scholarly journals An SIRS Epidemic Model with Vital Dynamics and a Ratio-Dependent Saturation Incidence Rate

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xinli Wang
Keyword(s):  
Computer Simulation ◽  
Periodic Solution ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Initial Position ◽  
Mass Action ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Sirs Epidemic Model ◽  
Disease Free

This paper presents an investigation on the dynamics of an epidemic model with vital dynamics and a nonlinear incidence rate of saturated mass action as a function of the ratio of the number of the infectives to that of the susceptibles. The stabilities of the disease-free equilibrium and the endemic equilibrium are first studied. Under the assumption of nonexistence of periodic solution, the global dynamics of the model is established: either the number of infective individuals tends to zero as time evolves or it produces bistability in which there is a region such that the disease will persist if the initial position lies in the region and disappears if the initial position lies outside this region. Computer simulation shows such results.

ANALYSIS ON AN EPIDEMIC MODEL WITH A RATIO-DEPENDENT NONLINEAR INCIDENCE RATE

2011 ◽  
Vol 04 (02) ◽  
pp. 227-239 ◽  
Author(s):  
BO LI ◽  
SANLING YUAN ◽  
WEIGUO ZHANG
Keyword(s):  
Qualitative Analysis ◽  
Incidence Rate ◽  
Computer Simulations ◽  
Epidemic Model ◽  
Mass Action ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Set Up ◽  
Disease Free

In this paper, we study the global dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectives to that of the susceptibles. The model has set up a challenging issue regarding its dynamics at the R-axis since it is not well defined on it. By carrying out a global qualitative analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

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.

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 Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongli Cai ◽  
Xixi Wang ◽  
Weiming Wang ◽  
Min Zhao
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Stochastic Dynamics ◽  
Complex Dynamics ◽  
Deterministic Model ◽  
Mass Action ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
The Stability ◽  
Disease Free

We investigate the complex dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectious individuals to that of susceptible individuals. We first deal with the boundedness, dissipation, persistence, and the stability of the disease-free and endemic points of the deterministic model. And then we prove the existence and uniqueness of the global positive solutions, stochastic boundedness, and permanence for the stochastic epidemic model. Furthermore, we perform some numerical examples to validate the analytical findings. Needless to say, both deterministic and stochastic epidemic models have their important roles.

Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate

2016 ◽  
Vol 17 (7-8) ◽  
pp. 401-412 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Epidemic Model ◽  
Local Stability ◽  
Host Population ◽  
Backward Bifurcation ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Disease Free

Abstract:In this work we study a fractional-order susceptible-infective-recovered-susceptible (SIRS) epidemic model with a nonlinear incidence rate. The incidence is assumed to be a convex function with respect to the infective class of a host population. Local and uniform stability analysis of the disease-free equilibrium is investigated. The conditions for the existence of endemic equilibria (EE) are given. Local stability of the EE is discussed. Conditions for the existence of Hopf bifurcation at the EE are given. Most importantly, conditions ensuring that the system exhibits backward bifurcation are provided. Numerical simulations are performed to verify the correctness of results obtained analytically.

Dynamic Behavior for an SIRS Model with Nonlinear Incidence Rate

2012 ◽  
Vol 479-481 ◽  
pp. 1495-1498 ◽  
Author(s):  
Jun Hong Li ◽  
Ning Cui ◽  
Hong Kai Sun
Keyword(s):  
Hopf Bifurcation ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Dulac Function ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Asymptotical Stable ◽  
Disease Free

An SIRS epidemic model with nonlinear incidence rate is studied. It is assumed that susceptible and infectious individuals have constant immigration rates. By means of Dulac function and Poincare-Bendixson Theorem, we proved the global asymptotical 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 analytical results.

scholarly journals Global Dynamics of an Epidemic Model with a Ratio-Dependent Nonlinear Incidence Rate

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Sanling Yuan ◽  
Bo Li
Keyword(s):  
Incidence Rate ◽  
Computer Simulations ◽  
Epidemic Model ◽  
Global Analysis ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Set Up ◽  
Disease Free

We study an epidemic model with a nonlinear incidence rate which describes the psychological effect of certain serious diseases on the community when the ratio of the number of infectives to that of the susceptibles is getting larger. The model has set up a challenging issue regarding its dynamics near the origin since it is not well defined there. By carrying out a global analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

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