A stochastic SICA model for HIV/AIDS transmission

Stability of a budworm growth model with random perturbations

The effect of environmental noise on threshold dynamics for a stochastic viral infection model with two modes of transmission and immune impairment

Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV

In daily lives, when emergencies occur, rumors will spread widely on the internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens’ behavior and attitude, which make the transmission rates of these information among social network groups be not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation. Firstly, increasing noise intensity can effectively reduce rumor propagation when . That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens’ opinions, thus reducing the degree of rumor propagation. Secondly, speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen (i.e., increase the value of β and decrease the value of α and γ), which can effectively curb the rumor propagation.

<abstract><p>In this paper, a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps is studied. Firstly, it is proved that the model has a unique global solution. Secondly, sufficient conditions for weak persistence in the mean and extinction of prey and predator populations are established. Finally, sufficient conditions for the existence and uniqueness of ergodic stationary distribution are established. Moreover, several numerical simulations are presented to illustrate the main results.</p></abstract>

This paper is concerned with a stochastic Leslie–Gower Holling-type II predator–prey system with regime switching and prey harvesting. Firstly, sharp sufficient criteria for the extinction and the existence of a unique ergodic stationary distribution of the system are established, and the convergence of transition probability of the solution to the stationary distribution is estimated. Then, sharp sufficient criteria for the existence of optimal harvesting strategy are obtained. Some critical influences of random perturbations on the existence of a unique ergodic stationary distribution, extinction and optimal harvesting strategy are revealed. Finally, several numerical simulations are presented to illustrate the theoretical results.

In this paper, we study the dynamical behavior of a stochastic two-compartment model of [Formula: see text]-cell chronic lymphocytic leukemia, which is perturbed by white noise. Firstly, by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then, conditions for extinction of the disease are derived. Furthermore, numerical simulations are presented for supporting the theoretical results. Our results show that large noise intensity may contribute to extinction of the disease.

Stochastic mutualism model with Lévy jumps

Modeling and dynamic analysis of a stochastic mutualism model with distributed delays

In this paper, we consider a stochastic food chain chemostat model with variable yields. First, we prove the stochastic model has a unique global positive solution. Second, by employing suitable Lyapunov functions, Itô[Formula: see text] formula and some other important inequalities, the existence of a unique ergodic stationary distribution of a stochastic food chain chemostat model is researched, which can help us better understand the statistical characteristics of stochastic food chain chemostat models. Second, we investigate the extinction of the microorganism and the bacteria. Moreover, the case of extinction for bacteria but persistence for microbial species is considered. Finally, some numerical simulations are carried out to illustrate our theoretical results and the influence of the variable yields on the microorganism and the bacteria.

A stochastic epidemic model with infectivity rate in incubation period and homestead–isolation on the susceptible is developed with the aim of revealing the effect of stochastic white noise on the long time behavior. A good understanding of extinction and strong persistence in the mean of the disease is obtained. Also, we derive sufficient criteria for the existence of a unique ergodic stationary distribution of the model. Our theoretical results show that the suitably large noise can make the disease extinct while the relatively small noise is advantageous for persistence of the disease and stationary distribution.

Dynamics of a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps

This paper deals with the stochastic Zika virus model within the human and mosquito population. Firstly, we prove that there exists a global positive solution. Further, we found the condition for a viral infection to be extinct. Besides that, we discuss the existence of a unique ergodic stationary distribution through a suitable Lyapunov function. The stationary distribution validates the occurrence of infection in the population. From that, we obtain the threshold value for prevail and disappear of disease within the population. Through the numerical simulations, we have verified the reproduction ratio R 0 S ${R}_{0}^{S}$ as stated in our theoretical findings.

Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation

A general stochastic compartment model for cholera with higher-order perturbation is proposed, which incorporates direct and indirect transmission by contaminated water. Nonlinear incidence, multiple stages of infection, multiple states of pathogen, and second-order white-noises perturbation are introduced into the model, which includes and extends the existing cholera model. The existence and ergodicity of the stationary distribution for the cholera system are obtained by constructing a suitable Lyapunov function, which determines a sharp critical value R0s corresponding to the basic productive number R0 of the ordinary differential equation. The results show that, if R0s>11$\\end{document}]]>, the system has a unique and ergodic stationary distribution, which implies the persistence of the diseases. Our general results are applied to a cholera system with a Holling type-II functional response.