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.

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 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.

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 Bifurcation of a Computer Virus Propagation Model with Delay and Incomplete Antivirus Ability

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jianguo Ren ◽  
Yonghong Xu
Keyword(s):  
Hopf Bifurcation ◽  
Threshold Value ◽  
Computer Virus ◽  
Propagation Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Virus Propagation ◽  
Globally Asymptotically Stable ◽  
Existence And Stability ◽  
Computer Virus Propagation Model

A new computer virus propagation model with delay and incomplete antivirus ability is formulated and its global dynamics is analyzed. The existence and stability of the equilibria are investigated by resorting to the threshold valueR0. By analysis, it is found that the model may undergo a Hopf bifurcation induced by the delay. Correspondingly, the critical value of the Hopf bifurcation is obtained. Using Lyapunov functional approach, it is proved that, under suitable conditions, the unique virus-free equilibrium is globally asymptotically stable ifR0<1, whereas the virus equilibrium is globally asymptotically stable ifR0>1. Numerical examples are presented to illustrate possible behavioral scenarios of the mode.

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 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 giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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.

Sign in / Sign up

Export Citation Format

Share Document