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

A fractional-order tumor-immune interaction model with immunotherapy is proposed and examined.-e existence, uniqueness, and nonnegativity of the solutions are proved. -e 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. -e results indicate that the fractional order has a stabilization effect, and it may help to control the tumor extinction.


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]. e 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 cellmediated immunity, and interleukin-2 (IL-2), produced mainly by CD4 + 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. ey proposed a model describing the interaction between the effector cells, tumor cells, and the cytokine (IL-2): dE dt � cT − μ 2 E + p 1 EI L g 1 + I L + s 1 , where E(t) 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; T(t) represents the tumor cells; and I L (t) represents the concentration of IL-2 in the single tumor-site compartment. e parameters and their biological interpretations are summarized in Table 1.
For the nondimensionalized model (1), we adopt the following scaling: (2) en model (1) is converted into the following form (dropping the tilde): 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. e 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].
is 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. e 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]. e relevant works related to the fractional modeling can be found in [10][11][12][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: 2 Complexity where α ∈ (0, 1) and c 0 D α t is the standard Caputo differentiation. e Caputo fractional derivative of order α is defined as [9,14] 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. e organization of this paper is as follows. In Section 2, the existence, uniqueness, and nonnegativity of the fractionalorder model (4) are presented. In Section 3, equilibria and (global) asymptotic stability analysis of the fractional-order model (4) are given. e 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.

Existence, Uniqueness, and Nonnegativity
is 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.
Definition 1 (see [16]). A point x * is called an equilibrium point of system (6) if and only if f(t, x * ) � 0.
exists a unique solution of the fractional-order model (4), which is defined for all t ≥ 0.
Proof. Let 0 < T < ∞. We seek a sufficient condition for existence and uniqueness of the solutions of the fractionalorder model (4) For any X, X ∈ Ω, it follows from (4) that us, F(X) satisfies the Lipschitz condition with respect to X. Consequently, it follows from Lemma 1 that there exists a unique solution of model (4).
+ , all the solutions of the fractional-order model (4) are nonnegative.
Proof. We will prove this theorem by contradiction. Suppose there exists t * ≥ 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: Using the standard comparison theorem for fractional order and the positivity of Mittag-Leffler erefore, the solution of model (4) will be nonnegative.

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: e 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.
us, 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 point E * � (u * , v * , w * ) to (4) and omit the proof process.
Lemma 2 (Lemma 2.1, see [18]). If one of the following inequalities holds, then (4) has a unique positive equilibrium point E * . Now, we determine the local stability of the equilibrium points of model (4) using the linearization method. e Jacobian matrix of the system evaluated at point X � (u, v, w) is given by where F(X) is defined in the proof of eorem 1. (4) is locally asymptotically stable if as 1 > gμ 2 and is unstable, which is a saddle point, if as 1 < gμ 2 .
erefore, according to Lemma 3, the equilibrium point E 2 is locally asymptotically stable.
erefore, according to Lemma 3, the equilibrium point E 4 is locally asymptotically stable.

Remark 2. It follows from Lemmas 2.2 and 2.3 in [18] that
So, the signs of some terms of a i , i � 1, 2, 3, could be determined. We next investigate the global stability of the positive equilibrium point E * by introducing the following Lyapunov function: for the solution (u, v, w) to (4). Note that E(t) ≥ 0 for all t ≥ 0, and thus, if c 0 D α t E(t) ≤ 0 can be derived, then we obtain the desired result from the well-known Lyapunov stability.
For better visualization of the impact of α on the asymptotic rate of convergence of the realistic tumor-free equilibria E 2 and E 4 , Figure 3 indicates that with the higher value of α, the asymptotic rate of convergence of E 2 and E 4 will be larger.
Note that v represents the tumor cells and s 1 and s 2 represent the treatment by an external source of effector  (4) with c � 0.9, μ 2 � 1, p 1 � 0.5, s 1 � 3, b � 3, a � 1, g � 2.5, p 2 � 1, μ 3 � 1, α � 0.9, and s 2 � 0.5.  Figure 4 implies the former case, Figures 5 and 6 imply the latter case. e results show (1) Tumor treatment by an external source of effector cells, i.e., s 2 � 0 with different s 1 . Figure 4 shows that the higher the value of s 1 , the asymptotic rate of convergence of v or the rate of tumor extinction will be larger; however, the variations are not obvious when s 1 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., s 1 � 3, s 2 � 0 or s 1 � 3, s 2 � 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., s 1 � 3 with different s 2 . Figure 6 shows that with the same value of s 1 and higher value of s 2 , the asymptotic rate of convergence of v or the rate of tumor extinction will be larger; however, the  (4) with c � 0.9, μ 2 � 1, p 1 � 0.5, s 1 � 3, b � 3, a � 1, g � 2.5, p 2 � 1, μ 3 � 1, α � 0.9, and the same treatment by an external input of IL-2 into the system s 1 � 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., s 2 � 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., s 2 � 0.5). (b) Drawing of partial enlargement of 5(a).
variations are not obvious when s 2 reaches a critical value.
In other words, the desired best effects can be achieved by combining the two types of immunotherapy.
with the results of eorem 6. is situation means that the tumor will exist indefinitely, which will be incurable in medicine.

Concluding Remarks
In this paper a fractional-order tumor-immune interaction model with immunotherapy is discussed. e existence, uniqueness, and nonnegativity of the solutions are proved. e 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. is 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request

Conflicts of Interest
e authors declare that they have no conflicts of interest.