# Asymptotic Behavior Analysis of a Fractional-Order Tumor-Immune Interaction Model with Immunotherapy

Authors: Wenbin Yang; Xiaozhou Feng; Shuhui Liang; Xiaojuan Wang
Journal: Complexity
 (2020)

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## Abstract

A fractional-order tumor-immune interaction model with immunotherapy is proposed and examined. The existence, uniqueness, and nonnegativity of the solutions are proved. The local and global asymptotic stability of some equilibrium points are investigated. In particular, we present the sufficient conditions for asymptotic stability of tumor-free equilibrium. Finally, numerical simulations are conducted to illustrate the analytical results. The results indicate that the fractional order has a stabilization effect, and it may help to control the tumor extinction.


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## Body

## 1. Introduction

Tumor or tumour is a term used to describe the name for a swelling or lesion formed by an abnormal growth of cells. A tumor can be benign, premalignant, or malignant, whereas cancer is by definition malignant and is used to describe a disease in which abnormal cells divide without control and are able to invade other tissues. Cancer cells can spread to other parts of the body through blood and lymph systems [1], and so cancer is known as the leading cause of death in the world. During the last four decades, a large body of evidence has accumulated to provide support for the concept that the host immune system interacts with developing tumors and may be responsible for the arrest of tumor growth and for tumor regression [2].Immunotherapy holds much promise for the treatment option and considered the fourth-line cancer therapy [3] by using cytokines and adoptive cellular immunotherapy (ACI) since adoptive immunotherapy using lymphokine-activated killer (LAK) cells or tumor-infiltrating lymphocytes (TIL) plus IL-2 has yielded positive results both in experimental tumor models and clinical trials [4].The most current terminology used to describe cytokines is “immunomodulating agents” which are important regulators of both the innate and adaptive immune response. Examples of cytokines are protein hormones produced mainly by activated T cells (lymphocytes) in cell-mediated immunity, and interleukin-2 (IL-2), produced mainly byCD4+ T cells, is the main cytokine responsible for lymphocyte activation, growth, and differentiation. ACI refers to the injection of cultured immune cells that have antitumor reactivity into the tumor-bearing host, which is typically achieved in conjunction with large amounts of IL-2 by using the following two methods: LAK therapy and TIL therapy. For more information on cytokines and ACI, the reader is referred to [5] and the references therein.By applying each therapy separately or by applying both therapies simultaneously, Kirschner and Panetta [6] considered a model describing tumor-immune dynamics together with the feature of IL-2 dynamics. They proposed a model describing the interaction between the effector cells, tumor cells, and the cytokine (IL-2):(1)dEdt=cT−μ2E+p1EILg1+IL+s1,dTdt=r2T1−bT−aETg2+T,dILdt=p2ETg3+T−μ3IL+s2,E0=E0,T0=T0,IL0=IL0,where Et represents the activated immune system cells (commonly called effector cells) such as cytotoxic T cells, macrophages, and natural killer cells that are cytotoxic to the tumor cells; Tt represents the tumor cells; and ILt represents the concentration of IL-2 in the single tumor-site compartment. The parameters and their biological interpretations are summarized in Table 1.Table 1
Parameters and their biological meanings.
Parameter
Biological meaning
c
Antigenicity of the tumor
1
/
μ
2
Natural average lifespan
p
1
,
a
,
p
2
Uptake velocity when all sites are saturated by the substrate
s
1
Treatment by an external source of effector cells (ACI)
r
2
Net per capita growth rate
1
/
b
Environmental carrying capacity
g
i
i
=
1,2,3
Half saturation constant
μ
3
Loss/degraded rate of IL-2
s
2
Treatment by an external input of IL-2 into the system (IL-2)For the nondimensionalized model (1), we adopt the following scaling:(2)t˜=r2t,u=E,v=Tg3,w=ILg1,b˜=bg3,c˜=cg3r2,μ˜2=μ2r2,p˜1=p1r2,s˜1=s1r2,s˜2=s2g1r2,μ˜3=μ3r2,a˜=ag3r2,g˜=g2g3,p˜2=p2g1r2.Then model (1) is converted into the following form (dropping the tilde):(3)dudt=cv−μ2u+p1uw1+w+s1,dvdt=v1−bv−auvg+v,dwdt=p2uv1+v−μ3w+s2,u0=u0,v0=v0,w0=w0.In recent years, fractional-order differential equations have attracted the attention of researchers due to their ability to provide a good description of certain nonlinear phenomena. The fractional-order differential equations are generalizations of ordinary differential equations to arbitrary (noninteger) orders. Some researchers studied the fractional-order differential equations to describe complex systems in different branches of physics, chemistry, and engineering [7]. In the last few years, many researchers have also employed fractional-order biological models [8]. This is because fractional-order differential equations are naturally related to systems with memory [8]. Many biological systems possess memory, and the conception of the fractional-order system may be closer to real-life situations than integer-order systems. The advantages of fractional-order systems are that they describe the whole time domain for physical processes, while the integer-order model is related to the local properties of a certain position, and they allow greater degrees of freedom in the model [9]. The relevant works related to the fractional modeling can be found in [10–13] and the references therein.To the best of the authors’ knowledge, the dynamical analysis of a fractional-order tumor-immune interaction system with immunotherapy has not been performed before. Motivated by the above considerations, in this paper, we study a fractional-order tumor-immune interaction system by extending the integer order model (3) as follows:(4)D0ctαut=cv−μ2u+p1uw1+w+s1,D0ctαvt=v1−bv−auvg+v,D0ctαwt=p2uv1+v−μ3w+s2,u0=u0≥0,v0=v0≥0,w0=w0≥0,where α∈0,1 and D0ctα is the standard Caputo differentiation. The Caputo fractional derivative of order α is defined as [9, 14](5)Dt0ctαft=1Γn−α∫t0tt−sn−α−1fnsds,n−1<α<n,n∈ℕ.In this paper, we consider immunotherapy to be ACI and/or IL-2 delivery either separately or in combination in the interaction site among effector cells, the tumor, and IL-2. The organization of this paper is as follows. In Section2, the existence, uniqueness, and nonnegativity of the fractional-order model (4) are presented. In Section 3, equilibria and (global) asymptotic stability analysis of the fractional-order model (4) are given. The numerical simulations are provided to verify the theoretical results of the fractional-order model (4) in Section 4. Finally, the study concludes with a brief discussion in Section 5.


## 2. Existence, Uniqueness, and Nonnegativity

This section studies the existence, uniqueness, and nonnegativity of the solutions of the fractional-order model (4). To prove the existence and uniqueness of the solution for model (4), we need the following lemma.Lemma 1 (see [8, 15]).
Consider the system(6)Dt0ctαxt=ft,x,with initial condition xt0, where α∈0,1 and f:t0,∞×Ω⟶ℝN,Ω∈ℝN, if ft,x satisfies the locally Lipschitz condition with respect to x, then there exists a unique solution of (6) on t0,∞×Ω.Definition 1.
(see [16]). A point x∗ is called an equilibrium point of system (6) if and only if ft,x∗=0.Theorem 1.
LetΩ=u,v,w∈ℝ3:maxu,v,w≤M. For each initial condition X0=u0,v0,w0∈Ω, there exists a unique solution of the fractional-order model (4), which is defined for all t≥0.Proof.
Let0<T<∞. We seek a sufficient condition for existence and uniqueness of the solutions of the fractional-order model (4) in the region Ω×0,T. We denote X=u,v,wandX¯=u¯,v¯,w¯. Consider a mapping FX=F1X,F2X,F3X, where(7)F1X=cv−μ2u+p1uw1+w+s1,F2X=v1−bv−auvg+v,F3X=p2uv1+v−μ3w+s2.
For anyX,X¯∈Ω, it follows from (4) that(8)FX−FX¯=cv−μ2u+p1uw1+w+s1−cv¯+μ2u¯−p1u¯w¯1+w¯−s1+v1−bv−auvg+v−v¯1−bv¯+au¯v¯g+v¯+p2uv1+v−μ3w+s2−p2u¯v¯1+v¯+μ3w¯−s2+1−bv+v¯−agu¯g+ug+u¯v−v¯+avg+vu−u¯=cv−v¯+p1w1+w−μ2u−u¯+p1u¯1+w1+w¯w−w¯+p2v1+vu−u¯+p2u¯1+v1+v¯v−v¯−μ3w−w¯≤cv−v¯+p1+μ2u−u¯+p1Mw−w¯+1+2bM+aMgv−v¯+au−u¯+p2u−u¯+p2Mv−v¯+μ3w−w¯=p1+μ2+a+p2u−u¯+c+1+2bM+aMg+p2Mv−v¯+p1M+μ3w−w¯≤LX−X¯,where L=maxp1+μ2+a+p2,c+1+2bM+aM/g+p2M,p1M+μ3. Thus, FX satisfies the Lipschitz condition with respect to X. Consequently, it follows from Lemma 1 that there exists a unique solution of model (4).Theorem 2.
Letℝ+=x∈ℝ:x≥0 and Ω+=u,v,w∈Ω:u∈ℝ+,v∈ℝ+andw∈ℝ+. For each initial condition X0=u0,v0,w0∈ℝ+3, all the solutions of the fractional-order model (4) are nonnegative.Proof.
We will prove this theorem by contradiction. Suppose there existst∗≥0 at which the solutions of model (4) passes through either the u-axis, v-axis, or w-axis. Let α∈0,1, then there are three possibilities:(1)
Assume thatut∗=0, vt∗>0, and wt∗>0. Then, there exists t∗>t∗ with t∗−t∗≪1, such that ut<0,vt>0 and wt>0 when t∈t∗,t∗. By the first equation of model (4), we have D0ctαu/>p1−μ2u for all t∈t∗,t∗, and then ut>u0Eαp1−μ2tα for all t∈t∗,t∗. Recall that Eα,βt=∑k=0∞tk/Γkα+β is the Mittag-Leffler function with α,β>0 [17] and Eα=Eα,1. Using the standard comparison theorem for fractional order and the positivity of Mittag-Leffler function Eαt>0 [17], for any α∈0,1, ut>0 for all t∈t∗,t∗, which is a contradiction.(2)
Assume thatvt∗=0, ut∗>0, and wt∗>0. Then, there exists t∗>t∗ with t∗−t∗≪1, such that ut>0,vt<0, and wt>0 when t∈t∗,t∗. By the second equation of model (4), we have D0ctαvt>v for all t∈t∗,t∗, and then vt>v0Eαtα for all t∈t∗,t∗. So, vt>0 for all t∈t∗,t∗, which is a contradiction.(3)
Assume thatwt∗=0, ut∗>0, and vt∗>0. Then, there exists t∗>t∗ with t∗−t∗≪1, such that ut>0,vt>0 and wt<0 when t∈t∗,t∗. By the second equation of model (4), we have D0ctαwt>−μ3w for all t∈t∗,t∗, and then wt>w0Eα−μ3tα for all t∈t∗,t∗. So, wt>0 for all t∈t∗,t∗, which is a contradiction.
Therefore, the solution of model (4) will be nonnegative.


## 3. Equilibria Analysis and Asymptotic Stability

We investigate all nonnegative constant equilibrium points to (4). First, according to Definition 1, model (4) has the following four nonnegative equilibrium points, which have at least one component zero:(1)
E1=0,0,0, if s1=0 and s2=0;(2)
E2=s1/μ2,0,0, if s1>0 and s2=0;(3)
E3=0,0,s2/μ3, if s1=0, s2>0, and p1s2/μ3+s2≠μ2;(4)
E4=s1μ3+s2/μ2μ3+s2μ2−p1,0,Is2/μ3, if s1>0, s2>0, and μ2μ3+s2μ2−p1>0.The cases (2) and (4) are realistic tumor-free equilibrium points. On the other hand, (1) and (3) are not realistic because the effector (or immune) cells do not disappear although the immune system can be weak. Thus, in this section, to investigate the tumor-free equilibrium points in (1), we examine the asymptotically stable behavior at the equilibrium points provided in the cases (2) and (4).Next, we only provide the sufficient conditions of the existence of a unique positive equilibrium pointE∗=u∗,v∗,w∗ to (4) and omit the proof process.Lemma 2 (Lemma 2.1, see [18]).
If one of the following inequalities(1)
gc≥1−bgs1, μ2−p1/ag>s1 and μ2>p1,(2)
gc>s1, μ2g/a>s1s2/μ3+1 and μ2=p1,
holds, then (4) has a unique positive equilibrium point E∗.Lemma 3.
LetJx∗ denote the Jacobian matrix of system (6) evaluated at equilibrium point x∗. The eigenvalues of Jx∗ are λi, where i=1,…,n. Then, equilibrium point x∗ is locally asymptotically stable if and only if all eigenvalues λi, i=1,…,n of Jx∗ satisfy argλi>απ/2; equilibrium point x∗ is a saddle point if some eigenvalues λi satisfy argλi>απ/2 and some others satisfy argλi>απ/2.
Now, we determine the local stability of the equilibrium points of model (4) using the linearization method. The Jacobian matrix of the system evaluated at point X=u,v,w is given by(9)JX≐FXX=−μ2+p1w1+wcp1u1+w2−avg+v1−2bv−agug+v20p2v1+vp2u1+v2−μ3,where FX is defined in the proof of Theorem 1.Theorem 3.
Equilibrium pointE2=s1/μ2,0,0 of (4) is locally asymptotically stable if as1>gμ2 and is unstable, which is a saddle point, if as1<gμ2.Proof.
By (9), the Jacobian matrix of model (4) evaluated at equilibrium point E2=s1/μ2,0,0 is given by(10)JE2=−μ2cp1s1μ201−as1gμ200p2s1μ2−μ3.
Hence, the eigenvalues ofJE2 are λ1=−μ2, λ2=−μ3, and λ3=1−as1/gμ2. Consequently, argλ1=argλ2=π and argλ3=π if as1>gμ2, which leads to argλi>απ/2,i=1,2,3, for 0<α<1. Therefore, according to Lemma 3, the equilibrium point E2 is locally asymptotically stable.
Ifas1<gμ2, then argλ3=0. Thus, argλ3<απ/2 for 0<α<1, which yields that the equilibrium point E2 is unstable and is a saddle point.Theorem 4.
Equilibrium pointE4=s1μ3+s2/μ2μ3+s2μ2−p1,0,Is2/μ3 of (4) is locally asymptotically stable if p1s2>μ2s2+μ3 and as1s2+μ3>gμ2μ3+s2μ2−p1; equilibrium point E4 is a saddle point, if p1s2<μ2s2+μ3 or as1s2+μ3<gμ2μ3+s2μ2−p1.Proof.
By (9), the Jacobian matrix of model (4) evaluated at equilibrium point E4=s1/μ2,0,0 is given by(11)JE2=−μ2+p1s2s2+μ3cp11+s2/μ32⋅s1μ3+s2μ2μ3+s2μ2−p101−ag⋅s1μ3+s2μ2μ3+s2μ2−p100p2⋅s1μ3+s2μ2μ3+s2μ2−p1−μ3.
Hence, the eigenvalues ofJE4 are λ1=−μ3, λ2=−μ2+p10s2/s2+μ3, and λ3=1−a/g⋅s1μ3+s2/μ2μ3+s2μ2−p1. Consequently, argλ1=π and argλ2=argλ3=π if p1s2>μ2s2+μ3 and as1s2+μ3>gμ2μ3+s2μ2−p1, which leads to argλi>απ/2,i=1,2,3, for 0<α<1. Therefore, according to Lemma 3, the equilibrium point E4 is locally asymptotically stable.
Ifp1s2<μ2s2+μ3 or as1s2+μ3>gμ2μ3+s2μ2−p1, then argλ2=0 or argλ3=0. Thus, argλ2 or argλ3<απ/2 for 0<α<1, which yields that the equilibrium point E2 is a saddle point.Remark 1.
Note that equilibrium pointsE1 and E3 are not of biological significance since the effector (or immune) cells do not disappear although the immune system can be weak. As a supplement, we point out a fact that E1 and E3 are saddle in mathematics since the Jacobian matrix of model (4) evaluated at equilibrium points E1 and E3 are as follows:(12)JE1=−μ2c001000−μ3,JE3=−μ2+p1s2s2+μ3c001000−μ3.
In what follows, the local stability of the unique interior equilibriumsE∗ is investigated. By (9), the Jacobian matrix of model (4) evaluated at equilibrium point E∗=u∗,v∗,w∗ is given by(13)JE∗=−μ2+p1w∗1+w∗cp1u∗1+w∗2−av∗g+v∗v∗−b+au∗g+v∗20p2v∗1+v∗p2u∗1+v∗2−μ3.
The eigenvalues of Jacobian matrixJE∗ are the roots of the following equation:(14)Pλ=λ3+a1λ2+a2λ+a3=0,where(15)a1=μ2−p1w∗1+w∗+v∗b−au∗g+v∗2+μ3,a2=v∗μ2−p1w∗1+w∗b−au∗g+v∗2+μ3v∗b−au∗g+v∗2+acv∗g+v∗+μ3μ2−p1w∗1+w∗−p1u∗1+w∗2p2v∗1+v∗,a3=v∗b−au∗g+v∗2μ3μ2−p1w∗1+w∗−p1u∗1+w∗2p2v∗1+v∗+p1u∗1+w∗2p2u∗1+v∗2av∗g+v∗+μ3acv∗g+v∗.
The discriminant ofDP (see [19], Definition 1) is(16)DP=18a1a2a3+a12a22−4a13a3−4a23−27a32.
By the Routh–Hurwitz conditions for fractional-order differential equations defined in [19], Proposition 1, we obtain the following results.Theorem 5.
(1)
IfDP>0 and a1>0,a3>0,a1a2>a3, then E∗ is locally asymptotically stable for α∈0,1.
(2)
IfDP<0 and a1≥0,a2≥0,a3>0, then E∗ is locally asymptotically stable for α∈0.5,2/3.
(3)
IfDP<0 and a1<0,a2<0, then E∗ is unstable for α>2/3.
(4)
IfDP<0 and a1>0,a2>0,a1a2=a3, then E∗ is locally asymptotically stable for α∈0,1.Remark 2.
It follows from Lemmas 2.2 and 2.3 in [18] that μ3μ2−p1w∗/1+w∗>p1u∗/1+w∗2p2v∗/1+v∗ and acv∗/g+v∗+v∗μ2−p1w∗/1+w∗b−au∗/g+v∗2>0. So, the signs of some terms of ai,i=1,2,3, could be determined.
We next investigate the global stability of the positive equilibrium pointE∗ by introducing the following Lyapunov function:(17)Et≐Eut,vt,wt=12u−u∗2+v−v∗−v∗lnvv∗+12w−w∗2,for the solution u,v,w to (4). Note that Et≥0 for all t≥0, and thus, if D0ctαEt≤0 can be derived, then we obtain the desired result from the well-known Lyapunov stability.Lemma 4 (see [20]).
Letxt∈ℝ+ be a continuous and derivable function. Then, for any time instant t>t0,(18)Dtαt0cxt−x∗−x∗lnxtx∗≤1−x∗xtDtαt0cxt,where x∗∈ℝ+ and α∈0,1.Lemma 5 (see [21, 22]).
Letα∈0,1, x∈C00,T:ℝ, and x′∈L10,T:ℝ. Then,(19)Dtαt0cx2t≤2xtDtαt0cxt,t∈0,T.Theorem 6.
Assume thatgc≥1−bgs1 and μ2−p1/ag>s1 where μ2>p1. Then, the positive equilibrium point E∗ to (4) is globally asymptotically stable if(20)μ2≥p1+c+a/g2+12p2+p1c/b+s1μ2−p1,b≥1g+c+a/g2+p2cc/b+s1μ2−p1,μ3≥12p2+p1c/b+s1μ2−p1+p22c/b+s1μ2−p1.Proof.
Calculating theα-order derivative of Et along the solution of model (4), it follows Lemmas 4 and 5 that(21)D0ctαEt≤u−u∗D0ctαut+1−v∗vtD0ctαvt+w−w∗D0ctαwt=u−u∗cv−μ2u+p1uw1+w+s1+v−v∗1−bv−aug+v+w−w∗p2uv1+v−μ3w+s2.
By the definition ofE∗=u∗,v∗,w∗, which is a coexistence equilibrium point of model (4), we have(22)D0ctαEt≤u−u∗2−μ2+p1+c+a/g2+12p2+p1c/b+s1μ2−p1+v−v∗2−b+1g+c+a/g2+p2cc/b+s1μ2−p1+w−w∗2−μ3+12p2+p1c/b+s1μ2−p1+p22c/b+s1μ2−p1.
The conditions given in (20) guarantee that D0ctαEt≤0 for all u,v,w∈ℝ+3, and D0ctαEt=0 implies that u,v,w=u∗,v∗,w∗.


## 4. Numerical Simulation

In this section, numerical simulations of the fractional-order tumor-immune interaction model (4) are conducted to illustrate the theoretical results obtained before. The predictor-corrector PECE method of Adams–Bashforth–Moulton type [23] and some implicit fractional linear multistep methods (FLMMs) of the second order [24] are applied in order to find an approximate solution for our fractional-order model.First, we choose the following set of parameters:(23)c=0.9,μ2=1,p1=0.5,b=3,a=1,g=2.5,p2=1,μ3=1,α=0.9,and consider the asymptotic stability of the realistic tumor-free equilibria E2 and E4. This yields that under some conditions, the tumor can be cured thoroughly, by the therapy (ACI or ACI plus IL-2). Following Theorem 3, when s1=3 and s2=0, the realistic tumor-free equilibrium E2=s1/μ2,0,0=3,0,0 of the fractional-order model (4) is locally asymptotically stable as indicated in Figure 1.Figure 1
Time series (a) and phase diagram (b) of the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,s1=3,b=3,a=1,g=2.5,p2=1,μ3=1,α=0.9, and s2=0.
(a)
(b)Following Theorem4, when s1=3 and s2=0.5, the realistic tumor-free equilibrium E4=s1μ3+s2/μ2μ3+s2μ2−p1,0,Is2/μ3=3.6,0,0.5 of the fractional-order model (4) is locally asymptotically stable as indicated in Figure 2.Figure 2
Time series (a) and phase diagram (b) of the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,s1=3,b=3,a=1,g=2.5,p2=1,μ3=1,α=0.9, and s2=0.5.
(a)
(b)For better visualization of the impact ofα on the asymptotic rate of convergence of the realistic tumor-free equilibria E2 and E4, Figure 3 indicates that with the higher value of α, the asymptotic rate of convergence of E2 and E4 will be larger.Figure 3
Time series of the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,s1=3,b=3,a=1,g=2.5,p2=1,μ3=1, and different values of the order α of the Caputo fractional derivative. (a)s2=0 and (b)s2=0.5.
(a)
(b)Note thatv represents the tumor cells and s1 and s2 represent the treatment by an external source of effector cells and the treatment by an external input of IL-2 into the system, respectively. For better visualization of the effects of two types of immunotherapy, we consider the rate of tumor extinction under two cases: s2 with different s1, or s1=3 with different s2. Figure 4 implies the former case, Figures 5 and 6 imply the latter case. The results show(1)
Tumor treatment by an external source of effector cells, i.e.,s2=0 with different s1. Figure 4 shows that the higher the value of s1, the asymptotic rate of convergence of v or the rate of tumor extinction will be larger; however, the variations are not obvious when s1 reaches a critical value.(2)
Tumor treatment by an external source of effector cells without or with an external input of IL-2 into the system, i.e.,s1=3,s2=0 or s1=3,s2=0.5. Figure 5 shows that the introduction of new immunotherapy methods has accelerated the asymptotic rate of convergence of v or the rate of tumor extinction.(3)
Tumor treatment by an external source of effector cells and an external input of IL-2 into the system, i.e.,s1=3 with different s2. Figure 6 shows that with the same value of s1 and higher value of s2, the asymptotic rate of convergence of v or the rate of tumor extinction will be larger; however, the variations are not obvious when s2 reaches a critical value.Figure 4
Time series of the immune system cellsv in the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,b=3,a=1,g=2.5,p2=1,μ3=1,s2=0,α=0.9, and different treatment by an external input of IL-2 into the system s1: (a) original drawing and (b) drawing of partial enlargement of 4(a).
(a)
(b)Figure 5
Time series of the immune system cellsv in the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,s1=3,b=3,a=1,g=2.5,p2=1,μ3=1,α=0.9, and the same treatment by an external input of IL-2 into the system s1=3. (a) Original drawing (the blue line represents that there is only the treatment by an external input of IL-2 into the system, i.e., s2=0, and the red line represents that besides the treatment by an external input of IL-2 into the system, there is also the treatment by an external source of effector cells, i.e., s2=0.5). (b) Drawing of partial enlargement of 5(a).
(a)
(b)Figure 6
Time series of the immune system cellsv in the fractional-order model (4) with c=0.9,μ2=1,p1=0.5,s1=3,b=3,a=1,g=2.5,p2=1,μ3=1,α=0.9, and different treatment by an external source of effector cells s2: (a) original drawing and (b) drawing of partial enlargement of 6(a).
(a)
(b)In other words, the desired best effects can be achieved by combining the two types of immunotherapy.Second, we choose the following set of parameters:(24)c=1,μ2=1,p1=0.5,s1=1,s2=0.5,and consider the asymptotic stability of the unique interior equilibrium E∗. This yields that under some conditions, combination therapy (ACI plus IL-2) can achieve satisfactory and stable tumor control; however, the tumor is incurable.Figures7 and 8 indicate that the unique interior equilibrium(25)E∗=7.2036,3.1197,5.9551or2.5191,0.7481,1.5781,is asymptotically stable when b=0.3,a=0.05,g=2.5,p2=1,μ3=1,andα=0.9 or b=0.3,a=1,g=2.5,p2=1,μ3=1,andα=0.6, respectively, concurring with the results of Theorem 5 (1) and (2).Figure 7
Time series of the fractional-order model (4) with c=1,μ2=1,p1=0.5,s1=1,b=0.3,a=0.05,g=2.5,p2=1,μ3=1,s2=0.5, and α=0.9.Figure 8
Time series of the fractional-order model (4) with c=1,μ2=1,p1=0.5,s1=1,b=0.3,a=1,g=2.5,p2=1,μ3=1,s2=0.5, and α=0.6.Figure9 indicates that all trajectories with different positive initial conditions converge to the unique interior equilibrium E∗=0.8463,0.5880,0.3283 when b=1.5,a=0.5,g=3,p2=0.5,μ3=2, and α=0.9, which indicates that E∗ is globally asymptotically stable, concurring with the results of Theorem 6. This situation means that the tumor will exist indefinitely, which will be incurable in medicine.Figure 9
Phase diagram of the fractional-order model (4) with c=1,μ2=2,p1=0.5,s1=1,b=1.5,a=0.5,g=3,p2=0.5,μ3=2,s2=0.5,α=0.9, and different initial conditions u0,v0,w0.


## 5. Concluding Remarks

In this paper a fractional-order tumor-immune interaction model with immunotherapy is discussed. The existence, uniqueness, and nonnegativity of the solutions are proved. The local and global asymptotic stability of some equilibrium points are investigated. Unfortunately, by the fractional calculation, we cannot obtain the boundedness of solutions to the fractional-order tumor-immune model (4) with α∈0,1.In addition, numerical simulations are conducted to illustrate the analytical results. This yields that under some conditions, the tumor can be cured thoroughly, by the therapy (ACI or ACI plus IL-2); under some other conditions, combination therapy (ACI plus IL-2) can achieve satisfactory and stable tumor control; however, the tumor is incurable. The results indicate that the sufficiently large orderα of the Caputo fractional derivative has a stabilization effect, and it may help to control the tumor extinction, in the tumor-immune model with immunotherapy.



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