Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 2979414, 10 pages

http://dx.doi.org/10.1155/2016/2979414

## State-Dependent Impulsive Control Strategies for a Tumor-Immune Model

^{1}Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea^{2}College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Received 23 May 2016; Accepted 1 September 2016

Academic Editor: Ryusuke Kon

Copyright © 2016 Kwang Su Kim 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

Controlling the number of tumor cells leads us to expect more efficient strategies for treatment of tumor. Towards this goal, a tumor-immune model with state-dependent impulsive treatments is established. This model may give an efficient treatment schedule to control tumor’s abnormal growth. By using the Poincaré map and analogue of Poincaré criterion, some conditions for the existence and stability of a positive order-1 periodic solution of this model are obtained. Moreover, we carry out numerical simulations to illustrate the feasibility of our main results and compare fixed-time impulsive treatment effects with state-dependent impulsive treatment effects. The results of our simulations say that, in determining optimal treatment timing, the model with state-dependent impulsive control is more efficient than that with fixed-time impulsive control.

#### 1. Introduction

Cancer is a class of diseases characterized by out of control cell growth. Abnormal cells divide without control and are able to invade other tissues. It can also spread to other parts of the body through the blood and lymph systems. Cancer is a leading cause of death worldwide. In the past decades, doctors are trying to cure cancer but it has been difficult. Among many treatments for cancer, immunotherapy is a treatment method by using immune system in a body to fight cancer cells. Immune system recognizes cancer cells and leads to destruction of cancer cells before cancer cells are big enough to see. However, since the mechanism of immune system in the body is very complex, it is difficult to find an efficient treatment schedule to eradicate some cancers. So, in order to find such schedule, a mathematical model describing dynamics of tumor-immune interactions would provide a new strategy for treatment of cancer.

In the past, mathematical modeling about tumor-immune interactions has been studied by many scholars (see [1–3]). Particularly, Gałach [4] provided three kinds of different models which describe a competition between the tumor and immune cells by using a mathematical model with delay. Actually they investigated the stability of the steady state and observed a state of the “returning" tumor in the model with time delay. In several papers, there are several mathematical models that describe treatment effects: immunotherapy, chemotherapy, radiation therapy, and so on [1, 5].

In real world, such treatments need not once but periodic schedules. In particular, the impulsive differential equations can describe well such periodic treatments and pulse vaccination. Disease mathematical models using time impulsive differential equations have been studied by many researchers. For example, Meng and Chen [6] suggested a delayed epidemic model with impulsive effect and analyzed the dynamic behaviors of the model. For model, Pei et al. [7] analyzed the dynamic behaviors by an impulsive delayed model. In Gao et al. [8], they proposed the periodic therapy from antiretroviral drugs for HIV by using impulsive differential equations. Qiao et al. [9] suggested a Hepatitis B virus infection model with impulsive vaccination and time delay. Also they showed the existence of a periodic solution of the model and derived sufficient conditions that Hepatitis B virus will be eliminated or persistent. Huang et al. [10] suggested two novel mathematical models by using an impulsive control term for injection of insulin and showed the existence and stability of the periodic solution of the model.

A noticed fact is that the treatment of different cancers has different ways. For some cancers that cannot be cured completely, a natural idea is therefore to keep the density of tumor cells at a low level to slow the progression of cancers. Based on this idea, Tang et al. [11] proposed an SIR epidemic model with state-dependent pulse control strategies and demonstrated that the combination of pulse vaccination and treatment is optimal in terms of cost under certain conditions, and the existence and stability of periodic solution with the maximum value of the infective no greater than threshold value are studied. Nie et al. [12] proposed an epidemic model with state-dependent pulse vaccination and proved the existence and the stability of a positive periodic solution for the model. Additionally, in tumor treating, people have high death rate if they have tumor cells too much. So it is important that we control to reduce the number of tumor cells below the value of hazardous. Therefore, to describe tumor-immune systems with treatments, state-dependent impulsive differential equation is more appropriate than fixed-time dependent impulsive differential equation. However, the mathematical models for tumor-immune interactions with state-dependent impulsive treatments are rare. Additionally, state-dependent impulsive control strategies are also used in the treatment and control of viral diseases. For example, Nie et al. [13] proposed two novel virus dynamics models with state-dependent control strategies and analytically proved the existence and orbit stability of semitrivial periodic solution and positive periodic solution. Theoretical results showed that the density of infected cells or free virus can be kept within a low level over a long period of time by adjusting control strength. Tang et al. [14] constructed a mathematical model with state-dependent impulsive interventions where the comprehensive therapy involving combining surgery with immunotherapy is considered. In it, authors examined the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semitrivial order-1 periodic solution and the positive order- periodic solution.

Inspired by the above discussion, the objective of this study is to develop a tumor-immune model with state-dependent impulsive treatment: immunotherapy and chemotherapy by using state-dependent impulsive differential equations, find some conditions for the existence and stability of periodic solution, and compare between state-dependent impulsive control and fixed-time impulsive control for control of tumor.

The rest of this paper is organized as follows: In Section 2, we give a mathematical model for the tumor-immune system with state-dependent impulsive control strategies, which is based on the Gałach’s model [4], and describe the parameters used in this model. In Section 3, for the main results, we show the existence and stability of a positive order-1 periodic solution. In Section 4, we carry out some numerical simulations to illustrate the main results and compare a fixed-time pulse treatment with a state-dependent pulse treatment. Finally, the conclusion is given in Section 5.

#### 2. Mathematical Model of Tumor-Immune Dynamics

Gałach’s autonomous tumor-immune dynamics model is given by the following differential equations [4]: Gałach changed the model proposed in [3] by replacing the Michaelis-Menten form of function with a Lotka-Volterra form. Here , represent the numbers of immune cells and tumor cells, respectively, and parameter denotes the rate of immune cell proliferation, parameter describes the immune response to the appearance of the tumor cells, parameter is a rate of immune cell turnover, parameter is a growth rate of tumor cells, and parameter denotes the inverse of tumor cell’s carrying capacity; is the mean tumor cell’s carrying capacity.

On the stability of an equilibrium, Gałach [4] has the following results.

Lemma 1. *(i) If and , then model (1) has a trivial equilibrium which is stable.**(ii) If and , then model (1) has an unstable equilibrium and a stable positive equilibrium , where **(iii) If , and , then model (1) has an unstable equilibrium and a stable positive equilibrium , where *

It is well known that immunotherapy and chemotherapy are effective to control the tumor cells. Chemotherapy attacks not only tumor cells but also normal cells and immunotherapy is applied by injection of immune cells. Therefore, cancer patients need periodic treatments for a long time. For the reality, a state-dependent impulsive control strategy for tumor abnormal growth is proposed here, rather than the usual continuous control or pulse fixed-time control strategy. The treatment strategy is based on the following assumption.When the amount of tumor cell reaches the hazardous threshold value at time at the th time, immunotherapy and chemotherapy are taken and the amounts of immune cells and tumor cells abruptly turn to and , respectively, where are the chemotherapy intensity and is the amounts of immunotherapy, respectively.

Under assumption , we build the control process that is modeled by the following differential equation with state-dependent impulsive effects:

*Remark 2. *We indicate that we do not know a priori time of pulse treatments of chemotherapy and immunotherapy since it is taken at the time when the number of tumor cells reaches the threshold value . So the pulse treatments of chemotherapy and immunotherapy time obviously depend on the solution and our model adapts state-dependent pulse treatments of chemotherapy and immunotherapy.

*Remark 3. *The control parameters , , and hazardous threshold rely heavily on the characteristics of various cancer types. The choices of these values are very important, which are closely geared to the growth or elimination of cancer cells.

Lemma 4. *Each component of solution of model (4) with nonnegative initial condition is positive for every .*

*Proof. *For any nonnegative initial condition , suppose that is a solution of model (4) with the positive initial conditions ; then relation of solution with line is one of the following cases.*(i) The Solution ** Intersects Line ** Finite Times*. Since intersects line finite times and the positive equilibrium is stable, , for every .*(ii) The Solution ** Intersects Line ** Infinite Times*. Suppose that solution intersects with line at times , so that . If the conclusion of Lemma 4 is not true, then there is a such that , and , for every . For this , there is a positive integer such that . (i)If , , then from the first and third equations of model (4), we obtain which leads to a contradiction to .(ii)If , , then from the second and fourth equations of model (4), we have which leads to a contradiction to .So we know that does not exist such that . Therefore, we can prove that for every .

Next, for the convenience of statement in the remainder of this paper, we introduce the following definitions. Let be an arbitrary set in and be an arbitrary point in . The distance between and is defined by

*Definition 5 (orbital stability [15]). *Trajectory is said to be orbitally stable if, given , there exists a constant such that, for any solution of model (4), when , one has for all .

*Definition 6 (orbitally asymptotical stability [15]). *Trajectory is said to be orbitally asymptotically stable if it is orbitally stable; there exists a constant such that, for any other solution of model (4), when , then .

To analyze the dynamic behaviors of model (4), we define two sections to the phase space of model (4) by For any point on section , suppose that trajectory starting from point intersects section infinitely many times. That is, trajectory jumps to point on section due to impulsive effects and . Furthermore, trajectory intersects section at point and then jumps to point on section again (see Figure 1(b)). Repeating this process, we have two impulsive point sequences and , where is determined by , , , , and . We then define the Poincaré map of section as follows: