Large Global Asymptotically Stability of an SEIR Epidemic Model with Distributed Time Delay

2011 ◽  
Vol 271-273 ◽  
pp. 428-434
Author(s):  
Jin Gao ◽  
Zhen Jin
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
State Space ◽  
Epidemic Model ◽  
Transmission Model ◽  
Force Of Infection ◽  
The Past ◽  
Distributed Time Delay ◽  
Globally Stable ◽  
Global Asymptotically Stability

An SEIR epidemic transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a desease free equilibrium is globally stable if no epidemic equilibrium point exists. Further the epidemic equilibrium (if it exists) is globally stable in the who;e state space except the neighborhood of the desease free equilibrium.

scholarly journals Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Shujing Gao ◽  
Zhidong Teng ◽  
Juan J. Nieto ◽  
Angela Torres
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Epidemic Model ◽  
Discrete Dynamical System ◽  
Vaccination Rate ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Distributed Time Delay ◽  
Large Pulse

Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.

PERMANENCE AND EXTINCTION OF A NONAUTONOMOUS STAGE-STRUCTURED EPIDEMIC MODEL WITH DISTRIBUTED TIME DELAY

2010 ◽  
Vol 18 (02) ◽  
pp. 377-398
Author(s):  
G. P. SAMANTA
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Functional Method ◽  
Analytical Technique ◽  
Distributed Time Delay ◽  
Lyapunov Functional Method ◽  
Varying Total Population Size ◽  
Model Computer ◽  
Two Stages

In this paper, we have considered a nonautonomous stage-structured epidemic model having two stages of the period of infection according to the progressing process of some infectious diseases (e.g. Chagas' disease, hepatitis C, etc.) with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R* and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

scholarly journals Dynamical Models For Prices With Distributed Delays

2015 ◽  
Vol 8 (1) ◽  
pp. 91-102
Author(s):  
Gabriela Mircea ◽  
Mihaela Neamţu ◽  
Laura Mariana Cismaş
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Supply And Demand ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Dynamical Models ◽  
Demand Functions ◽  
Distributed Time Delay ◽  
Commodity Demand ◽  
Positive Equilibrium Point

Abstract In the present paper we study some models for the price dynamics of a single commodity market. The quantities of supplied and demanded are regarded as a function of time. Nonlinearities in both supply and demand functions are considered. The inventory and the level of inventory are taken into consideration. Due to the fact that the consumer behavior affects commodity demand, and the behavior is influenced not only by the instantaneous price, but also by the weighted past prices, the distributed time delay is introduced. The following kernels are taken into consideration: demand price weak kernel and demand price Dirac kernel. Only one positive equilibrium point is found and its stability analysis is presented. When the demand price kernel is weak, under some conditions of the parameters, the equilibrium point is locally asymptotically stable. When the demand price kernel is Dirac, the existence of the local oscillations is investigated. A change in local stability of the equilibrium point, from stable to unstable, implies a Hopf bifurcation. A family of periodic orbits bifurcates from the positive equilibrium point when the time delay passes through a critical value. The last part contains some numerical simulations to illustrate the effectiveness of our results and conclusions.

scholarly journals Analisis dinamik model SVEIR pada penyebaran penyakit campak

2020 ◽  
Vol 1 (2) ◽  
pp. 57-64
Author(s):  
Sitty Oriza Sativa Putri Ahaya ◽  
Emli Rahmi ◽  
Nurwan Nurwan
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Model ◽  
Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this article, we analyze the dynamics of measles transmission model with vaccination via an SVEIR epidemic model. The total population is divided into five compartments, namely the Susceptible, Vaccinated, Exposed, Infected, and Recovered populations. Firstly, we determine the equilibrium points and their local asymptotically stability properties presented by the basic reproduction number R0. It is found that the disease free equilibrium point is locally asymptotically stable if satisfies R01 and the endemic equilibrium point is locally asymptotically stable when R01. We also show the existence of forward bifurcation driven by some parameters that influence the basic reproduction number R0 i.e., the infection rate α or proportion of vaccinated individuals θ. Lastly, some numerical simulations are performed to support our analytical results.

scholarly journals ANALYSIS OF A NONAUTONOMOUS HIV/AIDS EPIDEMIC MODEL WITH DISTRIBUTED TIME DELAY

2010 ◽  
Vol 15 (3) ◽  
pp. 327-347 ◽  
Author(s):  
Guruprasad P. Samanta
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Horizontal Transmission ◽  
Sufficient Conditions ◽  
Functional Method ◽  
Infected People ◽  
Analytical Technique ◽  
Aids Epidemic ◽  
Distributed Time Delay ◽  
Hiv Aids

In this paper, we have considered a nonautonomous stage‐structured HIV/AIDS epidemic model through vertical and horizontal transmissions of infections, having two stages of the period of infection according to the developing progress of infection before AIDS defined would be detected, with varying total population size and distributed time delay to become infectious (through horizontal transmission) due to intracellular delay between initial infection of a cell by HIV and the release of new virions. The infected people in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected people. We have introduced some new threshold values _Ro and R* and further obtained that the disease will be permanent when _Ro > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings.

Global asymptotic stability of an SIR epidemic model with distributed time delay

2001 ◽  
Vol 47 (6) ◽  
pp. 4107-4115 ◽  
Author(s):  
Edoardo Beretta ◽  
Tadayuki Hara ◽  
Wanbiao Ma ◽  
Yasuhiro Takeuchi
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Distributed Time Delay

scholarly journals Permanence and Extinction for a Nonautonomous Malaria Transmission Model with Distributed Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Xiaohong Zhang ◽  
Jianwen Jia ◽  
Xinyu Song
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Malaria Transmission ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Analytical Techniques ◽  
Functional Method ◽  
Transmission Model ◽  
Distributed Time Delay ◽  
Lyapunov Functional Method

We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.

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