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 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 Global Stability of an SEIS Epidemic Model with General Saturation Incidence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hui Zhang ◽  
Li Yingqi ◽  
Wenxiong Xu
Keyword(s):  
Latent Period ◽  
Epidemic Model ◽  
Convergence Theorem ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T0, and the disease spreads to be endemic.

COMPLEXITY AND ASYMPTOTICAL BEHAVIOR OF A SIRS EPIDEMIC MODEL WITH PROPORTIONAL IMPULSIVE VACCINATION

2005 ◽  
Vol 08 (04) ◽  
pp. 419-431 ◽  
Author(s):  
GUANG-ZHAO ZENG ◽  
LAN-SUN CHEN
Keyword(s):  
Epidemic Model ◽  
Impulsive Control ◽  
Periodic Oscillation ◽  
The Other ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sirs Epidemic Model ◽  
Other Hand ◽  
Ratio Dependent ◽  
Impulsive Vaccination

This paper considers an SIRS epidemic model with proportional impulsive vaccination, which may inherently oscillate. We study the ratio-dependent impulsive control and obtain the conditions about the basic reproductive number for which the epidemic-elimination solution is globally asymptotic. On the other hand, if the epidemic turns out to be endemic, we study numerically the influences of impulsive vaccination on the periodic oscillation of a system without impulsion and find sophisticated phenomena such as chaos in this case.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0<1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0>1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

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 AN SEIR MODEL WITH INFECTIOUS LATENT AND A PERIODIC VACCINATION STRATEGY

2021 ◽  
Vol 26 (2) ◽  
pp. 236-252
Author(s):  
Islam A. Moneim
Keyword(s):  
Infectious Diseases ◽  
Periodic Solutions ◽  
Disease Transmission ◽  
Vaccination Strategy ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
New Wave ◽  
Globally Asymptotically Stable ◽  
Seir Model ◽  
The Impact

An SEIR epidemic model with a nonconstant vaccination strategy is studied. This SEIR model has two disease transmission rates β1 and β2 which imitate the fact that, for some infectious diseases, a latent person can pass the disease into a susceptible one. Here we study the spread of some childhood infectious diseases as good examples of diseases with infectious latent. We found that our SEIR model has a unique disease free solution (DFS). A lower bound and an upper bound of the basic reproductive number, R0 are estimated. We show that, the DFS is globally asymptotically stable when and unstable if Computer simulations have been conducted to show that non trivial periodic solutions are possible. Moreover the impact of the contact rate between the latent and the susceptibles is simulated. Different periodic solutions with different periods including one, two and three years, are obtained. These results give a clearer view for the decision makers to know how and when they should take action against a possible new wave of these infectious diseases. This action is mainly, applying a suitable dose of vaccination just before a severe peak of infection occurs.

Modeling and simulation of the spread of H1N1 flu with periodic vaccination

2015 ◽  
Vol 09 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Islam A. Moneim
Keyword(s):  
Influenza A ◽  
Vaccination Strategy ◽  
Vaccination Rate ◽  
Influenza Viruses ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Influenza A H1n1 ◽  
Large Populations ◽  
Stability Results ◽  
Influenza H1n1

Influenza H1N1 has been found to exhibit oscillatory levels of incidence in large populations. Clear peaks for influenza H1N1 are observed in several countries including Vietnam each year [M. F. Boni, B. H. Manh, P. Q. Thai, J. Farrar, T. Hien, N. T. Hien, N. Van Kinh and P. Horby, Modelling the progression of pandemic influenza A (H1N1) in Vietnam and the opportunities for reassortment with other influenza viruses, BMC Med. 7 (2009) 43, Doi: 10.1186/1741-7015-7-43]. So it is important to study seasonal forces and factors which can affect the transmission of this disease. This paper studies an SIRS epidemic model with seasonal vaccination rate. This SIRS model has a unique disease-free solution (DFS). The value R0, the basic reproduction number is obtained when the vaccination is a periodic function. Stability results for the DFS are obtained when R0 < 1. The disease persists in the population and remains endemic if R0 > 1. Also when R0 > 1 existence of a nonzero periodic solution is proved. These results obtained for our model when the vaccination strategy is a non-constant time-dependent function.

scholarly journals A Node-Based SIRS Epidemic Model with Infective Media on Complex Networks

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Leyi Zheng ◽  
Longkun Tang
Keyword(s):  
Epidemic Model ◽  
Small World ◽  
Spreading Rate ◽  
Node Degree ◽  
Strong Positive Correlation ◽  
Globally Asymptotically Stable ◽  
Sirs Epidemic Model ◽  
Theoretical Investigations ◽  
Fully Connected ◽  
The Impact

We focus on the node-based epidemic modeling for networks, introduce the propagation medium, and propose a node-based Susceptible-Infected-Recovered-Susceptible (SIRS) epidemic model with infective media. Theoretical investigations show that the endemic equilibrium is globally asymptotically stable. Numerical examples of three typical network structures also verify the theoretical results. Furthermore, comparison between network node degree and its infected percents implies that there is a strong positive correlation between both; namely, the node with bigger degree is infected with more percents. Finally, we discuss the impact of the epidemic spreading rate of media as well as the effective recovered rate on the network average infected state. Theoretical and numerical results show that (1) network average infected percents go up (down) with the increase of the infected rate of media (the effective recovered rate); (2) the infected rate of media has almost no influence on network average infected percents for the fully connected network and NW small-world network; (3) network average infected percents decrease exponentially with the increase of the effective recovered rate, implying that the percents can be controlled at low level by an appropriate large effective recovered rate.

Global properties for an age-structured within-host model with Crowley–Martin functional response

2017 ◽  
Vol 10 (02) ◽  
pp. 1750030 ◽  
Author(s):  
Shaoli Wang ◽  
Xinyu Song
Keyword(s):  
Functional Response ◽  
Viral Infections ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Multi Scale ◽  
Global Properties ◽  
Age Structured

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

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