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Stochastic Differential Equations And Applications

Survival and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment

This paper considers a high-dimensional stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with quarantine-adjusted incidence and the imperfect vaccination. The main aim of this study is to investigate stochastic effects on the SEIQR epidemic model and obtain its thresholds. We first obtain the sufficient condition for extinction of the disease of the stochastic system. Then, by using the theory of Hasminskii and the Lyapunov analysis methods, we show there is a unique stationary distribution of the stochastic system and it has an ergodic property, which means the infectious disease is prevalent. This implies that the stochastic disturbance is conducive to epidemic diseases control. At last, computer numerical simulations are carried out to illustrate our theoretical results.

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Analysis and Numerical Simulations of a Stochastic SEIQR Epidemic System with Quarantine-Adjusted Incidence and Imperfect Vaccination

This study investigates the disease dynamics of a SIRS epidemic model as affected by environment fluctuations and media coverage. A unique global positive solution for the epidemic model is obtained. The stochastic endemic dynamics of the system is discussed by constructing suitable Lyapunov functions, and the ergodic stationary distribution of the stochastic model is established based on the method of Khasminskii. The extinction of the epidemic disease is also analyzed.

Ergodic stationary distribution of a stochastic SIRS epidemic model incorporating media coverage and saturated incidence rate

A class of SIR epidemic model with generalized nonlinear incidence rate is presented in this paper. Temporary immunity and stochastic perturbation are also considered. The existence and uniqueness of the global positive solution is achieved. Sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established. Moreover, the threshold behavior is discussed, and the threshold value R 0 is obtained. We show that if R 0 1 , the disease eventually becomes extinct with probability one, whereas if R 0 > 1 , then the system remains permanent in the mean.

A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity

In this article, we investigate a class of stochastic SIRS-type epidemic models with nonlinear incidence. The threshold value is identified. The existence of the unique global positive solution with any positive initial value is established. The sufficient conditions for the extinction of the disease and the permanence in the mean of the model with probability one are also established. That is, if , then the disease is extinct with probability one under certain parametric restrictions, and if , then the model is permanent in the mean with probability one. Furthermore, when there is not the disease-related death, if , then the existence of unique stationary distribution of the model is obtained.

Threshold Behavior in a Class of Stochastic SIRS Epidemic Models With Nonlinear Incidence

In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed.

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Tingting Tang

Papers

Threshold Behavior in a Class of Stochastic SIRS Epidemic Models With Nonlinear Incidence

Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination