Dynamical Behavior of a Stochastic SIRS Epidemic Model

Hieu, Nguyen Trong; Du, Nguyen Huu; Auger, Pierre; Dang, Nguyen Hai

doi:10.1051/mmnp/201510205

Cited by 41 publications

(27 citation statements)

References 31 publications

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“…Nevertheless, it is easy to see that the method used in this paper can be easily extended to the case where other parameters of model (1.2), such as the recruitment rate Λ and the natural mortality µ, can also change with the switching of environmental regimes, which may be more reasonable for the wildlife population. In addition, by this new method, under weaker conditions, we can directly extend the results in [11,12,15] from two environmental states to any finite ones.…”

Section: Discussionmentioning

confidence: 99%

“…By the modeling techniques of time discretization or branching processes, Bacäer et al [13,14] obtained very well results concerning the definition of the suitable basic reproduction number for the epidemic model with random switching of environmental regimes. Recently, Hieu and Du et al [15] and Greenhalgh, Liang and Mao [16] which was presented by Anderson and May [17] to study the influence of infectious diseases (caused by viruses or bacteria, etc.) on the density of host populations.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth mentioning that the modified method can be applied to the case where all parameters of the model considered in [7,15,16] were derived by the stochastic process r(t), however, here we only allow the transmission rate β of model (1.2) to be disturbed because in reality it may be more sensitive to environmental fluctuations than other parameters of model (1.1) for human populations. In particular, by the new method, under weaker conditions we can directly extend the results in [11,12,15] from two environmental states to any finite ones. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number R 0 is established: the disease can be eradicated almost surely if R 0 < 1, while the disease persists almost surely if R 0 > 1. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When R 0 > 1, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the Ω-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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“…Using the modeling techniques of branching processes, Bacäer et al [1,2] derived excellent results concerning the definition of the basic reproduction number for the epidemic model with random switching of environmental regimes. Recently, Hieu et al [15], Greenhalgh et al [11] and Li et al [24] investigated the effect of Markovian switching on the deterministic SIRS epidemic models, respectively.…”

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confidence: 99%

Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching

Liu¹,

Jiang²,

Hayat³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we consider a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Firstly, we obtain sufficient conditions for extinction of the diseases. Then we establish sufficient conditions for persistence in the mean of the diseases. Moreover, in the case of persistence, we derive sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.

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“…The contribution of Hieu et al [8] investigates the behaviour of the Kernack -MacKendrick epidemiological model in a noisy environment. The authors consider a particular case of noise: the telegraph noise, which switches between two values at random in both the transmission rate and the recovery rate.…”

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confidence: 99%

Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue

Морозов

2015

Math. Model. Nat. Phenom.

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Mathematical modelling in ecology, epidemiology and eco-epidemiology is a vast and constantly growing research field. This is perhaps unsurprising since mathematical models can provide a wide-ranging exploration of the biological problem without a need for experiments which are usually expensive and can be potentially dangerous to ecosystems. The current Special Issue of MMNP presents recent findings and developments in the following areas of theoretical ecology and epidemiology: (i) revealing the biological processes behind observed complex patterns; (iii) modelling the role of dispersal and spatial heterogeneity in species persistence; (ii) applications of bifurcation theory in population dynamics and epidemiology; (iv) the role of selection in self-replicating biological and ecological systems; (v) sensitivity, predictability and control of biological models in a noisy environment. It is important to emphasize that many of the contributions to this issue are not limited to one of the above topics, but rather lie at the interface of several of them.The work of D. Ahmed and S. Petrovskii [1] studies the movement of insects whose spatial dispersal is a random variable and is described by a certain probability distribution. Understanding the mechanisms of insects movement behaviour is a highly relevant problem from the practical point of view, since it can help us to correctly interpret the results of insect trapping, which is the basis of many current pest monitoring programs [17]. There is a lively ongoing debate in the literature regarding the possibility of Lévy-type movement behaviour by animals, as opposed to the Brownian motion, and various mechanisms have been proposed to explain the observed patterns [3,16,24]. Ahmed and Petrovskii provide an alternative description of Levy-type movement by introducing time-dependent diffusivity. In particular, the authors claim that the trap counts of insects whose dynamics is given by an example of a Lévy-type distribution -a Cauchy type random walk -can be successfully modelled via a different movement pattern: classical Brownian motion with a time-dependent diffusion coefficient. Thus, the authors provide an excellent example of how two completely different mechanisms can describe the same pattern of field observations.Among the central topics in theoretical ecology is the understanding of the role of space in population dynamics and species persistence across different time scales [6,7,11,12]. In this issue, three contributions [2,4,20] directly focus on the role of space in population dynamics.Modelling the mechanisms of biological aggregation and patchiness is a pertinent topic in theoretical ecology, with various models suggested [6,14,19]. In their work, R. Eftimie and A. Coulier [4] explore the effects of communication between organisms on the formation and structure of large biological agc EDP Sciences, 2015 Article published by EDP Sciences and available at

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Dynamical Behavior of a Stochastic SIRS Epidemic Model

Cited by 41 publications

References 31 publications

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching

Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue

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