Dynamics Analysis of an SEIQS Model with a Nonlinear Incidence Rate

2012 ◽  
Vol 157-158 ◽  
pp. 1220-1223
Author(s):  
Ning Cui ◽  
Jun Hong Li ◽  
Jiao Qu ◽  
Hong Dan Xue
Keyword(s):  
Lyapunov Function ◽  
Incidence Rate ◽  
Matrix Theory ◽  
Invariant Set ◽  
Dynamics Analysis ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Asymptotical Stable ◽  
Compound Matrix ◽  
Disease Free

This paper considers an SEIQS model with nonlinear incidence rate. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. It is then obtained the sufficent conditions for the global stability of the endemic equilibrium by the compound matrix theory. In addition, we also study the phenomena of limit cycle of the systems with the numerical simulations.

scholarly journals Dynamic Analysis of an SEIR Model with Distinct Incidence for Exposed and Infectives

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Matrix Theory ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Vaccination Strategy ◽  
Invariant Set ◽  
Incidence Rates ◽  
Seir Model ◽  
Asymptotical Stable ◽  
Compound Matrix ◽  
Disease Free

An SEIR model with vaccination strategy that incorporates distinct incidence rates for the exposed and the infected populations is studied. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The sufficient conditions for the global stability of the endemic equilibrium are obtained using the compound matrix theory. Furthermore, the method of direct numerical simulation of the system shows that there is a periodic solution, when the system has three equilibrium points.

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.

On global stability analysis for SEIRS models in epidemiology with nonlinear incidence rate function

2017 ◽  
Vol 10 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Lifei Zheng ◽  
Xiuxiang Yang ◽  
Liang Zhang
Keyword(s):  
Equilibrium Point ◽  
Incidence Rate ◽  
Threshold Value ◽  
Rate Function ◽  
Endemic Equilibrium ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Seirs Epidemic Model ◽  
Globally Stable ◽  
Disease Free

We study an SEIRS epidemic model with an isolation and nonlinear incidence rate function. We have obtained a threshold value [Formula: see text] and shown that there is only a disease-free equilibrium point, when [Formula: see text] and an endemic equilibrium point if [Formula: see text]. We have shown that both disease-free and endemic equilibrium point are globally stable.

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.

scholarly journals Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Lei Wang ◽  
Zhidong Teng ◽  
Long Zhang
Keyword(s):  
Incidence Rate ◽  
Endemic Equilibrium ◽  
Epidemic Models ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
General Incidence Rate ◽  
Special Cases ◽  
Disease Free

We study a class of discrete SIRS epidemic models with nonlinear incidence rateF(S)G(I)and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction numberℛ0<1, then the disease-free equilibrium is globally asymptotically stable, and ifℛ0>1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only whenℛ0>1, the disease in the model is permanent. Some special cases ofF(S)G(I)are discussed. Particularly, whenF(S)G(I)=βSI/(1+λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only ifℛ0>1. Furthermore, the numerical simulations show that for general incidence rateF(S)G(I)the endemic equilibrium may be globally asymptotically stable only asℛ0>1.

scholarly journals Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1829
Author(s):  
Ardak Kashkynbayev ◽  
Fathalla A. Rihan
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Incidence Rate ◽  
Fractional Order ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Theoretical Results ◽  
Disease Free

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

scholarly journals Dynamic Behavior for an SIRS Model with Nonlinear Incidence Rate and Treatment

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Incidence Rate ◽  
Dynamic Behavior ◽  
Limit Cycle ◽  
Nonlinear Incidence Rate ◽  
Existence Of Equilibrium ◽  
Nonlinear Incidence ◽  
Sirs Model ◽  
Asymptotical Stable

This paper considers an SIRS model with nonlinear incidence rate and treatment. It is assumed that susceptible and infectious individuals have constant immigration rates. We investigate the existence of equilibrium and prove the global asymptotical stable results of the endemic equilibrium. We then obtained that the model undergoes a Hopf bifurcation and existences a 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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