scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

Stability and Bifurcation in an SI Epidemic Model with Additive Allee Effect and Time Delay

2021 ◽  
Vol 31 (04) ◽  
pp. 2150060
Author(s):  
Yangyang Lv ◽  
Lijuan Chen ◽  
Fengde Chen ◽  
Zhong Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Allee Effect ◽  
Stability Of Equilibria ◽  
Existence And Stability ◽  
Vital Effects ◽  
Strong Allee Effect ◽  
The Stability

In this paper, we consider an SI epidemic model incorporating additive Allee effect and time delay. The primary purpose of this paper is to study the dynamics of the above system. Firstly, for the model without time delay, we demonstrate the existence and stability of equilibria for three different cases, i.e. with weak Allee effect, with strong Allee effect, and in the critical case. We also investigate the existence and uniqueness of Hopf bifurcation and limit cycle. Secondly, for the model with time delay, the stability of equilibria and the existence of Hopf bifurcation are discussed. All the above show that both additive Allee effect and time delay have vital effects on the prevalence of the disease.

scholarly journals Impulsive Vaccination SEIR Model with Nonlinear Incidence Rate and Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Dongmei Li ◽  
Chunyu Gui ◽  
Xuefeng Luo
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Uniform Persistence ◽  
Nonlinear Incidence Rate ◽  
Constant Input ◽  
Seir Model ◽  
Nonlinear Incidence ◽  
Impulsive Vaccination

This paper aims to discuss the delay epidemic model with vertical transmission, constant input, and nonlinear incidence. Some sufficient conditions are given to guarantee the existence and global attractiveness of the infection-free periodic solution and the uniform persistence of the addressed model with time delay. Finally, a numerical example is given to demonstrate the effectiveness of the proposed results.

scholarly journals A diffusive predator-prey model with generalist predator and time delay

2021 ◽  
Vol 7 (3) ◽  
pp. 4574-4591
Author(s):  
Ruizhi Yang ◽  
◽  
Dan Jin ◽  
Wenlong Wang
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Resource Limitation ◽  
Generalist Predator ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
Bifurcating Periodic Solution

<abstract><p>Time delay in the resource limitation of the prey is incorporated into a diffusive predator-prey model with generalist predator. By analyzing the eigenvalue spectrum, time delay inducing instability and Hopf bifurcation are investigated. Some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation. The results suggest that time delay can destabilize the stability of coexisting equilibrium and induce bifurcating periodic solution when it increases through a certain threshold.</p></abstract>

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yue Zhang ◽  
Xue Li ◽  
Xianghua Zhang ◽  
Guisheng Yin
Keyword(s):  
Fixed Point ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Theoretic Analysis ◽  
Simulation Results ◽  
The Stability ◽  
Positive Fixed Point

Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

scholarly journals An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Huitao Zhao ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Orbits ◽  
Epidemic Model ◽  
Media Coverage ◽  
Critical Value ◽  
Endemic Equilibrium ◽  
Sirs Epidemic Model ◽  
The Stability ◽  
Disease Free

An SIRS epidemic model incorporating media coverage with time delay is proposed. The positivity and boundedness are studied firstly. The locally asymptotical stability of the disease-free equilibrium and endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcate are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction numberR0<1. However, whenR0>1, the stability of the endemic equilibrium will be affected by the time delay; there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay increases through a critical value. Finally, some examples for numerical simulations are also included.

scholarly journals Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Changjin Xu ◽  
Xiaofei He
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form Theory ◽  
Neuron Networks ◽  
Form Theory ◽  
The Stability ◽  
Manifold Theory ◽  
Bifurcating Periodic Solution ◽  
Bilinear Terms ◽  
Zero Equilibrium

A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

HOPF BIFURCATION ANALYSIS FOR A DELAYED LESLIE–GOWER PREDATOR–PREY SYSTEM WITH DIFFUSION EFFECTS

2014 ◽  
Vol 07 (01) ◽  
pp. 1450007
Author(s):  
LIN-LIN WANG ◽  
BEI-BEI ZHOU ◽  
YONG-HONG FAN
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability ◽  
The Impact

A delayed predator–prey diffusion system with homogeneous Neumann boundary condition is considered. In order to study the impact of the time delay on the stability of the model, the delay τ is taken as the bifurcation parameter, the results show that when the time delay across some critical values, the Hopf bifurcations may occur. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution have been established. The effect of the diffusion on the bifurcated periodic solution is also considered. A numerical example is given to support the main result.

scholarly journals On the global stability of a delay epidemic model

1992 ◽  
Vol 34 (1) ◽  
pp. 26-34
Author(s):  
Xiaodong Lin
Keyword(s):  
Periodic Solution ◽  
Asymptotic Behavior ◽  
Time Delay ◽  
Global Stability ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Nonlinear Incidence Rate ◽  
Sirs Epidemic Model ◽  
Disease Free

AbstractIn this paper, we study the asymptotic behavior of an SIRS epidemic model with a time delay in the recovered class and a nonlinear incidence rate. A conjecture of Hethcote et al. [5] on the global stability of the disease-free equilibrium is solved. Moreover, we analyse the model when the contact number takes its threshold value. We show that solutions tend to either the disease-free equilibrium or to a unique positive endemic equilibrium, and there is no periodic solution.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

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