scholarly journals Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination

2017 ◽  
Vol 2017 ◽  
pp. 1-17
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Dianjie Bi ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Two Delays ◽  
The Stability ◽  
Computer Virus Propagation Model

A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical results.

Global stability and Hopf bifurcation of a delayed computer virus propagation model with saturation incidence rate and temporary immunity

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640009 ◽  
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Huitao Zhao ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Incidence Rate ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Computer Virus Propagation Model

In this paper, a delayed computer virus propagation model with a saturation incidence rate and a time delay describing temporary immune period is proposed and its dynamical behaviors are studied. The threshold value [Formula: see text] is given to determine whether the virus dies out completely. By comparison arguments and iteration technique, sufficient conditions are obtained for the global asymptotic stabilities of the virus-free equilibrium and the virus equilibrium. Taking the delay as a parameter, local Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stabilities of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, numerical simulations are carried out to illustrate the main theoretical results.

scholarly journals Dynamics of a Computer Virus Propagation Model with Delays and Graded Infection Rate

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Zizhen Zhang ◽  
Limin Song
Keyword(s):  
Hopf Bifurcation ◽  
Infection Rate ◽  
Computer Virus ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Computer Virus Propagation Model

A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results.

scholarly journals Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Propagation Model ◽  
Dynamical Analysis ◽  
Infection Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
The Stability

This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Junli Liu ◽  
Tailei Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Globally Asymptotically Stable ◽  
Center Manifold Theorem ◽  
The Stability ◽  
Delay Differential

To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.

scholarly journals Dynamical Analysis of a Computer Virus Propagation Model with Delay and Infectivity in Latent Period

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Dianjie Bi
Keyword(s):  
Hopf Bifurcation ◽  
Latent Period ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Propagation Model ◽  
Dynamical Analysis ◽  
Positive Equilibrium ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Computer Virus Propagation Model

A delayed SLB computer virus propagation model with infectivity in latent period is proposed in this paper. We establish sufficient conditions for local stability of the positive equilibrium and existence of Hopf bifurcation by analyzing distribution of the roots of the associated characteristic equation and applying the Hopf bifurcation theorem. Furthermore, properties of the Hopf bifurcation are determined by using the normal form theory and the center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are carried out.

scholarly journals Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zizhen Zhang ◽  
Fangfang Yang ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Transmission Model ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Synthetic Drug ◽  
The Stability

This paper is concerned with the Hopf bifurcation of a synthetic drug transmission model with two delays. Firstly, some sufficient conditions of delay-induced bifurcation for such a model are captured by using different combinations of the two delays as the bifurcation parameter. Secondly, based on the center manifold theorem and normal form theory, some sufficient conditions determining properties of the Hopf bifurcation such as the direction and the stability are established. Finally, to underline the effectiveness of the obtained results, some numerical simulations are ultimately addressed.

scholarly journals Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Tao Zhao ◽  
Dianjie Bi
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Infection Rates ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Internet Worms ◽  
Form Theory ◽  
The Stability

A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.

scholarly journals Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Long Li ◽  
Yanxia Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Stability Method ◽  
The Stability ◽  
Delay Differential

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

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