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

Controlling the number of tumor cells leads us to expectmore efficient strategies for treatment of tumor. Towards this goal, a tumorimmune 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.


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 tumorimmune interactions has been studied by many scholars (see [1][2][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 SIR epidemic model with impulsive effect and analyzed the dynamic behaviors of the model.For SI model, Pei et al. [7] analyzed the dynamic behaviors by an impulsive delayed SI 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 RL are studied.Nie et al. [12] proposed an SIR 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 tumorimmune 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 statedependent 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 statedependent 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 fixedtime pulse treatment with a state-dependent pulse treatment.Finally, the conclusion is given in Section 5.

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  > 0 denotes the rate of immune cell proliferation, parameter  > 0 describes the immune response to the appearance of the tumor cells, parameter  > 0 is a rate of immune cell turnover, parameter  > 0 is a growth rate of tumor cells, and parameter  > 0 denotes the inverse of tumor cell's carrying capacity; 1/ is the mean tumor cell's carrying capacity.
(ii) If  > 0 and  > , then model ( 1) has an unstable equilibrium  0 and a stable positive equilibrium  1 ( 1 ,  1 ), where ( (iii) If  < 0,  >  and ( − ) 2 + 4 > 0, then model (1) has an unstable equilibrium  0 and a stable positive equilibrium  1 ( 1 ,  1 ), 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.
Under assumption (A), we build the control process that is modeled by the following differential equation with statedependent 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.
Next, we consider the following autonomous model with impulsive effects: where (, ) and (, ) are continuous differential functions defined on R 2 and (, ) is a sufficiently smooth function with ∇(, ) ̸ = 0. Let ((), ]()) be a positive periodic solution of the model (9).The following result comes from Corollary 2 of [16].

Main Results
The first result is on the existence of a positive order-1 periodic solution for model (4).Theorem 9.For any ,  ∈ (0, 1), model ( 4) admits a positive order-1 periodic solution.
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).
On the other hand, integrating both sides of the second equation of model ( 4) along the orbit c+ , we have ln From ( 18)-( 19), we can obtain (20) 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.4) has a positive order-1 periodic solution which is orbitally asymptotical stable.
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  0 and a stable positive equilibrium  1 , 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.

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 (0.8333, 3.1667).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  = 0.1,  = 0.2,  = 0.22, and  = 0.7; 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  = 0.1,  = 0.4,  = 0.3, and  = 0.7 and other model parameters are fixed in Table 1.From Theorem 9, model (4) admits a positive order-1 periodic solution ((), ()) starting from point ( This shows that the positive order-1 periodic solution is orbitally asymptotically stable.This is shown in Figure 2  stability of an order-1 periodic solution of model (4) for case (iii) in Lemma 1.
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 0.06, 0.08, 0.1 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  = 0.06, 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 0.235 and the fifth injection that the value of  is 0.195 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)).
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  = 1.665, 2.96) for this model with fixed-time dependent impulsive control and  = 0.1,  = 0.2,  = 0.22, (0) = 1.58, and (0) = 0.56 as other parameters are fixed in Table 1.When period  = 1.665, the tumor cells are controlled in lower than the hazardous threshold value  = 0.7 which is shown in Figure 5 by blue line.However, if period  = 2.96, the tumor control fails.That is, the number of tumor cells is higher than the hazardous threshold value  = 0.7 which is shown in Figure 5 by red line.On the other hand, model (4) with statedependent impulsive control with threshold value  = 0.7 has approximately  = 3.0 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.

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.
(b).The plots in Figures3(a) and 3(b) show that the existence and