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 A SIRS Epidemic Model Incorporating Media Coverage with Random Perturbation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Wenbin Liu
Keyword(s):  
Stochastic Model ◽  
Epidemic Model ◽  
Media Coverage ◽  
Complex Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Endemic Equilibrium ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
The Stability

We investigate the complex dynamics of a SIRS epidemic model incorporating media coverage with random perturbation. We first deal with the boundedness and the stability of the disease—free and endemic equilibria of the deterministic model. And for the corresponding stochastic epidemic model, we prove that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Furthermore, we perform some numerical examples to validate the analytical finding, and find that if the conditions of stochastic stability are not satisfied, the solution for the stochastic model will oscillate strongly around the endemic equilibrium.

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.

scholarly journals Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Caijuan Yan ◽  
Jianwen Jia
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Sir Epidemic Model ◽  
Medical Resources ◽  
Reproduction Ratio ◽  
Disease Free ◽  
Information Variable

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

Hopf bifurcation of a delay SIRS epidemic model with novel nonlinear incidence: Application to scarlet fever

2018 ◽  
Vol 11 (07) ◽  
pp. 1850091 ◽  
Author(s):  
Yong Li ◽  
Xianning Liu ◽  
Lianwen Wang ◽  
Xingan Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Scarlet Fever ◽  
Reproduction Number ◽  
Control Strategies ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Intervention Measures ◽  
Theoretical Results

An [Formula: see text] epidemic model incorporating incubation time delay and novel nonlinear incidence is proposed and analyzed to seek for the control strategies of scarlet fever, where the contact rate which can reflect the regular behavior and habit changes of children is non-monotonic with respect to the number of susceptible. The model without delay may exhibit backward bifurcation and bistable states even though the basic reproduction number is less than unit. Furthermore, we derive the conditions for occurrence of Hopf bifurcation when the time delay is considered as a bifurcation parameter. The data of scarlet fever of China are simulated to verify our theoretical results. In the end, several effective preventive and intervention measures of scarlet fever are found out.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

Dynamic Analysis of an SIRS Model with Nonlinear Incidence Rate

2012 ◽  
Vol 155-156 ◽  
pp. 23-26
Author(s):  
Jun Hong Li ◽  
Ning Cui ◽  
Liang Cui ◽  
Cai Juan Li
Keyword(s):  
Hopf Bifurcation ◽  
Dynamic Analysis ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Dulac Function ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Disease Free

In this paper, we study the global dynamics of an SIRS epidemic model with nonlinear inci- dence rate. By means of Dulac function and Poincare-Bendixson Theorem, we proved the global asy- mptotical stable results of the disease-free equilibrium. It is then obtained the model undergoes Hopf bifurcation and existence of one limit cycle. Some numerical simulations are given to illustrate the an- alytical results.

scholarly journals Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

2019 ◽  
Vol 2019 ◽  
pp. 1-21
Author(s):  
Fulgensia Kamugisha Mbabazi ◽  
Joseph Y. T. Mugisha ◽  
Mark Kimathi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Pneumococcal Pneumonia ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Reproduction Ratio ◽  
Transversality Conditions ◽  
The Stability ◽  
Delay Differential ◽  
Disease Free

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratioR0is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by 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.

Dynamics of an Innovation Diffusion Model with Time Delay

2017 ◽  
Vol 7 (3) ◽  
pp. 455-481 ◽  
Author(s):  
Rakesh Kumar ◽  
Anuj K. Sharma ◽  
Kulbhushan Agnihotri
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Population Density ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Innovation Diffusion ◽  
Critical Value ◽  
Normal Form Theory ◽  
Interior Equilibrium ◽  
The Stability

AbstractA nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

2010 ◽  
Vol 03 (03) ◽  
pp. 299-312 ◽  
Author(s):  
SHU-MIN GUO ◽  
XUE-ZHI LI ◽  
XIN-YU SONG
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

Sign in / Sign up

Export Citation Format

Share Document