DYNAMICS OF SIS EPIDEMIC MODEL WITH THE STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260003 ◽  
Author(s):  
JINGJING WEI ◽  
JING-AN CUI
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Recruitment Rate ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Two Factors ◽  
Saturated Treatment ◽  
Disease Free ◽  
Sis Epidemic

An SIS epidemic model with the standard incidence rate and saturated treatment function is proposed. The dynamics of the system are discussed, and the effect of the capacity for treatment and the recruitment of the population are also studied. We find that the effect of the maximum recovery per unit of time and the recruitment rate of the population over some level are two factors which lead to the backward bifurcation, and in some cases, the model may undergo the saddle-node bifurcation or Bogdanov–Takens bifurcation. It is shown that the disease-free equilibrium is globally asymptotically stable under some conditions. Numerical simulations are consistent with our obtained results in theorems, which show that improving the efficiency and capacity of treatment is important for control of disease.

scholarly journals Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Yanju Xiao ◽  
Weipeng Zhang ◽  
Guifeng Deng ◽  
Zhehua Liu
Keyword(s):  
Sufficient Conditions ◽  
Global Dynamics ◽  
Sis Model ◽  
Delayed Treatment ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Lyapunov Coefficient ◽  
Saturated Treatment ◽  
Theoretical Results ◽  
Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

scholarly journals Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mouhcine Naim ◽  
Fouad Lahmidi
Keyword(s):  
Epidemic Model ◽  
Deterministic Model ◽  
Sufficient Condition ◽  
Nonlinear Incidence Rate ◽  
Local Asymptotic Stability ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Stochastic Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

Probability analysis of a stochastic SIS epidemic model

2017 ◽  
Vol 17 (06) ◽  
pp. 1750041 ◽  
Author(s):  
Qing Ge ◽  
Zhiming Li ◽  
Zhidong Teng
Keyword(s):  
Positive Integer ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Number Formula ◽  
Model Transition ◽  
Stochastic Sis Epidemic Model ◽  
Sis Epidemic

In this paper, the probability properties are investigated for a stochastic SIS epidemic model. Transition probabilities of the susceptible process are obtained by using Laplace transform and perturbation variables. According to two cases: basic reproduction number [Formula: see text] and [Formula: see text], the dynamical behaviors in probability of the process are analyzed. It is shown that when [Formula: see text] the disease-free equilibrium is globally asymptotically stable with probability one, and when [Formula: see text] and [Formula: see text] is a positive integer, the endemic equilibrium is globally asymptotically stable with probability one. These results coincide with the corresponding deterministic SIS epidemic model. However, when [Formula: see text] and [Formula: see text] is not a positive integer, there are different properties between the deterministic and stochastic models. Numerical simulations are also performed to validate these results.

scholarly journals Bifurcation analysis of an SIR epidemic model with saturated treatment function

2021 ◽  
Author(s):  
Phuc Ngo
Keyword(s):  
Epidemic Models ◽  
Stable Node ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Sir Epidemic Models ◽  
Saturated Treatment ◽  
Disease Free

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

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