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.
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 by 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):where 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; represents the tumor cells; and represents the concentration of IL-2 in the single tumor-site compartment. The parameters and their biological interpretations are summarized in Table 1.
For the nondimensionalized model (1), we adopt the following scaling:
Then 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. 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:where and is the standard Caputo differentiation. The Caputo fractional derivative of order 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. The organization of this paper is as follows. In Section 2, 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 systemwith initial condition , where and , if satisfies the locally Lipschitz condition with respect to , then there exists a unique solution of (6) on .
Definition 1. (see [16]). A point is called an equilibrium point of system (6) if and only if .
Theorem 1. Let . For each initial condition , there exists a unique solution of the fractional-order model (4), which is defined for all .
Proof. Let . We seek a sufficient condition for existence and uniqueness of the solutions of the fractional-order model (4) in the region . We denote . Consider a mapping , whereFor any , it follows from (4) thatwhere . Thus, satisfies the Lipschitz condition with respect to . Consequently, it follows from Lemma 1 that there exists a unique solution of model (4).
Theorem 2. Let and . For each initial condition , all the solutions of the fractional-order model (4) are nonnegative.
Proof. We will prove this theorem by contradiction. Suppose there exists at which the solutions of model (4) passes through either the -axis, -axis, or -axis. Let , then there are three possibilities:(1)Assume that , , and . Then, there exists with , such that and when . By the first equation of model (4), we have for all , and then for all . Recall that is the Mittag-Leffler function with [17] and . Using the standard comparison theorem for fractional order and the positivity of Mittag-Leffler function [17], for any , for all , which is a contradiction.(2)Assume that , , and . Then, there exists with , such that , and when . By the second equation of model (4), we have for all , and then for all . So, for all , which is a contradiction.(3)Assume that , , and . Then, there exists with , such that and when . By the second equation of model (4), we have for all , and then for all . So, for all , 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), if and ;(2), if and ;(3), if , , and ;(4), if , , and .
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 point to (4) and omit the proof process.
Lemma 2 (Lemma 2.1, see [18]). If one of the following inequalities(1), and ,(2), and ,holds, then (4) has a unique positive equilibrium point .
Lemma 3. Let denote the Jacobian matrix of system (6) evaluated at equilibrium point . The eigenvalues of are , where . Then, equilibrium point is locally asymptotically stable if and only if all eigenvalues , of satisfy ; equilibrium point is a saddle point if some eigenvalues satisfy and some others satisfy .
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 is given bywhere is defined in the proof of Theorem 1.
Theorem 3. Equilibrium point of (4) is locally asymptotically stable if and is unstable, which is a saddle point, if .
Proof. By (9), the Jacobian matrix of model (4) evaluated at equilibrium point is given byHence, the eigenvalues of are , , and . Consequently, and if , which leads to , for . Therefore, according to Lemma 3, the equilibrium point is locally asymptotically stable.
If , then . Thus, for , which yields that the equilibrium point is unstable and is a saddle point.
Theorem 4. Equilibrium point of (4) is locally asymptotically stable if and ; equilibrium point is a saddle point, if or .
Proof. By (9), the Jacobian matrix of model (4) evaluated at equilibrium point is given byHence, the eigenvalues of are , , and . Consequently, and if and , which leads to , for . Therefore, according to Lemma 3, the equilibrium point is locally asymptotically stable.
If or , then or . Thus, or for , which yields that the equilibrium point is a saddle point.
Remark 1. Note that equilibrium points and 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 and are saddle in mathematics since the Jacobian matrix of model (4) evaluated at equilibrium points and are as follows:In what follows, the local stability of the unique interior equilibriums is investigated. By (9), the Jacobian matrix of model (4) evaluated at equilibrium point is given byThe eigenvalues of Jacobian matrix are the roots of the following equation:whereThe discriminant of (see [19], Definition 1) isBy the Routh–Hurwitz conditions for fractional-order differential equations defined in [19], Proposition 1, we obtain the following results.
Theorem 5. (1)If and , then is locally asymptotically stable for .(2)If and , then is locally asymptotically stable for .(3)If and , then is unstable for .(4)If and , then is locally asymptotically stable for .
Remark 2. It follows from Lemmas 2.2 and 2.3 in [18] that and . So, the signs of some terms of , could be determined.
We next investigate the global stability of the positive equilibrium point by introducing the following Lyapunov function:for the solution to (4). Note that for all , and thus, if can be derived, then we obtain the desired result from the well-known Lyapunov stability.
Lemma 4 (see [20]). Let be a continuous and derivable function. Then, for any time instant ,where and .
Lemma 5 (see [21, 22]). Let , , and . Then,
Theorem 6. Assume that and where . Then, the positive equilibrium point to (4) is globally asymptotically stable if
Proof. Calculating the -order derivative of along the solution of model (4), it follows Lemmas 4 and 5 thatBy the definition of , which is a coexistence equilibrium point of model (4), we haveThe conditions given in (20) guarantee that for all , and implies that .
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:and consider the asymptotic stability of the realistic tumor-free equilibria and . 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 and , the realistic tumor-free equilibrium of the fractional-order model (4) is locally asymptotically stable as indicated in Figure 1.
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Following Theorem 4, when and , the realistic tumor-free equilibrium of the fractional-order model (4) is locally asymptotically stable as indicated in Figure 2.
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For better visualization of the impact of on the asymptotic rate of convergence of the realistic tumor-free equilibria and , Figure 3 indicates that with the higher value of , the asymptotic rate of convergence of and will be larger.
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Note that represents the tumor cells and and 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: with different , or with different . 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., with different . Figure 4 shows that the higher the value of , the asymptotic rate of convergence of or the rate of tumor extinction will be larger; however, the variations are not obvious when 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., or . Figure 5 shows that the introduction of new immunotherapy methods has accelerated the asymptotic rate of convergence of 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., with different . Figure 6 shows that with the same value of and higher value of , the asymptotic rate of convergence of or the rate of tumor extinction will be larger; however, the variations are not obvious when reaches a critical value.
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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:and consider the asymptotic stability of the unique interior equilibrium . This yields that under some conditions, combination therapy (ACI plus IL-2) can achieve satisfactory and stable tumor control; however, the tumor is incurable.
Figures 7 and 8 indicate that the unique interior equilibriumis asymptotically stable when or , respectively, concurring with the results of Theorem 5 (1) and (2).
Figure 9 indicates that all trajectories with different positive initial conditions converge to the unique interior equilibrium when , and , which indicates that 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.
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 .
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 order 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by National Natural Science Foundation of China (no. 81571731), Key Research and Development Program of Shaanxi (no. 2018SF-161), Natural Science Basic Research Plan in Shaanxi Province of China (no. 2020JM-569), and Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 19JK0792)