scholarly journals Modeling and Analysis of Bifurcation in a Delayed Worm Propagation Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Yao ◽  
Nan Zhang ◽  
Wenlong Xiang ◽  
Ge Yu ◽  
Fuxiang Gao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Simulation Experiments ◽  
Death Rates ◽  
Worm Propagation ◽  
Modeling And Analysis ◽  
Internet Worm ◽  
The Stability

A delayed worm propagation model with birth and death rates is formulated. The stability of the positive equilibrium is studied. Through theoretical analysis, a critical valueτ0of Hopf bifurcation is derived. The worm propagation system is locally asymptotically stable when time delay is less thanτ0. However, Hopf bifurcation appears when time delayτpasses the thresholdτ0, which means that the worm propagation system is unstable and out of control. Consequently, time delay should be adjusted to be less thanτ0to ensure the stability of the system stable and better prediction of the scale and speed of Internet worm spreading. Finally, numerical and simulation experiments are presented to simulate the system, which fully support our analysis.

scholarly journals Analysis of a Delayed Internet Worm Propagation Model with Impulsive Quarantine Strategy

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Yu Yao ◽  
Xiaodong Feng ◽  
Wei Yang ◽  
Wenlong Xiang ◽  
Fuxiang Gao
Keyword(s):  
Time Delay ◽  
Propagation Model ◽  
The Novel ◽  
Worm Propagation ◽  
Internet Worm ◽  
Internet Worms ◽  
The Stability ◽  
And Control ◽  
Theoretical Results ◽  
Significant Attention

Internet worms exploiting zero-day vulnerabilities have drawn significant attention owing to their enormous threats to Internet in the real world. To begin with, a worm propagation model with time delay in vaccination is formulated. Through theoretical analysis, it is proved that the worm propagation system is stable when the time delay is less than the thresholdτ0and Hopf bifurcation appears when time delay is equal to or greater thanτ0. Then, a worm propagation model with constant quarantine strategy is proposed. Through quantitative analysis, it is found that constant quarantine strategy has some inhibition effect but does not eliminate bifurcation. Considering all the above, we put forward impulsive quarantine strategy to eliminate worms. Theoretical results imply that the novel proposed strategy can eliminate bifurcation and control the stability of worm propagation. Finally, simulation results match numerical experiments well, which fully supports our analysis.

scholarly journals Bifurcation Analysis of a Delayed Worm Propagation Model with Saturated Incidence

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Worm Propagation ◽  
Center Manifold Theorem ◽  
Saturated Incidence ◽  
Parameter Values

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical results.

scholarly journals Hopf bifurcation in an Internet worm propagation model with time delay in quarantine

2013 ◽  
Vol 57 (11-12) ◽  
pp. 2635-2646 ◽  
Author(s):  
Yu Yao ◽  
Xiao-wu Xie ◽  
Hao Guo ◽  
Ge Yu ◽  
Fu-Xiang Gao ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Propagation Model ◽  
Worm Propagation ◽  
Internet Worm

scholarly journals An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Yao ◽  
Qiang Fu ◽  
Wei Yang ◽  
Ying Wang ◽  
Chuan Sheng
Keyword(s):  
Time Delay ◽  
Infection Rate ◽  
Numerical Experiments ◽  
Rapid Development ◽  
Nonlinear Process ◽  
Existence Condition ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Worm Propagation ◽  
Security Issues

With rapid development of Internet, network security issues become increasingly serious. Temporary patches have been put on the infectious hosts, which may lose efficacy on occasions. This leads to a time delay when vaccinated hosts change to susceptible hosts. On the other hand, the worm infection is usually a nonlinear process. Considering the actual situation, a variable infection rate is introduced to describe the spread process of worms. According to above aspects, we propose a time-delayed worm propagation model with variable infection rate. Then the existence condition and the stability of the positive equilibrium are derived. Due to the existence of time delay, the worm propagation system may be unstable and out of control. Moreover, the threshold τ0 of Hopf bifurcation is obtained. The worm propagation system is stable if time delay is less than τ0. When time delay is over τ0, the system will be unstable. In addition, numerical experiments have been performed, which can match the conclusions we deduce. The numerical experiments also show that there exists a threshold in the parameter a, which implies that we should choose appropriate infection rate β(t) to constrain worm prevalence. Finally, simulation experiments are carried out to prove the validity of our conclusions.

scholarly journals Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

Periodic Oscillations in the Quorum-Sensing System with Time Delay

2020 ◽  
Vol 30 (09) ◽  
pp. 2050127
Author(s):  
Menghan Chen ◽  
Jinchen Ji ◽  
Haihong Liu ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Quorum Sensing ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Sensing System ◽  
Quorum Sensing System ◽  
The Stability ◽  
System With Time Delay

The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system.

scholarly journals Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunru Li ◽  
Zujun Ma
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Rumor Propagation ◽  
Boundary Equilibrium ◽  
Normal Form Method ◽  
Positive Equilibrium Point ◽  
The Stability

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.

Oscillatory behavior of p53-Mdm2 system driven by transcriptional and translational time delays

2020 ◽  
Vol 13 (05) ◽  
pp. 2050034
Author(s):  
Chunyan Gao ◽  
Haihong Liu ◽  
Zengrong Liu ◽  
Yuan Zhang ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Cell Fate ◽  
Time Delays ◽  
Oscillatory Behavior ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Positive Equilibrium Point ◽  
The Stability

Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.

Modeling and analysis of a predator–prey system with time delay

2017 ◽  
Vol 10 (03) ◽  
pp. 1750032 ◽  
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
Time Delay ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Modeling And Analysis ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Explicit Formulae ◽  
The Stability ◽  
System With Time Delay

In this paper, a differential-algebraic predator–prey system with time delay is investigated, where the time delay is regarded as a parameter. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium and the existence of Hopf bifurcation are demonstrated. Furthermore, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are obtained by applying the normal form theory and the center manifold argument. At last, some numerical simulations are carried out to illustrate the feasibility of our main results.

scholarly journals Modeling and Bifurcation Research of a Worm Propagation Dynamical System with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Yu Yao ◽  
Zhao Zhang ◽  
Wenlong Xiang ◽  
Wei Yang ◽  
Fuxiang Gao
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Theoretical Analysis ◽  
Time Window ◽  
Detection System ◽  
Positive Equilibrium ◽  
Worm Propagation ◽  
Local Hopf Bifurcation ◽  
System With Time Delay

Both vaccination and quarantine strategy are adopted to control the Internet worm propagation. By considering the interaction infection between computers and external removable devices, a worm propagation dynamical system with time delay under quarantine strategy is constructed based on anomaly intrusion detection system (IDS). By regarding the time delay caused by time window of anomaly IDS as the bifurcation parameter, local asymptotic stability at the positive equilibrium and local Hopf bifurcation are discussed. Through theoretical analysis, a thresholdτ0is derived. When time delay is less thanτ0, the worm propagation is stable and easy to predict; otherwise, Hopf bifurcation occurs so that the system is out of control and the containment strategy does not work effectively. Numerical analysis and discrete-time simulation experiments are given to illustrate the correctness of theoretical analysis.

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