A stochastic SIRI epidemic model with Lévy noise

Berrhazi, Badr-eddine; Fatini, Mohamed El; Caraballo, Tomás; Pettersson, Roger

doi:10.3934/dcdsb.2018057

Cited by 31 publications

(27 citation statements)

References 45 publications

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“…For practical and realistic reasons, we consider that at the initial state, each compartment of our model is not empty. As in similar work done in [6] and [7] our study is organized as follows. In Section 2, we prove that the equation (13) has a unique global and positive solution.…”

Section: X(t-)mentioning

confidence: 99%

A Stochastic Model with Jumps for Smoking

COULIBALY¹,

N'zi²

2019

JAMCS

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Some stochastic epidemiological models are less significant. They do not take into account some sudden events that could disrupt the behavior of the studied phenomenon. In this work, we introduce a white noise and jumps in a deterministic SIRS model for smoking to take into account of the effects of randomly fluctuation and such sudden factors respectively. First of all we prove that the solution of the stochastic differential equation with jumps of the new modelis positive. Then we study the asymptotic behavior around the smoking-free equilibrium state and the smoking-present equilibrium state of the original deterministic model. Under certain conditions, we show that the solution oscillate respectively around these equilibrium states. We prove that the intensity of these oscillations depends on the magnitude of noise and the jump diffusion coefficient of our stochastic differential equation with jumps. To support our theoretical results, we realise numerical simulations. The observations confirm our conclusions.

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Section: X(t-)mentioning

confidence: 99%

A Stochastic Model with Jumps for Smoking

COULIBALY¹,

N'zi²

2019

JAMCS

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“…The dynamic effects of time delays and stochastic noise on disease outcomes in populations are important research themes in mathematical epidemiology (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Models incorporating systems of delay differential equations have been shown to exhibit more complex dynamics and capture more of the observed biology underlying disease transmission and persistence (see [11,[15][16][17] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Stochastic dynamics in a delayed epidemic system with Markovian switching and media coverage

Liu

Heffernan

2020

Adv Differ Equ

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A stochastic SIR system with Lévy jumps and distributed delay is developed and employed to study the combined effects of Markovian switching and media coverage on stochastic epidemiological dynamics and outcomes. Stochastic Lyapunov functions are used to prove the existence of a stationary distribution to the positive solution. Sufficient conditions for persistence in mean and the extinction of an infectious disease are also shown.

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“…The effect of environmental noise on epidemic models has been studied intensively. For instance, in [6], authors studied the dynamical properties of a stochastic SIRI epidemic model with relapse and media coverage around both equilibria points and proved the existence of a stationary distribution (see also [2,10,11,12]). In [18], authors introduced a stochastic SIRS epidemic model with general incidence rate in a population of varying size and developed sufficient conditions for the extinction and the existence of a unique stationary distribution.…”

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confidence: 99%

On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times

Settati¹,

Lahrouz²,

Jarroudi³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - B

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As it is well known, the dynamics of the stochastic SIRS epidemic model with mass action is governed by a threshold R S . If R S < 1 the disease dies out from the population, while if R S > 1 the disease persists. However, when R S = 1, classical techniques used to study the asymptotic behaviour do not work any more. In this paper, we give answer to this open problem by using a new approach involving some adequate stopping times. Our results show that if R S = 1 then, small noises promote extinction while the large one promote persistence. So, it is exactly the opposite role of the noises in case when R S = 1.2010 Mathematics Subject Classification. 92B05, 60G51, 60H30, 60G57.

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A stochastic SIRI epidemic model with Lévy noise

Cited by 31 publications

References 45 publications

A Stochastic Model with Jumps for Smoking

A Stochastic Model with Jumps for Smoking

Stochastic dynamics in a delayed epidemic system with Markovian switching and media coverage

On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times

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