The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence

Yang, Qingshan; Jiang, Daqing; Shi, Ning; Ji, Chunyan

doi:10.1016/j.jmaa.2011.11.072

Cited by 189 publications

(124 citation statements)

References 46 publications

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“…It includes a study of the asymptotic probability distribution of the variables S k . In the paper [17] is proved a theorem on the asymptotic value of the time average of the deviation from the disease-free equilibrium, subject to certain upper bounds on the perturbation parameters and with R 0 assumed to not exceed unity. The same paper gives results on limits of time averages, in particular, m value.…”

Section: Resultsmentioning

confidence: 99%

“…The model used in this paper can be obtained as special cases of models in Refs. [17,18]. We prove a theorem on almost sure exponential stability, which shows that the stochastic perturbation enhances stability of the disease-free equilibrium.…”

Section: Introductionmentioning

confidence: 95%

“…[7, p. 39] has only two of the four state variables carrying stochastic perturbations while in the model of Ref. [17] every variable is perturbed. There are models in which the transmission rate from susceptible to infected is perturbed such as in the Refs.…”

Section: The Seir Modelmentioning

confidence: 99%

“…[17,18], the non-zero values for the diffusion parameters associated with the S-classes, complicate the convergence possibilities. Since in our model we have no perturbation to S, convergence of S becomes much more tractable.…”

Section: S(t) E(t) I(t) R(t))mentioning

confidence: 99%

“…In particular, for this type of model, the total population size itself is perturbed directly by white noise. Examples of such models can be found in the works [15,16,10,[17][18][19] for instance. An sde model of an SEIR disease is studied in the paper [17] of Yang et al The papers [19] of Yuan et al and [18] of Yang and Mao study stability of sde multi-group SEIR models.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Stability of an SEIR epidemic model with independent stochastic perturbations

Witbooi

2013

Physica A: Statistical Mechanics and its Applications

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Stability of an SEIR epidemic model with independent stochastic perturbationsPeter J. Witbooi AbstractFor an epidemic model of the type mentioned, we prove a theorem on almost sure exponential stability of the disease-free equilibrium. For small values of the diffusion parameter, σ , we describe the stability of the disease free equilibrium point in terms of an appropriate analogue, R σ , of the basic reproduction number R 0 of the deterministic special case. Whenever σ > 0 then R σ < R 0 . For small values of σ , the stability theorem guarantees almost sure exponential stability whenever R σ < 1. We also discuss the effect of increasing σ .

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: The Seir Modelmentioning

confidence: 99%

Section: S(t) E(t) I(t) R(t))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stability of an SEIR epidemic model with independent stochastic perturbations

Witbooi

2013

Physica A: Statistical Mechanics and its Applications

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Dynamics of DS‐I‐A epidemic model with multiple stochastic perturbations

Liu

Jiang

et al. 2018

Math Methods in App Sciences

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In this paper, we investigate the dynamics behavior of DS-I-A epidemic model with multiple stochastic perturbations. Sufficient conditions for extinction of disease are established. Especially, we conclude that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. At last, some examples and simulations are provided to illustrate our results.KEYWORDS ergodicity, extinction, infectious disease models, multiple stochastic perturbations, stationary Markov process 6024

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New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Lévy disturbance

Kiouach

Sabbar

El-idrissi

2021

Math Methods in App Sciences

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The spread of infectious diseases is a major challenge in our contemporary world, especially after the recent outbreak of Coronavirus disease 2019 (COVID‐19). The quarantine strategy is one of the important intervention measures to control the spread of an epidemic by greatly minimizing the likelihood of contact between infected and susceptible individuals. In this study, we analyze the impact of various stochastic disturbances on the epidemic dynamics during the quarantine period. For this purpose, we present an SIQS epidemic model that incorporates the stochastic transmission and the Lévy noise in order to simulate both small and massive perturbations. Under appropriate conditions, some interesting asymptotic properties are proved, namely, ergodicity, persistence in the mean, and extinction of the disease. The theoretical results show that the dynamics of the perturbed model are determined by parameters that are closely related to the stochastic noises. Our work improves many existing studies in the field of mathematical epidemiology and provides new techniques to predict and analyze the dynamic behavior of epidemics.

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The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence

Cited by 189 publications

References 46 publications

Stability of an SEIR epidemic model with independent stochastic perturbations

Stability of an SEIR epidemic model with independent stochastic perturbations

Dynamics of DS‐I‐A epidemic model with multiple stochastic perturbations

New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Lévy disturbance

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