scholarly journals A Feedback Control Model of Comprehensive Therapy for Treating Immunogenic Tumours

2016 ◽  
Vol 26 (03) ◽  
pp. 1650039 ◽  
Author(s):  
Biao Tang ◽  
Yanni Xiao ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Tumour Size ◽  
Global Dynamics ◽  
Immunologic Memory ◽  
Treatment Frequency ◽  
Positive Order ◽  
State Dependent ◽  
Comprehensive Therapy ◽  
Tumour System

Surgery is the traditional method for treating cancers, but it often fails to cure patients for complex reasons so new therapeutic approaches that include both surgery and immunotherapy have recently been proposed. These have been shown to be effective, clinically, in inhibiting cancer cells while allowing retention of immunologic memory. This comprehensive strategy is guided by whether a population of tumour cells has or has not exceeded a threshold density. Conditions for successful control of tumours in an immune tumour system were modeled and the related dynamics were addressed. A mathematical model with state-dependent impulsive interventions is formulated to describe combinations of surgery with immunotherapy. By analyzing the properties of the Poincaré map, we examine the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semi-trivial order-1 periodic solution and the positive order-[Formula: see text] periodic solution. The main results showed that surgery alone can only control the tumour size below a certain level while there is no immunologic memory. If comprehensive therapy involving combining surgery with immunotherapy is considered, then not only can the cancers be controlled below a certain level, but the immune system can also retain its activity. The existence of positive order-[Formula: see text] periodic solutions implies that periodical therapy is needed to control the cancers. However, choosing the treatment frequency and the strength of the therapy remains challenging, and hence a strategy of individual-based therapy is suggested.

scholarly journals The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-19
Author(s):  
Y. Tian ◽  
H. M. Li
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Prey Population ◽  
Global Dynamics ◽  
Predator Population ◽  
Positive Order ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
The Stability

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q ( q ≥ 1 ) periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.

scholarly journals The Effect of Pulse Vaccination and Treatment on SIR Epidemic Model with Media Impact

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Zhi-Long He ◽  
Lin-Fei Nie
Keyword(s):  
Periodic Solution ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Positive Order ◽  
Media Impact ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
State Dependent ◽  
Control Threshold

We propose a novel SIR epidemic dynamical control model with media impact, where the state dependent pulse vaccination and medication treatment control strategies are being introduced to prevent the spread of disease at different control threshold values. By using the geometry theory of differential equation and method of successor function, the existence of positive order-1 periodic solution is studied. Further, some sufficient conditions of the orbitally asymptotical stability for positive order-1 periodic solution are given by the analog Poincaré criterion. Furthermore, numerical simulations are carried to illustrate the feasibility of our main results presented here.

Microbial insecticide model and homoclinic bifurcation of impulsive control system

2021 ◽  
pp. 2150043
Author(s):  
Tieying Wang
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Homoclinic Orbit ◽  
State Feedback ◽  
Homoclinic Bifurcation ◽  
State Feedback Control ◽  
Global Dynamics ◽  
Impulsive State Feedback Control ◽  
Microbial Insecticide ◽  
Existence And Stability

A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor’s theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system.

scholarly journals Global dynamics of a state-dependent feedback control system

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Sanyi Tang ◽  
Wenhong Pang ◽  
Robert A Cheke ◽  
Jianhong Wu
Keyword(s):  
Control System ◽  
Feedback Control ◽  
Global Dynamics ◽  
Feedback Control System ◽  
State Dependent

Dynamical Behavior and Bifurcation Analysis of the SIR Model with Continuous Treatment and State-Dependent Impulsive Control

2019 ◽  
Vol 29 (10) ◽  
pp. 1950131 ◽  
Author(s):  
Qian Li ◽  
Yanni Xiao
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Reproduction Number ◽  
Continuous Treatment ◽  
Positive Order ◽  
Susceptible Population ◽  
State Dependent ◽  
Existence And Stability ◽  
Impulsive Model ◽  
Disease Free

In this study, we propose a state-dependent impulsive model describing the susceptible individuals-triggered interventions. We find that the model with susceptible individuals-guided impulsive interventions can exhibit very complex dynamical behaviors with rich biological meanings. We note that this formulated impulsive model has disease-free periodic solution, and we can investigate the threshold dynamics by defining the control reproduction number. We study the existence and stability of the disease-free periodic solution (DFPS) for [Formula: see text]. Our results show that, even if the basic reproduction number [Formula: see text], the DFPS can still be stable when the threshold level of susceptible population [Formula: see text], indicating that with a proper chosen [Formula: see text], the state-dependent impulsive strategy can effectively control the development of the infectious disease and eradicate the disease eventually. By employing the bifurcation theory, we investigate the bifurcation phenomenon near the DFPS with respect to some key parameters, and observe that a positive order-1 periodic solution can bifurcate from the DFPS via a transcritical bifurcation. By utilizing numerical simulation, we further explore the existence and stability of the positive order-[Formula: see text] periodic solutions, and found the feasibility of stable positive order-1, order-2 and order-3 periodic solutions, that imply the existence of chaos. In particular, we find that there can be three positive order-1 periodic solutions simultaneously, in which one is stable and the other two are unstable. Our finding indicates that the comprehensive strategy combining continuous treatment with state-dependent impulsive vaccination and isolation plays a crucial role in controlling the prevalence and further spread of the infectious diseases.

scholarly journals Dynamics and bifurcation analysis of a state-dependent impulsive SIS model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jinyan Wang
Keyword(s):  
Periodic Solution ◽  
Control Measures ◽  
Global Dynamics ◽  
Sis Model ◽  
Susceptible Population ◽  
Ode System ◽  
State Dependent ◽  
Existence And Stability ◽  
Modelling Studies ◽  
The Stability

AbstractRecently, considering the susceptible population size-guided implementations of control measures, several modelling studies investigated the global dynamics and bifurcation phenomena of the state-dependent impulsive SIR models. In this study, we propose a state-dependent impulsive model based on the SIS model. We firstly recall the complicated dynamics of the ODE system with saturated treatment. Based on the dynamics of the ODE system, we firstly discuss the existence and the stability of the semi-trivial periodic solution. Then, based on the definition of the Poincaré map and its properties, we systematically investigate the bifurcations near the semi-trivial periodic solution with all the key control parameters; consequently, we prove the existence and stability of the positive periodic solutions.

scholarly journals Impulsive State Feedback Control of the Rhizosphere Microbial Degradation in the Wetland Plant

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Zhong Zhao ◽  
Liuyong Pang ◽  
Zhanping Zhao ◽  
Chengguang Luo
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Microbial Degradation ◽  
State Feedback ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
State Feedback Control ◽  
Control Measures ◽  
Positive Order ◽  
Impulsive State Feedback Control

The rhizosphere microbe plays an important role in removing the pollutant generated from industrial and agricultural production. To investigate the dynamics of the microbial degradation, a nonlinear mathematical model of the rhizosphere microbial degradation is proposed based on impulsive state feedback control. The sufficient conditions for existence of the positive order-1 or order-2 periodic solution are obtained by using the geometrical theory of the semicontinuous dynamical system. We show the impulsive control system tends to an order-1 periodic solution or order-2 periodic solution if the control measures are achieved during the process of the microbial degradation. Furthermore, mathematical results are justified by some numerical simulations.

scholarly journals Threshold Dynamics and Bifurcation of a State-Dependent Feedback Nonlinear Control Susceptible–Infected–Recovered Model1

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Tianyu Cheng ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Reproduction Number ◽  
Threshold Level ◽  
Sir Model ◽  
Threshold Dynamics ◽  
Threshold Condition ◽  
Susceptible Population ◽  
State Dependent ◽  
Nonlinear State

A classic susceptible–infected–recovered (SIR) model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease-free periodic solution (DFPS) are addressed, and the threshold condition is provided, which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincaré map of the proposed model, and the main results indicate that under certain conditions, a stable or unstable interior periodic solution could be generated through transcritical, pitchfork, and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve, which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency, and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.

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