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 Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Xiaoguang Zhang ◽  
Rui Song ◽  
Gui-Quan Sun ◽  
Zhen Jin
Keyword(s):  
Complex Networks ◽  
Reproduction Number ◽  
Epidemic Models ◽  
Infectious Period ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Different Types ◽  
Theoretical Results ◽  
Disease Free

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

scholarly journals Global Dynamics of an SIS Model on Metapopulation Networks with Demographics

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Maoxing Liu ◽  
Xinjie Fu ◽  
Jie Zhang ◽  
Donghua Zhao
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Active Role ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Sis Model ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
The Basic Reproduction Number

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

scholarly journals Qualitative and Bifurcation Analysis of an SIR Epidemic Model with Saturated Treatment Function and Nonlinear Pulse Vaccination

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Xiangsen Liu ◽  
Binxiang Dai
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Existence And Stability ◽  
The Stability ◽  
Saturated Treatment ◽  
Disease Free

An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.

scholarly journals Survival and Stationary Distribution in a Stochastic SIS Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanli Zhou ◽  
Weiguo Zhang ◽  
Sanling Yuan
Keyword(s):  
Stationary Distribution ◽  
Deterministic Model ◽  
Deterministic System ◽  
Initial Value ◽  
Sis Model ◽  
Sis Epidemic Model ◽  
Stochastic Sis Epidemic Model ◽  
Theoretical Results ◽  
Disease Free

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

N-intertwined SIS epidemic model with Markovian switching

2019 ◽  
Vol 19 (04) ◽  
pp. 1950031 ◽  
Author(s):  
Xiaochun Cao ◽  
Zhen Jin
Keyword(s):  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Sis Model ◽  
Color Noise ◽  
Epidemic Dynamics ◽  
Sis Epidemic Model ◽  
Finite State ◽  
Global Positive Solution ◽  
Time Average ◽  
Theoretical Results

Epidemic dynamics is often subject to environmental noise and uncertainty. In this paper, we investigate the effect of color noise on the spread of epidemic in complex networks, which is modeled by stochastic switched differential equations based on the [Formula: see text]-intertwined SIS model using a continuous time finite-state Markov chain. Applying Lyapunov functions, we prove that the model has a unique global positive solution and establish sufficient conditions for stochastic extinction and permanence of the epidemic. We also show that the solution is stochastically ultimately bounded and the variance of the solution is bounded too. Furthermore, we discuss the limit of the time average of the solution. Finally, numerical simulations are carried out to illustrate our theoretical results.

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 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.

scholarly journals Global exponential stability of positive periodic solutions for a cholera model with saturated treatment

2018 ◽  
Vol 23 (5) ◽  
pp. 619-641
Author(s):  
Hongzheng Quana ◽  
Xueyong Zhoub ◽  
Jianzhou Liua
Keyword(s):  
Exponential Stability ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Incidence Rate ◽  
Global Exponential Stability ◽  
Positive Periodic Solutions ◽  
Treatment Function ◽  
Novel Method ◽  
Saturated Treatment ◽  
Theoretical Results

In this paper, we consider a cholera model with periodic incidence rate and saturated treatment function. Under certain conditions, we establish a criterion on the global exponential stability of positive periodic solutions for this model by using a novel method. We illustrate our theoretical results with numerical simulations by using Matlab.

scholarly journals Global Dynamics of a Discrete-Time MERS-Cov Model

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 563
Author(s):  
Mahmoud H. DarAssi ◽  
Mohammad A. Safi ◽  
Morad Ahmad
Keyword(s):  
Discrete Time ◽  
Global Attractivity ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Threshold Conditions ◽  
Theoretical Results ◽  
Disease Free

In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. We proved that the DFE is globally asymptotically stable when R0≤1. Whenever R˜0>1, the proposed model has a unique endemic equilibrium that is globally asymptotically stable. The theoretical results are illustrated by a numerical simulation.

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