Computational and Mathematical Methods in Medicine
Volume 2017 (2017), Article ID 1642976, 9 pages
https://doi.org/10.1155/2017/1642976

Research Article

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

Ping Bi,¹ Zijian Liu,² Mutei Damaris Muthoni,¹ and Jianhua Pang³

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

Correspondence should be addressed to Ping Bi; pbi@math.ecnu.edu.cn

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 as

where 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 is

Thus, 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) becomes

Obviously, 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.

(a)

(b)

Figure 1: (a) Stable equilibrium . (b) Saddle-node equilibrium .

(a)

(b)

Figure 2: (a) Stable focus equilibrium . (b) Saddle node () and stable focus ().

3. Dynamical Behaviors of the Model with Delay

In this section, we shall consider model (5) with delay

The existence of the equilibria is the same as those of (5). The dynamical behaviors of the trivial equilibria and semitrivial equilibria are not difficult; we only study the positive equilibrium here. The characteristic equation of linearized system for (25) at takes the following form:

where

Similar to the proof in [7], then the following results can be obtained easily.

Theorem 9. (1) If

then all roots of (26) have negative real parts for all .
(2) If or and , then (26) has a pair of purely imaginary roots at .
(3) If and , then (26) has a pair of purely imaginary roots , resp.) at , where

Theorem 10. (1) If the conditions of (1) in Theorem 9 hold, then is asymptotically stable for any .
(2) If or the conditions of (3) in Theorem 9 hold, then undergoes Hopf bifurcation as
(3) If , and , then undergoes a Hopf bifurcation as

Proof. From the analysis of the above, we only need to compute

In third case, , then we need the sign of the second derivative of of (26) at the point where is passing through
We can easily obtain

with

Since , Thus it is easy to see that This finishes the proof.

In the above, we obtained the conditions under which a family of periodic solutions bifurcated from the positive equilibrium at . In the following, we derive the explicit formulae for determining the properties of the Hopf bifurcated solution by using the normal form and the center manifold theory. Throughout this section, we always assume that the system (25) undergoes Hopf bifurcations at the positive equilibrium for and are the corresponding pure imaginary roots.

Let , . Then (25) becomes

Set , . Then is a Hopf bifurcation value for (33). Equation (5) can be written as

where , , and , are given, respectively, by

with and

By Riesz representation theorem, there exists a function of bounded variation for such that

For , define

Then system (34) is equivalent to

where for For , define

and a bilinear inner product

where Then and are joint operators.

Suppose that is the eigenvector of corresponding to , with Then is the eigenvector of with , such that , .

In the following, we use the ideas in Adam and Bellomo [1] to compute the coordinates describing center manifold at Define

On the center manifold , we have

where and are local coordinates for the center manifold in the direction of and Note that is real if is real. For of (39), we have

where

From (43) and (44), we have

and Comparing the coefficients of the above equality with (45), we obtain

In order to determine , we also need to compute and as follows:

where

On the center manifold near the origin,

Comparing the coefficients of , we obtain

From (52) and the definition of , one has

Solving the above ODE, then

where is a constant vector. Similarly, we obtain

where is a constant vector.

From the definition of and the above equations, we can obtain

with

In the same way, it is easy to get

In order to determine and , we can calculate and from (54) and (55). Furthermore, we can also determine . Thus we can be able to compute the following values:

Theorem 11. Equation (25) undergoes Hopf bifurcation as . If (), then the bifurcated periodic solution is supercritical (subcritical); if (), then the bifurcating solution is stable (unstable). The period of the bifurcated periodic solution is .

In the following, we present some numerical results of system (25) at different values of We choose the parameters the same as before; the system (25) becomes

Obviously, system (60) has three positive equilibria: , , and Equilibrium is stable for all , while equilibrium is unstable saddle node for all . When , the positive equilibrium is a stable focus, as is growing to , When passes the critical value , loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from This is clearly demonstrated by Figure 3.

(a)

(b)

Figure 3: (a) Stable equilibrium as . (b) Bifurcated solution around as .

4. Conclusions

In 1994, Kuznetsov et al. [11] introduced a model, which describes competition between the tumor and immune cells. This mode describes the response of effector cells to the growth of tumor cells. Later, the model is simplified to a two-dimensional differential equation with Lotka-Volterra form immune response function ; the rich dynamical behaviors are studied such as stability, qualitative behaviors, and bifurcation behaviors.

In this paper, we studied the model with an immune response function with Michaelis-Menten form . In this case, the properties of system (4) are more complicated. There are so many tedious computations even for the existence of the equilibria. The dynamical behaviors of the model are more rich.

We have studied the nonlinear dynamics of a two-dimensional general differential system. We first provided linear analysis of the system at the possible equilibria, namely, the semitrivial and positive equilibria, and discussed the existence of Hopf bifurcation at the positive equilibrium. Then we consider the system with delay; we investigated the Hopf bifurcation of the system. The existence and stability of periodic solutions were given. Numerical simulations were presented to illustrate the theoretical analysis and results. Cancer immunosurveillance functions are taken as an important defense to cancer; this is the elimination process. In fact, the existence and stability of the semitrivial equilibrium correspond to the elimination process. Our results on the existence and stability of the Hopf bifurcated periodic solutions describe the equilibrium process. If a stable periodic orbit exists, then the tumor and the immune system can coexist for a long time, although the cancer cannot be eliminated. Furthermore, the parameters are important in controlling the development and progression of the tumor, which is decided by the conditions of the existence of the bifurcations.

The existence of oscillatory solutions in the tummor and immune system interaction models demonstrates that the phenomenon has been observed in some related models [7, 10, 13–15]. The initial values and delay are also important in the oscillatory coexistence of the tumor cells and the effector cells. Numerical simulations indicate this information well.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This paper was supported by National Natural Science Foundation of China (11401217, 11671142, 11401060, and 11401117), Shanghai Leading Academic Discipline Project (B407), and 211 Project of Key Academic Discipline of East China Normal University.

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