Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 1642976, 9 pages

https://doi.org/10.1155/2017/1642976

## The Dynamical Behaviors for a Class of Immunogenic Tumor Model with Delay

^{1}Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, China^{2}College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China^{3}School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China

Correspondence should be addressed to Ping Bi

Received 18 May 2017; Revised 13 July 2017; Accepted 9 August 2017; Published 25 October 2017

Academic Editor: Po-Hsiang Tsui

Copyright © 2017 Ping Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer et al., time delay is introduced in the general model. The dynamic behaviors of this model are studied, which include the existence and stability of the equilibria and Hopf bifurcation of the model with discrete delays. The properties of the bifurcated periodic solutions are studied by using the normal form on the center manifold. Numerical examples and simulations are given to illustrate the bifurcation analysis and the obtained results.

#### 1. Introduction

For the longest time, malignant tumors have posed a threat to human lives. Effective strategies based on the immune system have been championed to complement traditional methods of cancer treatment, such as surgery, radiation, and chemotherapy. Cancer immunotherapy is the use of the immune system to treat cancer. Immunotherapy is used to provoke the immune system into attacking the tumor cells by using these cancer antigens as targets. Cell types that can be used in this way are natural killer cells, lymphokine-activated killer cells, cytotoxic T cells, and dendritic cells. There has been much interest in mathematical models describing the interaction between tumor cells and effector cells of the immune system; see [1–6]. An ideal model can provide insight into the dynamics of interactions of the immune response with the tumor and may play an important role in understanding of cancer and developing effective drug therapies. Developing ideal models for such complex processes is not easy. Simple models which display some useful immunological phenomena have been proposed and studied extensively. See Bi and Xiao [7], Galach [4], and Yafia [8–10], and the references cited therein. In , Kuznetsov et al. [11] introduced a model, which describes competition between the tumor and immune cells. It also describes the response of effector cells (ECs) to the growth of tumor cells (TCs). It is assumed that the tumor cells are “immunogenic” and thus subject to immune attack by cytotoxic (CT) or natural killer (NK) cells. This model studies the infiltration of TCs by ECs and also the possibility of ECs inactivation.* It is assumed that interactions between ECs and TCs in vitro can be described by the following kinetic scheme describing interactions between ECs and TCs:*where are the local concentrations of , , - complexes, inactivated ECs, and TCs, respectively. The parameters and are nonnegative constants, which describe the conversion rates of differential cells. Then Kuznetsov and Taylor’s model is as follows:where is the normal rate of the flow of adult into the tumor site, describes the accumulation of effector cells in the tumor cells localization due to the presence of the tumor. , ( are constants). If , it is reasonable to assume , that is, , then . Then we only need to analyze the dynamical behavior of the first two equations.

In , Galach [4] suggested that the function is in the Michaelis-Menten form , and thus (2) can be simplified aswhere is the local concentrations of , and is the local concentrations of , , , and all the parameters are positive. The properties of the model (3) is studied in [7–9]. The dynamical behaviors and the bifurcation of the model with delay are also studied by [7, 10].

In this paper, we analyze the dynamics of an immune response function with Michaelis-Menten form , where and are positive constants, that is,In order to simplify the original model, we nondimensionalize (4) by choosing scale for and cell population, respectively. Let , and replace with ; thus the model can be written as follows:where , , and all the above parameters are positive.

This paper is organized as follows: In Section 2, the model without delay is considered, and the conditions for existence and stability of equilibria are given. In Section 3, the model with delay is studied. The existence of Hopf bifurcation is obtained; the direction and stability of bifurcated periodic solutions are also given with the help of center manifold and bifurcation theories. At the end of this paper, numerical results are given to illustrate the main results of this paper.

#### 2. Existence and Stability of the Equilibria

It is easy to see that (5) has a tumor-free equilibrium . In order to find the positive equilibria , we need to solve the following equations:Equation (6) is reduced to the following cubic polynomial:where , , , Obviously, system (5) in is as “well behaved” just as in biology. We have the following lemma by qualitative analysis.

Lemma 1. *If , then the solutions of (5) are invariable in : *

*Proof. *If and , then Similarly, if and , one has Also, if and , we have and, then, there exists a positive number , which is given by Such that when . Similarly, when On the other hand, one has with the help of . That is

When and , we can prove the result in a similar way. This finishes the proof.

It is easy to see that (5) has one tumor-free equilibrium . With the distribution of the eigenvalues, we can easily obtain the following results.

Theorem 2. *If , then is unstable. If , is stable. If , is asymptotically stable.*

Theorem 3. *If , , then is globally asymptotically stable.*

*Proof. *When , , then it is easy to see that This implies that is the only positive equilibrium.

Noting , we know that is locally stable with the help of Theorem 2. In the remaining part of the proof, we only need to prove the global stability. By Lemma 1, we know that the following domain is invariable: Then it is easy to prove that as . In addition, when , This shows that the vector fields are moving towards as increases.

Let Dulac function . Then Hence there are no closed trajectory surrounds in field . That is, the result follows.

Using the original parameters, we can give the results as follows.

Theorem 4. *If , , system (5) has only one critical equilibrium . Furthermore, is globally asymptotically stable.*

*Remark 5. *Theorem 4 is an instructive results to kill the tumor cells. The tumor cells will be killed out by the immune cells sooner or later under the above conditions; then we only need to take the necessary measures to control the parameters to satisfy the inequity in Theorem 4.

In the following, we will study the existence and stability of the tumor-present equilibrium Similar to the proof of Lemma 2.4 in [12], we can easily obtain the following results.

Theorem 6. *For the number of positive equilibria, we can get the following results:*(1)*If , , and , then (5) has three distinct positive roots.*(2)*If one of the following conditions is satisfied, then (5) has two distinct positive roots:(a) , , and .(b), and *(3)

*Assume one of the following conditions is satisfied, then (5) has one positive root:(a)*

*, , and .*(b)*, , and .*(c)*, , , , and .*(d)*, , and .**The corresponding Jacobian matrix at isThus, we can give the following results.*

*Theorem 7. If , then is stable. And no Hopf bifurcation appears around the equilibrium .*

*Proof. *It is easy to see Noting , it is easy to see that Thus the results are proved.

*Theorem 8. Let be the coordinate of the positive equilibrium . Then the following results hold: (1)If , then is stable.(2)If , then is stable for and unstable for .*

*Proof. *It is easy to see where

Let Then the sign of the is the same as that of Let If for any , then Hence, Then the first result can be obtained easily.

If , that is, , , then for any . That is to say, is stable as . On the other hand, one knows as . Noting , then Then has roots and . From , we have as and as . That is to say, the second result holds.

*In the following, we give some simulation results of the above results. We consider the system (5) and the parameters suggested by V. A. Kuznetsov et al. Then system (5) becomesObviously, system (24) has three positive equilibria: , , and By simple computation, it is easy to know that the eigenvalues of are , and has eigenvalues and , and has eigenvalues and . These results are represented in Figures 1 and 2.*