scholarly journals Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Deterministic Models ◽  
Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

We introduce stochasticity into an SIS epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction numberR0is a threshold which determines the persistence or extinction of the disease. When the perturbation and the disease-related death rate are small, we carry out a detailed analysis on the dynamical behavior of the stochastic model, also regarding of the value ofR0. IfR0≤1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, ifR0>1, there is a stationary distribution, which means that the disease will prevail. The results are illustrated by computer simulations.

The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Number Formula ◽  
Disease Free

In this paper, we introduce stochasticity into an SIR epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number [Formula: see text]. If [Formula: see text], the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If [Formula: see text], there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.

Equilibriums of an SIS Epidemic Model

2014 ◽  
Vol 678 ◽  
pp. 103-106
Author(s):  
Jing Hai Wang
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Death Rate ◽  
Birth Rate ◽  
Reproduction Number ◽  
Nonlinear Incidence Rate ◽  
Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic ◽  
The Basic Reproduction Number

An SIS epidemic model with nonlinear incidence rate is considered. At the same time we decide that the birth rate is equal to death rate. We get the basic reproduction number . We analyze the existences of the equilibriums of the system. There are some endemic equilibriums and one disease-free equilibrium of the system.

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.

scholarly journals An SIS epidemic model with time delay and stochastic perturbation on heterogeneous networks

2021 ◽  
Vol 18 (5) ◽  
pp. 6790-6805
Author(s):  
Meici Sun ◽  
◽  
Qiming Liu
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Scale Free ◽  
Sis Epidemic Model ◽  
The Mean ◽  
Sis Epidemic

<abstract><p>An SIS epidemic model with time delay and stochastic perturbation on scale-free networks is established in this paper. And we derive sufficient conditions guaranteeing extinction and persistence of epidemics, respectively, which are related to the basic reproduction number $ R_0 $ of the corresponding deterministic model. When $ R_0 &lt; 1 $, almost surely exponential extinction and $ p $-th moment exponential extinction of epidemics are proved by Razumikhin-Mao Theorem. Whereas, when $ R_0 &gt; 1 $, the system is persistent in the mean under sufficiently weak noise intensities, which indicates that the disease will prevail. Finally, the main results are demonstrated by numerical simulations.</p></abstract>

Modelling the role of opportunistic diseases in coinfection

2018 ◽  
Vol 13 (3) ◽  
pp. 28
Author(s):  
Marcos Marvá ◽  
Rafael Bravo de la Parra ◽  
Ezio Venturino
Keyword(s):  
Epidemic Model ◽  
Recovery Rate ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Treatment Rate ◽  
Opportunistic Disease ◽  
Sis Epidemic Model ◽  
Primary Disease ◽  
Sis Epidemic

In this paper, we formulate a model for evaluating the effects of an opportunistic disease affecting only those individuals already infected by a primary disease. The opportunistic disease act on a faster time scale and it is represented by an SIS epidemic model with frequency-dependent transmission. The primary disease is governed by an SIS epidemic model with density-dependent transmission, and we consider two different recovery cases. The first one assumes a constant recovery rate whereas the second one takes into account limited treatment resources by means of a saturating treatment rate. No demographics is included in these models.Our results indicate that misunderstanding the role of the opportunistic disease may lead to wrong estimates of the overall potential amount of infected individuals. In the case of constant recovery rate, an expression measuring this discrepancy is derived, as well as conditions on the opportunistic disease imposing a coinfection endemic state on a primary disease otherwise tending to disappear. The case of saturating treatment rate adds the phenomenon of backward bifurcation, which fosters the presence of endemic coinfection and greater levels of infected individuals. Nevertheless, there are specific situations where increasing the opportunistic disease basic reproduction number helps to eradicate both diseases.

scholarly journals A Stochastic SIS Epidemic Model with Ornstein-Uhlenbeck Process and Brown Motion

2021 ◽  
Author(s):  
Xingzhi Chen ◽  
Baodan Tian ◽  
Xin Xu ◽  
Ruoxi Yang ◽  
Shouming Zhong
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Global Solutions ◽  
Uhlenbeck Process ◽  
Sis Epidemic Model ◽  
Ornstein Uhlenbeck Process ◽  
Brown Motion ◽  
Sis Epidemic ◽  
The Basic Reproduction Number

Abstract This paper studies a stochastic differential equation SIS epidemic model, disturbed randomly by the mean-reverting Ornstein-Uhlenbeck process and Brownian motion. We prove the existence and uniqueness of the positive global solutions of the model and obtain the controlling conditions for the extinction and persistence of the disease. The results show that when the basic reproduction number Rs0 < 1, the disease will extinct, on the contrary, when the basic reproduction number Rs0 > 1, the disease will persist. Furthermore, we can inhibit the outbreak of the disease by increasing the intensity of volatility or decreasing the speed of reversion ϑ, respectively. Finally, we give some numerical examples to verify these results.

scholarly journals Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

F1000Research ◽  
2021 ◽  
Vol 10 ◽  
pp. 518
Author(s):  
Christopher Saaha Bornaa ◽  
Baba Seidu ◽  
Yakubu Ibrahim Seini
Keyword(s):  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Probability ◽  
Transmission Dynamics ◽  
Early Interventions ◽  
Coronavirus Infection ◽  
The Impact ◽  
Disease Free

A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

THRESHOLD DYNAMICS IN A PERIODIC SVEIR EPIDEMIC MODEL

2011 ◽  
Vol 04 (04) ◽  
pp. 493-509 ◽  
Author(s):  
JINLIANG WANG ◽  
SHENGQIANG LIU ◽  
YASUHIRO TAKEUCHI
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Threshold Dynamics ◽  
Globally Asymptotically Stable ◽  
Infectious Risk ◽  
Persistence Theory ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

scholarly journals The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Time Average ◽  
The Difference ◽  
Disease Free

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction numberR0. We deduce the globally asymptotic stability of the disease-free equilibrium whenR0≤ 1and the perturbation is small, which means that the disease will die out. WhenR0>1, we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

scholarly journals Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

F1000Research ◽  
2021 ◽  
Vol 10 ◽  
pp. 518
Author(s):  
Christopher Saaha Bornaa ◽  
Baba Seidu ◽  
Yakubu Ibrahim Seini
Keyword(s):  
Death Rate ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Transmission Probability ◽  
Transmission Dynamics ◽  
Early Interventions ◽  
Coronavirus Infection ◽  
The Impact ◽  
Disease Free

A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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