scholarly journals Hopf Bifurcation Analysis of a Two-Delay HIV-1 Virus Model with Delay-Dependent Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Yu Xiao ◽  
Yunxian Dai ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Reproduction Rate ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Delay Dependent ◽  
Theoretical Results ◽  
Hiv 1

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

HOPF BIFURCATION AND STABILITY ANALYSIS FOR A STAGE-STRUCTURED SYSTEM

2010 ◽  
Vol 03 (01) ◽  
pp. 21-41 ◽  
Author(s):  
JUNLI LIU ◽  
TAILEI ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Structured System ◽  
Explicit Formulae ◽  
Bifurcation And Stability ◽  
Theoretical Results

In this paper, we considered a time-delay predator–prey system, in which the prey has two life stages, juvenile and mature. Delay was regarded as the bifurcation parameter, we analyzed the characteristic equation of the system at the positive equilibrium, stability of the positive equilibrium and existence of Hopf bifurcation with delay τ in the term of degree are investigated. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. To verify our theoretical results, a numerical example is also included.

scholarly journals Hopf bifurcation of an HIV-1 virus model with two delays and logistic growth

2020 ◽  
Vol 15 ◽  
pp. 16
Author(s):  
Caixia Sun ◽  
Lele Li ◽  
Jianwen Jia
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Humoral Immune ◽  
Virus Model ◽  
Pass Through ◽  
The Stability ◽  
Hiv 1

The paper establish and investigate an HIV-1 virus model with logistic growth, which also has intracellular delay and humoral immunity delay. The local stability of feasible equilibria are established by analyzing the characteristic equations. The globally stability of infection-free equilibrium and immunity-inactivated equilibrium are studied using the Lyapunov functional and LaSalles invariance principle. Besides, we prove that Hopf bifurcation will occur when the humoral immune delay pass through the critical value. And the stability of the positive equilibrium and Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem. Finally, we confirm the theoretical results by numerical simulations.

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.

scholarly journals Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Wanjun Xia ◽  
Soumen Kundu ◽  
Sarit Maitra
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Transmission Rate ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Ratio Dependent ◽  
Theoretical Results

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

Stability and Hopf Bifurcation in a Delayed SIS Epidemic Model with Double Epidemic Hypothesis

2018 ◽  
Vol 19 (6) ◽  
pp. 561-571
Author(s):  
Jiangang Zhang ◽  
Yandong Chu ◽  
Wenju Du ◽  
Yingxiang Chang ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Characteristic Roots ◽  
Sis Epidemic Model ◽  
Form Theory ◽  
The Stability ◽  
Sis Epidemic

AbstractThe stability and Hopf bifurcation of a delayed SIS epidemic model with double epidemic hypothesis are investigated in this paper. We first study the stability of the unique positive equilibrium of the model in four cases, and we obtain the stability conditions through analyzing the distribution of characteristic roots of the corresponding linearized system. Moreover, we choosing the delay as bifurcation parameter and the existence of Hopf bifurcation is investigated in detail. We can derive explicit formulas for determining the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by center manifold theorem and normal form theory. Finally, we perform the numerical simulations for justifying the theoretical results.

scholarly journals Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Zizhen Zhang ◽  
Yougang Wang
Keyword(s):  
Wireless Sensor Network ◽  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Wireless Sensor ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results.

scholarly journals Dynamical Analysis of a Computer Virus Model with Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Juan Liu ◽  
Carlo Bianca ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Characteristic Root ◽  
Computer Virus ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Virus Model ◽  
Bifurcation And Stability

An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of the theoretical results.

Hopf Bifurcation Analysis of KdV–Burgers–Kuramoto Chaotic System with Distributed Delay Feedback

2019 ◽  
Vol 29 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Jie Liu ◽  
Junbiao Guan ◽  
Zhaosheng Feng
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Equilibrium Points ◽  
Distributed Delay ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Mean ◽  
Mean Time ◽  
Theoretical Results

In this paper, the KdV–Burgers–Kuramoto chaotic system with distributed delay feedback is studied. The local stability of equilibrium points of this system is analyzed and the conditions under which Hopf bifurcation occurs are obtained by choosing the mean time delay as a bifurcation parameter. The direction and stability of bifurcating periodic solutions are derived by means of the normal form theory and the center manifold theorem. Numerical simulations are also illustrated which are in agreement with our theoretical results.

scholarly journals Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Center Manifold ◽  
Nonlinear Incidence Rate ◽  
Normal Form Theory ◽  
Nonlinear Incidence ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.

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