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.

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 Nonlinear state-dependent feedback control of a pest-natural enemy system

2018 ◽  
Vol 94 (3) ◽  
pp. 2243-2263 ◽  
Author(s):  
Yuan Tian ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Feedback Control ◽  
Natural Enemy ◽  
State Dependent ◽  
Nonlinear State

An early estimation of the number of affected people in South Asia due to Covid-19 pandemic using susceptible, infected and recover model

2020 ◽  
Vol 31 (10) ◽  
pp. 2050140
Author(s):  
Md. Enamul Hoque
Keyword(s):  
Simple Model ◽  
South Asia ◽  
South Asian ◽  
Basic Reproduction Number ◽  
Numerical Solutions ◽  
Reproduction Number ◽  
Sir Model ◽  
Asian Countries ◽  
Susceptible Population ◽  
The Basic Reproduction Number

The Susceptible, Infected and Recover (SIR) model is a very simple model to estimate the dynamics of an epidemic. In the current pandemic due to Covid-19, the SIR model has been used to estimate the dynamics of infection for Bangladesh, India, Pakistan and compared with that of China. Numerical solutions are used to obtain the value of parameters for the SIR model. It is predicted that the active case in Pakistan due to the SARS-CoV-2 will be comparable with that in China whereas it will be low for Bangladesh and India. The basic reproduction number, with fluctuations, for South Asian countries are predicted to be less than that of China. The susceptible population is also estimated to be under a million for Bangladesh and India but it becomes very large for Pakistan.

scholarly journals Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

2021 ◽  
Author(s):  
Jayanta Kumar Ghosh ◽  
Prahlad Majumdar ◽  
Uttam Ghosh
Keyword(s):  
Optimal Control ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Sir Model ◽  
Logistic Growth ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Susceptible Population ◽  
Saturated Treatment

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point ( DFE ) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov - Takens bifurcation ( BT ) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.

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.

Global threshold dynamics of SIQS epidemic model in time fluctuating environment

2017 ◽  
Vol 10 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Shang Li ◽  
Meng Fan ◽  
Xinmiao Rong
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Integral Operator ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Global Threshold ◽  
Linear Integral ◽  
Disease Free ◽  
The Basic Reproduction Number

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.

Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control

2017 ◽  
Vol 10 (07) ◽  
pp. 1750093 ◽  
Author(s):  
Hongjian Guo ◽  
Lansun Chen ◽  
Xinyu Song
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
State Feedback ◽  
Sir Model ◽  
Vaccination Rate ◽  
State Feedback Control ◽  
Dimensional System ◽  
Impulsive State Feedback Control ◽  
Dynamical Properties ◽  
The Government

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated.

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.

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