Quasi-stationary probability distribution for the SIR epidemic model

2021 ◽  
pp. 2150084
Author(s):  
Fahima Ouicher ◽  
Tewfik Kernane
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Stationary Probability ◽  
Positive Probability ◽  
Sir Epidemic Model ◽  
Stationary Probability Distribution ◽  
Sir Epidemic ◽  
Stochastic Epidemic ◽  
Number Formula ◽  
Quasi Stationary Distribution

In this paper, we propose two new approximations to the joint quasi-stationary distribution (QSD) of the number susceptible and infected individuals in the Susceptible-Infected-Recovered (SIR) stochastic epidemic model and we derive the marginal QSD of the infected individuals. These two approximations depend on the basic reproduction number [Formula: see text] and give a positive probability of the QSD to all the transient states. Numerical comparisons are presented to check the accuracy of these approximations.

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.

scholarly journals Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaodong Wang ◽  
Chunxia Wang ◽  
Kai Wang
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Media Coverage ◽  
Existence And Uniqueness ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stochastic Sir

AbstractIn this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number $R_{0}$ R 0 which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.

THE DYNAMICS OF THE CONSTANT AND PULSE BIRTH IN AN SIR EPIDEMIC MODEL WITH CONSTANT RECRUITMENT

2007 ◽  
Vol 15 (02) ◽  
pp. 203-218 ◽  
Author(s):  
WENJUN CAO ◽  
ZHEN JIN
Keyword(s):  
Epidemic Model ◽  
Disease Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Free Solution ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stable Period ◽  
Number Formula ◽  
Globally Stable

In this paper, an SIR epidemic model with constant recruitment is considered. The dynamic behavior of this disease model with constant and pulse birth are analyzed. With constant birth, the infection-free equilibrium is locally and globally stable when the basic reproductive number R0 < 1. However, with pulse birth the system converges to a stable period solution with the number of infectious individuals equal to zero. Furthermore, the local and global stability of the periodic infection-free solution is obtained if the basic reproductive number [Formula: see text]. Numerical simulation shows that the periodic infection-free solution is unstable and the disease will persist when [Formula: see text]. The effectiveness of the constant and pulse birth to eliminating the disease are compared.

scholarly journals Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
The Stability ◽  
The Basic Reproduction Number

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

Impact of media coverage on a fractional-order SIR epidemic model

2021 ◽  
Author(s):  
Chandan Maji
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Media Coverage ◽  
Lambert W Function ◽  
Order System ◽  
Media Effect ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Number Formula

In this work, we formulated and analyzed a fractional-order epidemic model of infectious disease (such as SARS, 2019-nCoV and COVID-19) concerning media effect. The model is based on classical susceptible-infected-recovered (SIR) model. Basic properties regarding positivity, boundedness and non-negative solutions are discussed. Basic reproduction number [Formula: see text] of the system has been calculated using next-generation matrix method and it is seen that the disease-free equilibrium is locally as well as globally asymptotically stable if [Formula: see text], otherwise unstable. The existence of endemic equilibrium point is established using the Lambert W function. The condition for global stability has been derived. Numerical simulation suggests that fractional order and media have a large effect on our system dynamics. When media impact is stronger enough, our fractional-order system stabilizes the oscillation.

DYNAMIC ANALYSIS OF AN SIR MODEL WITH STOCHASTIC PERTURBATIONS

2013 ◽  
Vol 06 (06) ◽  
pp. 1350041
Author(s):  
ZHENJIE LIU ◽  
JINLIANG WANG ◽  
YALAN XU ◽  
GUOQIANG LI
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Sir Model ◽  
Sir Epidemic Model ◽  
Stochastic Perturbations ◽  
Sir Epidemic ◽  
Stochastic Epidemic ◽  
Differential Infectivity ◽  
Global Positive Solution ◽  
Stochastic Epidemic Model

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution.

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 Global stability for a discrete SIR epidemic model with delay in the general incidence function

2019 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Guiro Aboudramane ◽  
Dramane Ouedraogo ◽  
Harouna Ouedraogo
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Step Size ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Sir Epidemic ◽  
Disease Free ◽  
Boundedness Of Solution

In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

PERMANENCE OF SOME DISCRETE EPIDEMIC MODELS

2009 ◽  
Vol 02 (04) ◽  
pp. 443-461 ◽  
Author(s):  
MASAKI SEKIGUCHI
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Epidemic Models ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Seirs Epidemic Model ◽  
Two Delays ◽  
Appl Math ◽  
The Basic Reproduction Number

In this paper, we discuss some discrete epidemic models, that is, discrete SIR epidemic model with no delay, discrete SIR epidemic model with one delay and discrete SEIRS epidemic model with two delays. By applying the method given in Wang, Appl. Math. Lett. 15(2002) 423–428, we prove the permanence of these discrete epidemic models. These sufficient conditions are similar to the continuous epidemic models, that is, the basic reproduction number of each model is larger than one.

scholarly journals Stability analysis in a delayed SIR epidemic model with a saturated incidence rate

2010 ◽  
Vol 15 (3) ◽  
pp. 299-306 ◽  
Author(s):  
A. Kaddar
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Basic Reproduction Number

We formulate a delayed SIR epidemic model by introducing a latent period into susceptible, and infectious individuals in incidence rate. This new reformulation provides a reasonable role of incubation period on the dynamics of SIR epidemic model. We show that if the basic reproduction number, denoted, R0, is less than unity, the diseasefree equilibrium is locally asymptotically stable. Moreover, we prove that if R0 > 1, the endemic equilibrium is locally asymptotically stable. In the end some numerical simulations are given to compare our model with existing model.

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