Abstract
A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.
1. Introduction
In this century, cancer remains one of the most dangerous killers of humankind; every year millions of people suffer from cancer and die from this disease throughout the world; see Boyle et al. [1]. Recently, there has been much interest in mathematical modeling of immune response with the intruder (see, e.g., Liu et al. [2, 3], Yafia [4], d’Onofrio et al. [5, 6], and the references cited therein). In fact, mathematical models are feasible to propose simple models which are capable of displaying some of the essential immunological phenomena. The delayed models of tumor and immune response interactions have been studied extensively; we refer to Bi and Ruan [7], Yafia [8], Mayer et al. [9], Yafia [10], and the references cited therein, which have shown that various bifurcations can occur in such models. It is interesting to consider the nonlinear dynamics of the delayed tumor-immune model.
In 1994, Kuznetsov et al. [11] took into account the penetration of tumor cells (TCs) by effector cells (ECs) and proposed a model describing the response of ECs to the growth of TCs. They assumed that interactions between ECs and TCs in vitro can be described by the kinetic scheme shown in Figure 1, where , and are the local concentrations of ECs, TCs, EC-TC complexes, inactivated ECs, and lethally hit TCs, respectively. Then the Kuznetsov and Taylor model is as follows: where is the normal rate of the flow of adult ECs into the tumor site, describes the accumulation of effector cells in the tumor cells localization due to the presence of the tumor, , and are the coefficients of the processes of destruction and migration for E, EC, and TC, respectively, is the coefficient of the maximal growth of tumor, and is the environment capacity. Kuznetsov et al. [11] claimed that experimental observations motivate the approximation ; therefore, it is reasonable to assume that with . Kuznetsov et al. [11] also suggested that the function is in the Michaelis-Menten form . In 2003, Gałach [12] suggested that the function should be in a Lotka-Volterra form ; then the model (1) can be reduced to where denotes the dimensionless density of ECs, stands for the dimensionless density of the population of TCs, , , and all coefficients are positive. Set , , , , . Replace with and with . Then (2) can be written as where , , , , , and .
Mayer et al. [9] and Asachenkov et al. [13] pointed out that the delays should be taken into account to describe the times necessary for molecule production, proliferation, differentiation of cells, transport, and so forth. In fact, the immune system needs time to develop a suitable response after the invasion of tumor cells; the binding of EC and TC also needs time. Therefore, we introduce time delays into the model of immune response. Integrating models [9–11], we will consider the model as follows: where ; if the stimulation coefficient of the immune system exceeds the neutralization coefficient of ECs in the process of the formation of EC-TC complexes, then . Yafia [4] studied the linear stability of the equilibria and the existence of Hopf bifurcation for model (4) with . Yafia [10] and Gałach [12] obtained similar results as those of Yafia [4] for (4) with . Recently, Bi and Xiao [14] give conditions for the properties of Hopf bifurcated periodic solution and existence of the global Hopf bifurcation for (4) with .
In this paper, we will consider the dynamical behaviors of model (4) with . The rest of this paper is organized as follows. In Section 2, the linear analysis of the model is carried out and local stability of the equilibria and the conditions of Hopf bifurcation are given. Section 3 is devoted to the analysis of Hopf, steady-state bifurcations, and B-T bifurcation. Numerical results and simulations are carried out to illustrate the main results. A brief discussion and more numerical simulations are given in Section 4.
2. Local Analysis
In this section, we will study the local stability of the equilibria and the Hopf bifurcations of system It is easy to obtain that system (5) have three equilibria , , and , where . It is easy to see that . Because the number of tumor cells or effect cells is positive, we only consider the dynamical behaviors of the equilibria (tumor-free point) and in the rest of the paper.
Let , . System (5) can be written as where , , , , , and is the coordinate of the equilibrium.
It is easy to see that the linear system of system (7) is where , , , , , and are the same as those in (7).
2.1. Tumor-Free Point
The characteristic equation of system (8) at the tumor-free equilibrium is Then we have the following results.
Lemma 1. (I) If , then(1)Equation (9) has a simple zero root, and all other roots have negative real parts as ;(2)Equation (9) has a double zero root, and all other roots have negative real parts as ;(3)Equation (9) has at least one root with positive real parts as .
(II) If , then(1)all roots of (9) have negative real parts as ;(2)Equation (9) has a pair of conjugate purely imaginary roots , and all other roots have negative real parts as ;(3)Equation (9) has at least one root with positive real parts as .
(III) Equation (9) has a negative root , and all other roots have positive real parts as .
Proof. is a root of (9) if and only if . If , (9) has two roots and . Then there are three cases: (1) as ; (2) as ; (3) as .
We will consider the case as follows. If , , then
hence, , and . Thus is the double zero root of (9).
If (9) has purely imaginary roots, then the roots must be the solution of
Assume that is the root of (11); that is,
that is . Hence (11) has a positive root if and only if , and the corresponding critical values are
Using Rouché theorem, we know that conclusions (II)(1), (II)(2), and (III) hold.
If , , we can obtain . Noting the continuous of the function , we know that there is at least a such that . On the other hand, it is easy to see that ; then there exists such that .
Differentiating both sides of (11) with respect to , we have
If , then
Using Rouché theorem, we know that the conclusions of (I)(1) and (I)(2) are true.
If , then
Thus
Hence the conclusion (I)(3) is true.
Noting
then (II)(3) is proved. Then all the proof is finished.
Thus the following results can be obtained by Lemma 1.
Theorem 2. (I) If , then(1)system (5) undergoes a codimension one steady-state bifurcation at the tumor-free equilibrium as ;(2)the tumor-free equilibrium is a B-T singular equilibrium as .
(II) If , then(1)the tumor-free equilibrium is asymptotically stable as ;(2)the tumor-free equilibrium is unstable as ;(3)system (5) undergoes Hopf bifurcation at the tumor-free equilibrium as .
(III) The tumor-free equilibrium is unstable as .
2.2. Positive Equilibrium
If , then the positive equilibrium exists. The characteristic equation of system (8) at the point is where
Lemma 3. If , , then(1)all roots of (19) have negative real parts as ;(2)Equation (19) has a pair of conjugate purely imaginary roots , and all other roots have negative real parts as ;(3)Equation (19) has at least one root with positive real parts as .
Proof. Noting , , one has
thus (19) has no zero root.
If , then (19) can be written as
It is easy to see
and then all roots of (22) have negative real parts.
If , we assume that (19) has a pair of purely imaginary roots ; thus
and hence
Noting , , then we have and . That is to say, (25) has only one positive root
and the corresponding critical value is
We can also give the following transversal condition:
Then all results of this theorem have been proven.
From Lemma 3, the following theorem can be obtained directly.
Theorem 4. Suppose that , ; then(1)the positive equilibrium is stable as ;(2)the positive equilibrium is unstable as ;(3)system (5) undergoes a Hopf bifurcation at the equilibrium as .
3. Direction and Stability of the Bifurcations
3.1. Hopf Bifurcation
In the previous section, we know that system (5) undergoes Hopf bifurcation at the tumor-free equilibrium and positive equilibrium under certain conditions. In this section, we will study the stability and direction of the Hopf bifurcated periodic solution by using the center manifold reduction and normal form theory of retarded functional differential equations due to the ideals of Faria and Magalhães [15, 16]. Throughout this section, we always assume that system (5) undergoes Hopf bifurcations at the equilibrium ( or ) as the critical parameter and the corresponding purely imaginary roots are .
Normalizing the delay in system (7) by the time-scaling , then (7) is transformed into This scaling is irrelevant for the study of the stability of the equilibrium but will be crucial for the Hopf bifurcation analysis.
Let . we transformed (29) into an FDE in : where , are given by where . Let . Setting the new parameter , then (30) can be written as where .
Assume that is the infinitesimal generator of satisfying with and has a pair of conjugate purely imaginary roots . Denote that is the invariant space of associated with ; then . We can decompose to by the formal adjoint theory for FDEs by Hale [17]. Considering complex coordinates, we still denote as . Let be the bases of , where is a vector in and .
Choose a basis for the adjoint space , such that , where is the bilinear form on associated with the adjoint equation. Thus, with such that , . Then and .
Take the enlarged phase space , defined as The projection is defined as thus we have the decomposition . Let , , we can decompose (32) to where We write the Taylor expansion as follows: where and are homogeneous polynomials in , , and of degree , , with coefficients in and , h.o.t. stands for higher order terms. The normal form method implies a normal form on the center manifold of the origin for (32) which is where and are homogeneous polynomials in and , respectively.
From (39), it follows that that is, where Thus
We will compute the cubic terms as follows.
Since are irrelevant to determine the generic Hopf bifurcation, then hence where . In order to obtain , we need to compute ; that is, , , and should be given; we will compute them as follows.
Firstly, knowing that then
Secondly, noting (46), we know that ; then .
Lastly, we will compute as follow.
Let . Noting , one has On the other hand, we know that If , then Let From we obtain where
In order to obtain , we need to compute , . From (53), it follows that where , , , and . Solving (59), we can obtain where Hence Thus, the normal form of system (42) has the form
Let , , , and . Then the normal form becomes where , .
Summarizing all above, we have the following theorem.
Theorem 5. The flow on the center manifold of the equilibrium at is given by (64). Also the following results hold:(1)the Hopf bifurcation is supercritical if and subcritical if ;(2)the bifurcated periodic solution is stable if and unstable if ;(3)the period of the bifurcated periodic solution is
In the following, we will give some simulations to illustrate the results of Theorems 4 and 5 for model (4). We cite the parameters in [11], that is, , and . Then (4) has a tumor-free equilibrium , which is unstable, and a positive equilibrium , which is locally asymptotically stable. We only simulate local properties of the stable equilibrium here in Figures 2(a) and 2(b).
(a)
(b)
(c)
(d)
Remark 6. From Figures 2(c) and 2(d), we can see that the amplitude vibration for is much bigger than that of ; also both and with respect to are not so smooth. Then the Hopf bifurcated periodic solution on plan is not given here. At the same time, we can see that the dynamical behaviors of the system have been changed although is small.
3.2. Steady-State Bifurcation
From Section 2, we know that system (5) undergoes a steady-state bifurcation at the tumor-free equilibrium as , . In this section, we will discuss the properties of the steady-state bifurcation by using the center manifold reduction and normal form theories of retarded functional differential equations.
At the tumor-free equilibrium , we write system (5) as an FDE: where Letting , then (66) can be written as where .
Assuming that is the infinitesimal generator of , then has a simple zero root. Set and we denote by the invariant space of associated with ; then . We can decompose to by the formal adjoint theory for FDEs by Hale [17]. Let be the bases for , where , which is a vector in satisfying Choose a basis for the adjoint space , where , which is a vector in satisfying . Thus we can obtain
According to the method of Faria and a similar computation in the last section, we can obtain Noting one has Thus, the normal form of system (5) is Then the following two results are obvious.
Theorem 7. If , then the tumor-free equilibrium is stable.
Theorem 8. If , and is small enough, then(1)the tumor-free equilibrium is stable as and unstable as ;(2)system (5) undergoes transcritical bifurcation at the tumor-free equilibrium .
3.3. Bogdanov-Takens Bifurcation
From Theorem 2 we know that the tumor-free equilibrium is a B-T singular equilibrium of the system (5) as , . In this section, we will discuss the bifurcations of the system (5) at .
At tumor-free equilibrium , we can write (5) as an FDE: where Let , . Then system (75) can be written as where , .
Assuming that is the infinitesimal generator of , then has double zero roots. Set and denote by the invariant space of associated with ; then . We can decompose as by the formal adjoint theory for FDEs by Hale [17]. Assume that and . On the other hand, we know that , where that is, , , and .
Let From Lemma 3.1 by Xu and Huang [18], we can get Using the method of Faria and the last section, we can obtainOn the other hand, the basis of is and then we can obtain Thus, the normal form of the system (5) is where
From above, we know that the following result can be obtained with the help of the theories of Xu and Huang [18] and Chow and Hale [19].
Theorem 9. Assume that , , , and is small enough; then system (5) undergoes Bogdanov-Takens bifurcation at the tumor-free equilibrium . Furthermore, on the -parameter plane, both and are located in the area , and is on the left of , where is Hopf bifurcation curve defined by is the homoclinic bifurcation curve defined by and is a continuously differentiable function with .
Take the same parameter in last section, that is, , and ; then the tumor-free equilibrium is . From Theorem 9, we can obtain that the Hopf bifurcation curve is and the homoclinic bifurcation is . Then on -parameter plane, the bifurcation diagram of system (5) at the equilibrium is in Figure 3.
4. Discussion
We have studied the nonlinear dynamics of Kuznetsov, Makalkin, and Taylor’s model with delay, which is a two-dimensional model of tumor cells and immune system. We first provided linear analysis of the model with delays at the possible equilibria, namely, the tumor-free and positive equilibria, and discussed the existence of Hopf bifurcation at the equilibria. We investigated the Hopf bifurcation, Bogdanov-Takens bifurcation, and steady-state bifurcation in the model. Numerical simulations were presented to illustrate the theoretical analysis and results.
Our analysis on the existence and stability of the tumor-free equilibrium corresponds to this elimination process and on the existence and stability of the positive equilibrium corresponds to coexistence of the immune system and the tumor system. Our results on the existence and stability of the Hopf bifurcated periodic solutions of describe the equilibrium process. When a stable periodic orbit exists, it can be understood that the tumor and the immune system can coexist although the cancer is not eliminated. The conditions for the parameters provide theories basis to control the development or progression of the tumors. The phenomena have been observed in some models as d’Onofrio [5], Kuznetsov et al. [11], and Bi and Xiao [14]. In particular, Bi and Ruan [7] have shown that various bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation, can occur in such models. Our results on the existence and stability of the bifurcated (Hopf, Bogdanov-Takens, and steady-state) periodic solutions describe rich dynamical behaviors of , which show that the elimination process is so complex and difficult to control.
Finally, we should point out that we have studied the local dynamical behaviors of and . As the example in our paper showed these two equilibria may coexist. Correspondingly, the system can exhibit more degenerate bifurcations including Hopf-Hopf and resonant higher codimension bifurcations. It would be interesting to consider these dynamics of the delayed model.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11171110), Shanghai Leading Academic Discipline Project (no. B407), and 211 Project of Key Academic Discipline of East China Normal University.