Sanling Yuan

Survival and stationary distribution of a SIR epidemic model with stochastic perturbations

Abstract In this paper, the dynamics of a SIR epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, when R 0 > 1 , we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. When R 0 ⩽ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic system is established under suitable conditions. Finally, numerical simulations are carried out to illustrate the theoretical results.

Threshold behavior of a stochastic SIS model with Levy jumps

In this paper, the dynamics of a stochastic SIS model with Levy jumps are investigated. We first prove that this model has a unique global positive solution starting from the positive initial value. Then, taking the accumulated jump size into account, we find a threshold of the model, denoted by R ? 0 , which completely determines the extinction and prevalence of the disease: if R ? 0 < 1 , the disease dies out exponentially with probability one; if R ? 0 1 , the solution of the model tends to a point in time average which leads to the stochastical persistence of the disease. From the view of epidemiology, the existence of threshold is useful in determining treatment strategies and forecasting epidemic dynamics. Moreover, we find that Levy noise can suppress disease outbreak. Finally, we introduce some numerical simulations to support the main results obtained.

Improved stability conditions for a class of stochastic Volterra-Levin equations

In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear stochastic Volterra-Levin equations by using fixed point theory. Several sufficient conditions are established for ensuring that the equation is mean square asymptotically stable as well as exponentially stable. The main results are new which generalize and improve some well-known results in Burton (2006) 4 and Luo (2010) 18. Finally, two examples are given to illustrate our results.

Survival and Stationary Distribution in a Stochastic SIS Model

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: when , we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; when , we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

A note on persistence and extinction of a randomized food-limited logistic population model

This paper addresses the issue of the asymptotic behavior for a non-autonomous randomized food-limited logistic population model. Several sufficient conditions are formulated and proved for p-moment persistence and extinction of the population, as well as in sense of almost sure. Results show that food-limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model. Some previously known results are improved. Numerical simulations are provided to support the results.

The effect of media coverage on threshold dynamics for a stochastic SIS epidemic model

Abstract Media coverage is one of the important measures for controlling infectious diseases, but the effect of media coverage on diseases spreading in a stochastic environment still needs to be further investigated. Here, we present a stochastic susceptible–infected–susceptible (SIS) epidemic model incorporating media coverage and environmental fluctuations. By using Feller’s test and stochastic comparison principle, we establish the stochastic basic reproduction number R 0 s , which completely determines whether the disease is persistent or not in the population. If R 0 s ≤ 1 , the disease will go to extinction; if R 0 s = 1 , the disease will also go to extinction in probability, which has not been reported in the known literatures; and if R 0 s > 1 , the disease will be stochastically persistent. In addition, the existence of the stationary distribution of the model and its ergodicity are obtained. Numerical simulations based on real examples support the theoretical results. The interesting findings are that (i) the environmental fluctuation may significantly affect the threshold dynamical behavior of the disease and the fluctuations in different size scale population, and (ii) the media coverage plays an important role in affecting the stationary distribution of disease under a low intensity noise environment.

Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat

Abstract This paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment. Based on the Feller property, sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in [11, 12] (See Lemma 2.4 for details), which closes the gap left by the Lyapunov function. Further, we obtain a sufficient condition for the extinction of the organism based on two noise-induced parameters: an analogue of the feed concentration S* and the break-even concentration λ. Results indicate that both noises have negative effects on persistence of the microorganism.