scholarly journals A SIRS Epidemic Model Incorporating Media Coverage with Random Perturbation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Wenbin Liu
Keyword(s):  
Stochastic Model ◽  
Epidemic Model ◽  
Media Coverage ◽  
Complex Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Endemic Equilibrium ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
The Stability

We investigate the complex dynamics of a SIRS epidemic model incorporating media coverage with random perturbation. We first deal with the boundedness and the stability of the disease—free and endemic equilibria of the deterministic model. And for the corresponding stochastic epidemic model, we prove that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Furthermore, we perform some numerical examples to validate the analytical finding, and find that if the conditions of stochastic stability are not satisfied, the solution for the stochastic model will oscillate strongly around the endemic equilibrium.

scholarly journals Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongli Cai ◽  
Xixi Wang ◽  
Weiming Wang ◽  
Min Zhao
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Stochastic Dynamics ◽  
Complex Dynamics ◽  
Deterministic Model ◽  
Mass Action ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
The Stability ◽  
Disease Free

We investigate the complex dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectious individuals to that of susceptible individuals. We first deal with the boundedness, dissipation, persistence, and the stability of the disease-free and endemic points of the deterministic model. And then we prove the existence and uniqueness of the global positive solutions, stochastic boundedness, and permanence for the stochastic epidemic model. Furthermore, we perform some numerical examples to validate the analytical findings. Needless to say, both deterministic and stochastic epidemic models have their important roles.

scholarly journals An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Huitao Zhao ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Orbits ◽  
Epidemic Model ◽  
Media Coverage ◽  
Critical Value ◽  
Endemic Equilibrium ◽  
Sirs Epidemic Model ◽  
The Stability ◽  
Disease Free

An SIRS epidemic model incorporating media coverage with time delay is proposed. The positivity and boundedness are studied firstly. The locally asymptotical stability of the disease-free equilibrium and endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcate are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction numberR0<1. However, whenR0>1, the stability of the endemic equilibrium will be affected by the time delay; there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay increases through a critical value. Finally, some examples for numerical simulations are also included.

scholarly journals Global Analysis of SIRS Epidemic Model With General Incidence Function and Incomplete Recovery Rates Stochastical Model

2020 ◽  
Vol 12 (6) ◽  
pp. 100
Author(s):  
Dramane Ouedraogo ◽  
Ali Traore ◽  
Aboudramane Guiro
Keyword(s):  
Stochastic Model ◽  
Global Stability ◽  
Stochastic Models ◽  
Epidemic Model ◽  
Global Analysis ◽  
Endemic Equilibrium ◽  
Epidemic Models ◽  
Local And Global Stability ◽  
Sirs Epidemic Model ◽  
Disease Free

In this paper, deterministic and stochastic models are developped for a class of SIRS epidemic models. Firstly, The conditions for the existence, local and global stability of the disease-free equilibrium and endemic equilibrium are obtained. Secondly, we built the stochastic model. The populations are computationally simulated under various conditions. Comparisons are made between the deterministic and stochastic model.

scholarly journals On the Stability of the Endemic Equilibrium of A Discrete-Time Networked Epidemic Model

2020 ◽  
Vol 53 (2) ◽  
pp. 2576-2581
Author(s):  
Fangzhou Liu ◽  
Shaoxuan CUI ◽  
Xianwei Li ◽  
Martin Buss
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Endemic Equilibrium ◽  
The Stability

Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Junna Hu ◽  
Buyu Wen ◽  
Ting Zeng ◽  
Zhidong Teng
Keyword(s):  
Epidemic Model ◽  
Regime Switching ◽  
Markov Switching ◽  
Threshold Value ◽  
Open Problems ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
Lévy Jumps ◽  
White Noises

Abstract In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

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 transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (3) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

General stochastic epidemic with recovery

1975 ◽  
Vol 12 (01) ◽  
pp. 29-38 ◽  
Author(s):  
L. Billard
Keyword(s):  
Stochastic Model ◽  
Finite Number ◽  
Epidemic Model ◽  
The State ◽  
Stochastic Epidemic ◽  
General Stochastic

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.

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.

Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Modeste N’zi ◽  
Jacques Tano
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Deterministic Model ◽  
Random Noise ◽  
Stochastic Version ◽  
Noise Perturbation ◽  
Sirs Epidemic Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free

AbstractIn this paper, we formulate an epidemic model for the spread of an infectious disease in a population of varying size. The total population is divided into three distinct epidemiological subclass of individuals (susceptible, infectious and recovered) and we study a deterministic and stochastic models with saturated incidence rate. The stochastic model is obtained by incorporating a random noise into the deterministic model. In the deterministic case, we briefly discuss the global asymptotic stability of the disease free equilibrium by using a Lyapunov function. For the stochastic version, we study the global existence and positivity of the solution. Under suitable conditions on the intensity of the white noise perturbation, we prove that there are a

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