The extinction and persistence of the stochastic SIS epidemic model with vaccination

Zhao, Yunlong; Jiang, Daqing; O’Regan, Donal

doi:10.1016/j.physa.2013.06.009

Cited by 130 publications

(60 citation statements)

References 22 publications

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“…The definition of the persistence in the mean was initially proposed for the deterministic system [25] and was also used for the stationary stochastic system with ergodicity [19,20]. According to the ergodicity for the stationary process, the time average in the long time limit is equal to the ensemble expectation over phase space, so we introduce the notation 2 ( + ) > 1,…”

Section: Persistencementioning

confidence: 99%

“…Ji et al investigated a stochastic multigroup SIR model with fluctuations around the transmission coefficient and the death rate of each subpopulation in [18,19] separately. Zhao et al [20] discussed the extinction and persistence of the stochastic SIS epidemic model with vaccination. However, among all these researches, there are few literatures considering the parameter perturbation in SIS epidemic model with saturated incidence.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Chen

Kang

2014

Abstract and Applied Analysis

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We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and its pth moment exponential stability, which indicates that the pth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

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Section: Persistencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Chen

Kang

2014

Abstract and Applied Analysis

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“…In this section, we use the Lyapunov analysis method to show that the solution of system (1.2) is positive and global as mentioned in Refs. [7][8][9].…”

Section: Existence and Uniqueness Of The Positive Solutionmentioning

confidence: 99%

Dynamics of the stochastic low concentration trimolecular chemical reaction model

Chen

Jiang

2014

J Math Chem

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In this paper, we will introduce a low concentration trimolecular stochastic chemical reaction model. We will prove that the solution of the system is positive and global. And then we draw a conclusion that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, we test our theory conclusion by simulations. It is interesting that no matter what states the unique equilibrium of ordinary differential equation model appears, and regardless of whether the limit cycle exists, our stochastic model is always ergodic.

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“…Recently, the inequalities techniques have been widely used to impulsive differential systems [6][7][8][9][10], stochastic differential systems [11][12][13][14][15][16] and impulsive stochastic differential systems [17][18][19][20], thus some new and interesting results have been obtained.…”

Section: Introductionmentioning

confidence: 99%

Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function

2018

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In this paper, we propose a new nonlinear stochastic SIRS epidemic model with standard incidence rate and saturated treatment function. The main purpose of this paper is to investigate the threshold dynamics of the nonlinear stochastic SIRS epidemic model by making use of stochastic inequality techniques. By using Lyapunov methods and Itô's formula, we first prove the existence and uniqueness of a global positive solution for the corresponding limiting system. Furthermore, we obtain sufficient conditions for the extinction and persistence in mean of the nonlinear stochastic SIRS epidemic model by using the techniques of a series of stochastic inequalities. Finally, we provide some numerical simulations to illustrate the performance of our theoretical findings.

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The extinction and persistence of the stochastic SIS epidemic model with vaccination

Cited by 130 publications

References 22 publications

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Dynamics of the stochastic low concentration trimolecular chemical reaction model

Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function

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