study on the dynamics of a breast cancer model with discrete-time delay, 2024, 99, 0031-8949, 035235, 10.1088/1402-4896/ad2753
    8. ZULQURNAIN SABIR, DUMITRU BALEANU, MUHAMMAD ASIF ZAHOOR RAJA, ALI S. ALSHOMRANI, EVREN HINCAL, MEYER WAVELET NEURAL NETWORKS PROCEDURES TO INVESTIGATE THE NUMERICAL PERFORMANCES OF THE COMPUTER VIRUS SPREAD WITH KILL SIGNALS, 2023, 31, 0218-348X, 10.1142/S0218348X2340025X
    9. Muhammad Farman, Maryam Batool, Kottakkaran Sooppy Nisar, Abdul Sattar Ghaffari, Aqeel Ahmad, Controllability and analysis of sustainable approach for cancer treatment with chemotherapy by using the fractional operator, 2023, 51, 22113797, 106630, 10.1016/j.rinp.2023.106630
    10. Parvaiz Ahmad Naik, Muhammad Owais Kulachi, Aqeel Ahmad, Muhammad Farman, Faiza Iqbal, Muhammad Taimoor, Zhengxin Huang, Modeling different strategies towards control of lung cancer: leveraging early detection and anti-cancer cell measures, 2024, 1025-5842, 1, 10.1080/10255842.2024.2404540
    11. Muhammad Farman, Cicik Alfiniyah, A constant proportional caputo operator for modeling childhood disease epidemics, 2024, 10, 27726622, 100393, 10.1016/j.dajour.2023.100393
    12. Aqeel Ahmad, Muhammad Owais Kulachi, Muhammad Farman, Moin-ud-Din Junjua, Muhammad Bilal Riaz, Sidra Riaz, Muntazir Hussain, Mathematical modeling and control of lung cancer with IL2 cytokine and anti-PD-L1 inhibitor effects for low immune individuals, 2024, 19, 1932-6203, e0299560, 10.1371/journal.pone.0299560
    13. ZULQURNAIN SABIR, DUMITRU BALEANU, MUHAMMAD ASIF ZAHOOR RAJA, ALI S. ALSHOMRANI, EVREN HINCAL, COMPUTATIONAL PERFORMANCES OF MORLET WAVELET NEURAL NETWORK FOR SOLVING A NONLINEAR DYNAMIC BASED ON THE MATHEMATICAL MODEL OF THE AFFECTION OF LAYLA AND MAJNUN, 2023, 31, 0218-348X, 10.1142/S0218348X23400169
    14. Kaushik Dehingia, Salah Boulaaras, The Stability of a Tumor–Macrophages Model with Caputo Fractional Operator, 2024, 8, 2504-3110, 394, 10.3390/fractalfract8070394
    15. Rongrong Qiao, Yuhan Hu, Dynamic analysis of a SI1I2ADSI1I2AD information dissemination model considering the word of mouth, 2023, 111, 0924-090X, 22763, 10.1007/s11071-023-09021-5
    16. Kottakkaran Sooppy Nisar, Muhammad Owais Kulachi, Aqeel Ahmad, Muhammad Farman, Muhammad Saqib, Muhammad Umer Saleem, Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy, 2024, 14, 2045-2322, 10.1038/s41598-024-66593-x
    17. Kaushik Dehingia, Sana Abdulkream Alharbi, Awatif Jahman Alqarni, Mounirah Areshi, Mona Alsulami, Reima Daher Alsemiry, Reem Allogmany, Homan Emadifar, Mati ur Rahman, Kavikumar Jacob, Exploring the combined effect of optimally controlled chemo-stem cell therapy on a fractional-order cancer model, 2025, 20, 1932-6203, e0311822, 10.1371/journal.pone.0311822
    18. Kaushik Dehingia, Bhagya Jyoti Nath, 2025, Chapter 5, 978-981-97-8714-2, 99, 10.1007/978-981-97-8715-9_5
    19. Chinwe Peace Igiri, Samuel Shikaa, Monkeypox Transmission Dynamics Using Fractional Disease Informed Neural Network: A Global and Continental Analysis, 2025, 13, 2169-3536, 77611, 10.1109/ACCESS.2025.3559005
    20. Jianping Li, Nan Liu, Danni Wang, Hongli Yang, Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay, 2025, 19, 1751-3758, 10.1080/17513758.2025.2508240
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2266) PDF downloads(139) Cited by(20)

Figures and Tables

Figures(14)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog