MIXED VACCINATION STRATEGY IN SIRS EPIDEMIC MODEL WITH SEASONAL VARIABILITY ON INFECTION

2011 ◽  
Vol 04 (04) ◽  
pp. 473-491 ◽  
Author(s):  
SHUJING GAO ◽  
HONGSHUI OUYANG ◽  
JUAN J. NIETO
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Epidemic Model ◽  
Vaccination Strategy ◽  
Contact Rate ◽  
Seasonal Fluctuations ◽  
Pulse Vaccination ◽  
Sirs Epidemic Model ◽  
Theoretical Results ◽  
Disease Free

In many diseases seasonal fluctuations are observed. SIRS epidemic model with seasonal varying contact rate and mixed vaccination strategy (including first vaccination and pulse vaccination strategy) is investigated. The effects of the variation of dependent on the season of the contact rate and the vaccination strategy to eradicate infectious diseases are studied and discussed. A threshold for a disease to be extinct or endemic is established. The existence and global asymptotic stability of disease-free periodic solution and the permanence of the disease are illustrated. Finally, our theoretical results are confirmed by numerical simulations.

scholarly journals Dynamical Analysis of SIR Epidemic Model with Nonlinear Pulse Vaccination and Lifelong Immunity

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Wencai Zhao ◽  
Juan Li ◽  
Xinzhu Meng
Keyword(s):  
Periodic Solution ◽  
Epidemic Model ◽  
Fixed Point Theory ◽  
Positive Periodic Solution ◽  
Point Theory ◽  
Dynamical Analysis ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Disease Free

SIR epidemic model with nonlinear pulse vaccination and lifelong immunity is proposed. Due to the limited medical resources, vaccine immunization rate is considered as a nonlinear saturation function. Firstly, by using stroboscopic map and fixed point theory of difference equations, the existence of disease-free periodic solution is discussed, and the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Moreover, by using the bifurcation theorem, sufficient condition for the existence of positive periodic solution is obtained by choosing impulsive vaccination period as a bifurcation parameter. Lastly, some simulations are given to validate the theoretical results.

scholarly journals Dynamics for a stochastic delayed SIRS epidemic model

2020 ◽  
Vol 25 (5) ◽  
Author(s):  
Xiangyun Shi ◽  
Yimeng Cao ◽  
Xueyong Zhou
Keyword(s):  
Periodic Solution ◽  
Seasonal Variation ◽  
Global Existence ◽  
Positive Solutions ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sirs Epidemic Model ◽  
Nontrivial Periodic Solution ◽  
Well Posed ◽  
Disease Free

In this paper, we consider a stochastic delayed SIRS epidemic model with seasonal variation. Firstly, we prove that the system is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution. Secondly, some sufficient conditions on the permanence and extinction of the positive solutions with probability one are presented. Thirdly, we show that the solution of the system is asymptotical around of the disease-free periodic solution and the intensity of the oscillation depends of the intensity of the noise. Lastly, the existence of stochastic nontrivial periodic solution for the system is obtained.

THRESHOLD AND STABILITY RESULTS FOR AN SIRS EPIDEMIC MODEL WITH A GENERAL PERIODIC VACCINATION STRATEGY

2005 ◽  
Vol 13 (02) ◽  
pp. 131-150 ◽  
Author(s):  
I. A. MONEIM ◽  
D. GREENHALGH
Keyword(s):  
Epidemic Model ◽  
Vaccination Strategy ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Contact Rate ◽  
Globally Asymptotically Stable ◽  
Epidemic Dynamics ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Periodic Disease

An SIRS epidemic model with general periodic vaccination strategy is analyzed. This periodic vaccination strategy is discussed first for an SIRS model with seasonal variation in the contact rate of period T = 1 year. We start with the case where the vaccination strategy and the contact rate have the same period and then discuss the case where the period of the vaccination strategy is LT, where L is an integer. We investigate whether a periodic vaccination strategy may force the epidemic dynamics to have periodic behavior. We prove that our SIRS model has a unique periodic disease free solution (DFS) whose period is the same as that of the vaccination strategy, which is globally asymptotically stable when the basic reproductive number R0 is less than or equal to one in value. When R0 > 1, we prove that there exists a non-trivial periodic solution of period the same as that of the vaccination strategy. Some persistence results are also discussed. Threshold conditions for these periodic vaccination strategies to ensure that R0 ≤ 1 are derived.

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.

Global threshold dynamics of SIQS epidemic model in time fluctuating environment

2017 ◽  
Vol 10 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Shang Li ◽  
Meng Fan ◽  
Xinmiao Rong
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Integral Operator ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Global Threshold ◽  
Linear Integral ◽  
Disease Free ◽  
The Basic Reproduction Number

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.

GLOBAL ASYMPTOTIC STABILITY FOR THE SIV EPIDEMIC MODEL WITH IMPULSIVE VACCINATION AND INFECTION AGE

2006 ◽  
Vol 14 (01) ◽  
pp. 43-51 ◽  
Author(s):  
HELONG LIU ◽  
HOUBAO XU ◽  
JINGYUAN YU ◽  
GUANGTIAN ZHU
Keyword(s):  
Asymptotic Stability ◽  
Hepatitis B ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Vaccination Strategy ◽  
Infection Age ◽  
Pulse Vaccination ◽  
Impulsive Vaccination ◽  
Time Lead ◽  
Stability Of The Solution

In this paper, we study the application of a pulse vaccination strategy to eradicate hepatitis B and C modeled by SIV epidemic model. Since infection age is an important factor of hepatitis progression, we incorporate the infection age into the model. In this model, we consider the infectiousness of latent individuals. We derive the condition in which eradication solution is a global attractor, this condition depends on pulse vaccination proportion p and interpulse time T. We also obtain the condition of the global asymptotic stability of the solution. The condition shows that large enough pulse vaccination proportion and relatively small interpulse time lead to the eradication of hepatitis B and C. Moreover the results of the theoretical study might be instructive to the epidemiology of HIV.

scholarly journals On the global stability of a delay epidemic model

1992 ◽  
Vol 34 (1) ◽  
pp. 26-34
Author(s):  
Xiaodong Lin
Keyword(s):  
Periodic Solution ◽  
Asymptotic Behavior ◽  
Time Delay ◽  
Global Stability ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Nonlinear Incidence Rate ◽  
Sirs Epidemic Model ◽  
Disease Free

AbstractIn this paper, we study the asymptotic behavior of an SIRS epidemic model with a time delay in the recovered class and a nonlinear incidence rate. A conjecture of Hethcote et al. [5] on the global stability of the disease-free equilibrium is solved. Moreover, we analyse the model when the contact number takes its threshold value. We show that solutions tend to either the disease-free equilibrium or to a unique positive endemic equilibrium, and there is no periodic solution.

scholarly journals The Global Behavior of a Periodic Epidemic Model with Travel between Patches

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Luosheng Wen ◽  
Bin Long ◽  
Xin Liang ◽  
Fengling Zeng
Keyword(s):  
Epidemic Model ◽  
Transmission Rate ◽  
Global Behavior ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Periodic Transmission ◽  
Theoretical Results ◽  
Disease Free

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratioR0and two theorems on the global behavior: ifR0< 1 the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then it is unstable; ifR0> 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

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