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.

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 Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Jihad Adnani ◽  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Endemic Equilibrium ◽  
Random Perturbations ◽  
Sir Epidemic Model ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sir Epidemic ◽  
Stochastic Sir ◽  
The Stability

We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical 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 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.

scholarly journals ANALISIS TITIK EKUILIBRIUM ENDEMIK MODEL EPIDEMI SEIV DENGAN LAJU PENULARAN NONLINEAR

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
Nurul Hikmah
Keyword(s):  
Equilibrium Point ◽  
Key Words ◽  
Incidence Rate ◽  
Nonlinear Model ◽  
Epidemic Model ◽  
Behavioral Change ◽  
Global Analysis ◽  
Endemic Equilibrium ◽  
Nonlinear Incidence Rate ◽  
Globally Stable

Abstrak. Pada paper ini diberikan model epidemi SEIV dengan laju penularan nonlinear. Model ini menjelaskan tentang efek psikologi dari perubahan perilaku individu yang rentan ketika jumlah individu yang terinfeksi mengalami peningkatan. Dalam paper ini akan dilakukan analisis global dari model epidemi SEIV dan menyelidiki kestabilan global titik ekuilibrium endemik , selanjutnya diperoleh bahwa titik ekuilibrium endemik model epidemi SEIV stabil global. Kata Kunci : SEIV, titik ekuilibrium, kestabilan Abstract. In this paper, we consider a SEIV epidemic model with nonlinear incidence rate. This model describes the psychological effect of the behavioral change of susceptible individuals when the number of infectious individuals increases. By carrying out a global analysis of the model and studying the globally stability of the endemic equilibrium in this paper, we show that the endemic equilibrium of a SEIV epidemic model is globally stable. Key words: SEIV, equilibrium point, stability

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.

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