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:
(a)
(b)
Definition 7. A trajectory of model (4) is said to be order- periodic if there exists a positive integer such that is the smallest integer for .
Next, we consider the following autonomous model with impulsive effects:where and are continuous differential functions defined on and is a sufficiently smooth function with . Let be a positive -periodic solution of the model (9). The following result comes from Corollary 2 of [16].
Lemma 8 (analogue of Poincaré criterion). If the Floquet multiplier satisfies condition , where with and , , , , , , , and have been calculated at the points , , , and is the time of the th jump, then, is orbitally asymptotically stable.
3. Main Results
The first result is on the existence of a positive order-1 periodic solution for model (4).
Theorem 9. For any , model (4) admits a positive order-1 periodic solution.
Proof. Let the point for sufficiently small with . In view of the geometrical structure of the phase space of model (4), trajectory of model (4) starting from initial point intersects section at point . At point , the trajectory jumps to point on section due to impulsive effects and . Furthermore, the trajectory intersects section at point .
Since , then . It follows that is on the left of . We claim that is also on the left of . In fact, if is on the right of , then orbits and intersect at the point . This shows that there are two different solutions which start from the point . This contradicts with the uniqueness of solutions for model (4). So by (8), we have and On the other hand, suppose that the curve intersects section at point . Then trajectory starting from the initial point intersects section at point and then jumps to point on section and finally reaches a point in section again. If there is a positive constant such that , then coincides with for . That is, coincides with . Otherwise, is on the left of for and is on the right of for . However, from the geometrical structure of a phase space of model (4), is on the left of for any .
To sum up, we get, from the above discussions, that(i)when , model (4) has a positive order-1 periodic solution;(ii)when , By (12) and (13), it follows that the Poincaré map (8) has a fixed point. This means that model (4) has a positive order-1 periodic solution. The proof is complete.
The next result is on the orbital stability of a positive order-1 periodic solution for model (4).
Theorem 10. Let be a positive order-1 periodic solution of model (4) with period . Ifwhere then is locally orbitally asymptotically stable.
Proof. Suppose that intersects sections and at points and , respectively. Comparing with model (9), we have and , , , , and . ThusFurthermore, it follows from (17) thatOn the other hand, integrating both sides of the second equation of model (4) along the orbit , we haveFrom (18)-(19), we can obtain By condition (14), we see that model (4) satisfies all conditions of Lemma 8. It then follows from Lemma 8 that the order-1 periodic solution of model (4) is locally orbitally asymptotically stable and has asymptotic phase property. This completes the proof.
From the proof of Theorem 10, integrating both sides of the first equation of model (4) along orbit , it follows that
Therefore, we have the consequence of Theorem 10 as Corollary 11 below.
Corollary 11. Let be a positive order-1 periodic solution of model (4) with periodic . If then is locally orbitally asymptotically stable.
Theorem 12. If , then model (4) has a positive order-1 periodic solution which is orbitally asymptotical stable.
Proof. Suppose that the curve intersects section at point . Let . Then, since , there is a positive constant such that . For any two points and on section , and . In view of the impulsive treatment effects, point is on the left of point . Therefore, from the geometrical construction of the phase space of model (4) we have Then, for any , from the Poincaré map (8) of section we have , , and . In particular, if , then model (4) has a positive order-1 periodic solution.
Next, we discuss the general circumstance; that is, (1)If , then it follows that , . Repeating the above process, we have (2)If , then it follows that , . Repeating the above process, we have Further in Case (1) it follows , where . Similarly, in Case , where . So model (4) has orbitally asymptotically stable positive order-1 periodic solution.
Remark 13. Obviously, the treatment strength of chemotherapy is closely related to the stable region of an order-1 solution. If is close to 1, then is large, so by Theorem 12, the local asymptotic orbital stability region is large. Otherwise, if is close to 0, then the local asymptotic orbital stability region is small. That is, the local asymptotic orbital stability region depends on value of .
Remark 14. We study the existence and the stability of a positive order-1 periodic solution for the case (ii) of Lemma 1. For the case (ii) of Lemma 1, model (4) has an unstable equilibrium and a stable positive equilibrium , so that we can obtain similar results of case (ii) of Lemma 1 by using the same methods. Therefore, we only consider numerical simulations for case (iii) of Lemma 1 in next section.
4. Numerical Results
In this work, we investigate a tumor-immune model with state-dependent impulsive immunotherapy and chemotherapy. By the Poincaré map, analogue of Poincaré criterion, and qualitative analysis method, we give the criteria for the existence and stability of the positive periodic solution for this model. In order to illustrate the implications of our results and the feasibility of the state-dependent pulse feedback control strategy, we consider a tumor-immune model with state-dependent impulsive immunotherapy and chemotherapy. For simulation, we use parameter values in Table 1.
Firstly, it is easy to obtain that the model (4) without impulsive effects has a stable focus . The plots in Figure 1(a) show the tumor-immune dynamical behaviors in patients without therapies, which implies that cancer is deadly. Now, we apply chemotherapy and immunotherapy to control tumor cells. That is, we choose control parameters , , , and ; the plots in Figure 1(b) show that model (4) has a positive order-1 periodic solution. This indicates that tumor cells might be controlled below the threshold value by state-dependent impulsive control strategies.
Nextly, we choose control parameters , , , and and other model parameters are fixed in Table 1. From Theorem 9, model (4) admits a positive order-1 periodic solution starting from point and intersection section at point . This is shown in Figure 2(a). Further, this can be easily calculated by Corollary 11: This shows that the positive order-1 periodic solution is orbitally asymptotically stable. This is shown in Figure 2(b). The plots in Figures 3(a) and 3(b) show that the existence and stability of an order-1 periodic solution of model (4) for case (iii) in Lemma 1.
(a)
(b)
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Thirdly, we investigate how the strength of chemotherapy and immunotherapy affects the control of tumor cell numbers and existence and stability of the periodic solutions. The value means the side effects of chemotherapy to immune cells. Now, we change only the values of such that but the other parameter values are fixed. In Figure 3(a), our simulation results show that if the value of increases, then periodic of positive order-1 periodic solution for model (4) decreases. In particular, if , then we can see that the tumor cell numbers fall off more than the other cases. Secondly, let denote amounts of injected immune cells for immunotherapy. In Figure 3(b), we can see that the periodic of an order-1 periodic solution decreases with decreasing value of . The fourth injection that the value of is and the fifth injection that the value of is happen simultaneously. Thirdly, the value implies the strength of chemotherapy to tumor cells. Then there are some differences of simulation results between the value and the values and . Numerical simulation shows that the period of positive order-1 periodic solution for model (4) increases with the increase of the strength of chemotherapy to tumor cell (see Figure 3(c)). And if the value of increases, tumor cell’s lower bounded line is lowered in Figure 4(a). These show that the recurrence time for tumor cells can be prolonged by increasing strength of chemotherapy to tumor cells , amounts of injected immune cells , and decreasing side effects of chemotherapy to immune cells by through simulation results (see Figures 4(b) and 4(c)).
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(b)
(c)
Finally, we investigate the effect of the model with two types of controls: fixed-time dependent impulsive control and state-dependent impulsive control. We first consider two treatment (or injection) period scenarios (the periods ) for this model with fixed-time dependent impulsive control and , , , , and as other parameters are fixed in Table 1. When period , the tumor cells are controlled in lower than the hazardous threshold value which is shown in Figure 5 by blue line. However, if period , the tumor control fails. That is, the number of tumor cells is higher than the hazardous threshold value which is shown in Figure 5 by red line. On the other hand, model (4) with state-dependent impulsive control with threshold value has approximately as treatment period and we can control the number of tumor cells in the prescribed fixed boundary which is shown in Figure 5 by black line. As we know, the proper treatment schedule is very important to control tumor cells because of the costs for treatment.
5. Conclusion
In this paper, a mathematical model of tumor and immune cells with state-dependent impulsive chemotherapy and immunotherapy was considered. This is the first attempt to investigate a tumor-immune model with state-dependent impulsive treatments. The criterion for the existence and stability of the positive periodic solution of tumor-immune model, based on Poincaré map and analogue of Poincaré criterion, was presented. We proved existence of positive order-1 periodic solutions and obtained condition of orbitally asymptotical stability of positive order-1 periodic solution for our model. From the results, we were able to see that local asymptotic orbital stability region is large as treatment strength of chemotherapy is increasing. Also, increasing strength of chemotherapy and immunotherapy and decreasing side effects of chemotherapy lengthened tumor recurrence time. The simulation results indicate that in determining optimal treatment timing the model with state-dependent impulsive control is more efficient than fixed-time impulsive control.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank antonymous referees for their constructive suggestions and comments that improve substantially the original manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2015R1D1A1A02062131). This work was supported in part by the National Natural Science Foundation of China (Grant no. 11461067) and the Natural Science Foundation of Xinjiang (Grant no. 2016D01C046).