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In this paper, by using Itô's formula, we study the dynamic behaviors of a stochastic human immune deficiency virus (HIV) model, which describes Acquired Immune Deficiency Syndrome (AIDS) with the effect of treatment. First, we discuss the global positivity of the solution to such a model with positive initial value. Then, we consider the limiting behavior of the solution. Moreover, we obtain the sufficient condition to guarantee the extinction of AIDS. Meanwhile, by constructing an appropriate Lyapunov function, we prove that the solution has a unique ergodic stationary distribution under a certain assumptions. Finally, we illustrate the validity of our results by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a periodic stochastic Lotka‐Volterra model with distributed delay and dispersal is investigated. First, by Itô's formula, we show that the solution with any positive initial value is global and positive. Then, we obtain the sufficient conditions guaranteeing the extinction of the system. Furthermore, we discuss the existence of the nontrivial periodic solution by Khasminskii theory. Finally, numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Motivated by P.L. Lions' seminal course on Mean Field Games (MFG) and the associated Master equation at Collège de France [19] , we investigate in the case of absence of control mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process. This BDSDE can be interpreted as randomly perturbed recursive cost functional which we associate with a forward mean-field SDE. Defining the value function V = V (t , x , μ) through this BDSDE, we prove that V = V (t , x , μ) is the unique classical solution of a backward stochastic PDE over [ 0 , T ] × R d × P 2 (R d). This backward SPDE extends in the non-control case the Master equation studied in earlier works by different authors. The associated MFG can be interpreted as a forward dynamics endowed with a randomly perturbed recursive cost functional, the solution of the MFG system (u (t , x) , m (t)) remains also in this extended case related with V = V (t , x , μ) by the relation u (t , x) = V (t , x , m (t)). Our main objective is to characterise the random field V as classical solution of the backward SPDE. For this we study the regularity of V over [ 0 , T ] × R d × P 2 (R d). However, unlike the pioneering paper on BDSDEs by Pardoux-Peng [25] where V does not depend on the measure, here we have only an L 2 -regularity of V with respect to its variables. Other difficulties to overcome and subtle technical proofs are related with the study of the L 2 -regularity and, hence, that of the solution of the underlying mean-field BDSDE. It adds that our BDSDE needs weaker assumptions on the coefficients than in [25]. Finally, let us point out that for our approach we extend the classical mean-field Itô formula to smooth functions of solutions of mean-field BDSDEs. This formula contains earlier (mean-field) Itô formulas as special cases. [ABSTRACT FROM AUTHOR]

We derive a change of variable formula for C 1 functions U : R + × R m → R whose second order spatial derivatives may explode and not be integrable in the neighbourhood of a surface b : R + × R m − 1 → R that splits the state space into two sets C and D. The formula is tailored for applications in problems of optimal stopping where it is generally very hard to control the second order derivatives of the value function near the optimal stopping boundary. Differently to other existing papers on similar topics we only require that the surface b be monotonic in each variable and we formally obtain the same expression as the classical Itô's formula. [ABSTRACT FROM AUTHOR]

Consider the fluctuation of environmental noise and the incubation period of virus, a stochastic vector-host model with incubation delay is proposed to describe the transmission of Dengue fever virus (DFVs) between humans and mosquitoes. By constructing suitable stochastic Lyapunov functional, the existence and uniqueness of global positive solutions of this model are proved. Furthermore, some sufficient conditions are obtained for the asymptotic pathwise estimation of solutions. In addition, we also consider the asymptotic pathwise estimation of solutions, the existence and uniqueness of stationary distribution of this model without delay, which imply that disease is stochastic persistent. Finally, numerical simulations explain the theoretical results and discuss the effects of stochastic perturbations and delays on DFVs transmission. Our results imply that the spread of DFVs presents many unpredictability under the influence of stochastic perturbation, and the reproduction number can no longer be used as a threshold condition for the extinction or persistence of disease. Moreover, ignoring the incubation delay will not only overestimate the risk of disease outbreaks but also underestimate the duration of diseases. Rational application of stochastic perturbations on disease transmission and incubation periods will play a very important role in vector-borne disease control. [ABSTRACT FROM AUTHOR]

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 ′ s 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. [ABSTRACT FROM AUTHOR]

We establish Itô's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on Itô processes. Our approach is to first establish Itô's formula for cylindrical functions and then extend it to the general case via function approximation and localization techniques. This general form of Itô's formula enables the derivation of dynamic programming equations and verification theorems for McKean–Vlasov controls with jump diffusions and for McKean–Vlasov mixed regular–singular control problems. It also allows for generalizing the classical relationship between the maximum principle and the dynamic programming principle to the McKean–Vlasov singular control setting, where the adjoint process is expressed in terms of the derivative of the value function with respect to the probability measures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Stratonovich-Henstock integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also formulate a version of Itô's formula for this integral. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear stochastic eco-epidemic model is presented to study the dynamics of harvested prey–predator model with group defence. One of the key roles of forming a group is to defend themselves from the attack of the predators. Since prey population spreads infection through direct contact; we consider two groups of prey population. Using some certain assumptions, a stochastic model is formulated by considering noisy intrinsic growth rate of susceptible pest population, death rate of infected pest and predator populations, respectively. Next, the existence and uniqueness of positive solution are performed using comparison theorem of stochastic differential equations (SDEs). The boundedness and uniform continuity of the system are analyzed. Then, the bionomic equilibrium and optimal harvesting policy are obtained by using stochastic maximum principle. We verify the conditions of extinction and persistence in mean with the numerical results. To verify our consideration with the reality, the effects of harvesting effort and catchability coefficient are observed. In addition, infected and uninfected regions are reported to specify the infection free system. [ABSTRACT FROM AUTHOR]

By introducing some parameters perturbed by white noises, we propose a class of stochastic inertial neural networks in random environments. To describe dynamics behaviours of this stochastic system accurately, we introduce the concept of mean-square exponential stability. By constructing two Lyapunov–Krasovskii functionals, we establish the mean-square exponential stability on the addressed system, which extend and improve the latest results. In addition, an example with numerical simulation is provided to support the theoretical findings. [ABSTRACT FROM AUTHOR]

The double epidemic study is one of the critical studies in recent times as humankind experiences various simultaneous spreads of diseases. Reproduction numbers are important, which helps to derive sufficient conditions for extinction, persistence and co-persistence of diseases. This work aims to establish the stationary distribution of a stochastic double epidemic model comprised of SIR and SIRS transmission hypothesis through saturated incidence rate, with the aid of Markov semigroup theory. By constructing a suitable Lyapunov functional, the required result is sufficient by conditions on reproduction numbers. • Stationary distribution of a double epidemic model using Markov Semigroup Theory. • Long-term behaviour of a double epidemic model with SIR and SIRS transmission mechanisms. [ABSTRACT FROM AUTHOR]

A class of stochastic SIRS infectious disease models with both logistic birth and Markov switching were investigated. The uniqueness of the existence of a globally positive solution to the stochastic infectious disease model was first analyzed through construction of suitable V functions and then by means of Itô’s formula. Afterwards, the results of the existence of an ergodic smooth distribution for the solution of the model and the sufficient conditions for the extinction of the disease were discussed. Finally, numerical examples were given to illustrate the conclusions. [ABSTRACT FROM AUTHOR]

By introducing some parameters perturbed by white noises, we propose a class of stochastic inertial neural networks in random environments. Constructing two Lyapunov–Krasovskii functionals, we establish the mean-square exponential input-to-state stability on the addressed model, which generalizes and refines the recent results. In addition, an example with numerical simulation is carried out to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with vaccination effect is investigated. Our purpose is to study the dynamical behaviors of the model. First, the existence and uniqueness of the positive solution of the model is proved. Then, the dynamic properties of the model, such as extinction and persistence, are analyzed by regarding certain conditions on the parameters. Some sufficient conditions for extinction and persistence are given by using Itô's formula and martingale methods. The results show that low-intensity white noise may cause disease to spread, while high-intensity white noise may drive infective, quarantined individuals to extinction. Finally, numerical simulations are illustrated to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The estimates of N. V. Krylov for distributions of stochastic integrals by means of the Ld-norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W2d. We also show how to use these estimates for the investigation of L2-convergence of solutions of optional SDE's. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a two-group SVIR epidemic model with random effect. Firstly, the two-group SVIR epidemic model with random perturbation of natural death rate is established. The existence and uniqueness of positive solution are proved by using stopping time theory and the Lyapunov analysis method. Secondly, a property of the system solution is obtained by using the law of strong numbers and the continuous local martingale. Finally, a new combination of Lyapunov functions is applied. The solution of the model we obtained is oscillating around a steady state if the basic reproduction number is less than one, which is the disease-free equilibrium of the corresponding deterministic model. A numerical simulation is presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of an invariant measure for a stochastic reaction–diffusion equation of the type d u = (Δ u + f (x , u)) d t + σ (x , u) d W (x , t) , where f and σ are nonlinear maps and W is an infinite dimensional Q-–Wiener process. Our emphasis is on unbounded domain R d . Under a very mild dissipation assumption, we show the existence of a solution which is bounded in probability using a general Ito's formula. In addition, we investigate a type of equation for which bounded solutions may be obtained as a limit of an iteration scheme. Together with the compactness property of the heat equation semigroup, stochastic continuity and Feller property of the transition semigroup, the existence of such solution implies the existence of an invariant measure which is an important step in establishing the ergodic behaviour of the underlying physical system. [ABSTRACT FROM AUTHOR]

A stochastic SIQR epidemic model with non-monotonic transmission rate is formulated. The exist ence and uniqueness of global solution to the model are obtained by using the stopping time theory. By using the Liapunov function method and the Ito's formula,the oscillation behavior of the solutions of the stochastic model near the disease-free equilibrium point of the corresponding deterministic model is discussed. Sufficient conditions for the extinction and the existence of a unique stationary distribution are obtained. Finally, the numerical simulations results show the difference of asymptotic behavior between the model solution and the corresponding deterministic model solution. [ABSTRACT FROM AUTHOR]

Robust analysis in the stochastic sense, including robust boundedness and robust stability, has become a momentous problem of stochastic systems. Up to now, almost all existing works about robust boundedness and robust stability require that stochastic perturbations are rooted in continuous time observations of systems states. However, compared with the continuous time noises case, stochastic perturbations of discrete time noises are not only appropriate but also reasonable. Hence, this brief proposes and explores the problems of robust boundedness and robust stability of discrete time noises for stochastic systems satisfying the linear growth condition. The destination of this brief is to answer the question: how much stochastic perturbations of discrete time noises can a bounded or stable system tolerate guaranteeing stochastically perturbed system remains asymptotically bounded or stable. In addition, this brief discusses the application of our results in neural networks. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics property of a stochastic HIV model with Beddington–DeAngelis functional response. It has a unique uninfected steady state. We prove that the model has a unique global positive solution. Furthermore, if the basic reproductive number is not larger than 1, the asymptotic behavior of the solution is stochastically stable. Otherwise, it fluctuates randomly around the infected steady state of the corresponding deterministic HIV model. Finally, some numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇ -delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations. [ABSTRACT FROM AUTHOR]

This paper, studies the class of diffusion processes generated by a first-order continuous-time bilinear processes (COBL(1, 1)) with time-varying coefficients. So, we used the It o ̂ formula approach for examining the L 2 − structure of the process and its powers. In time-invariant case, an expression of the moments of any order are given and the continuous autoregressive (C A R) representation of such version is given, in particular the moments properties of some specifications are however derived. Based on these results we are able to examine the statistical properties as well as we develop an estimation method of the process via the so-called Yule-Walker (YW) type algorithm which relates with unknown parameters of CAR representation. The method is illustrated by a Monte Carlo study and applied to modeling the electricity consumption sampled at each 15 mn in Algeria. [ABSTRACT FROM AUTHOR]

Let L = { L (t , x) , t ≥ 0 , x ∈ R } be the local time of a G-Brownian motion B on a sublinear expectation space (Ω , H , E ˆ). In this paper, we show that the local time L is a rough path of roughness p quasi-surely for any 2

In this paper, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and propose a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. For each species, sharp sufficient conditions for persistence in the mean and extinction are respectively obtained. The results demonstrate that the persistence and extinction of the species have close relationships with the environmental stochasticity. In addition, the theoretical results are numerically illustrated by some simulations. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a version of Itô's formula for the backwards Itô-Henstock integral of an operator-valued stochastic process. Itô's formula is the stochastic analogue of the change of variable for deterministic integrals. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of nonlinear rough paths. Following the ideas of Lyons and Gubinelli, we define the nonlinear rough integral ∫tsW(dr; Yr), where W and Y are only α-Hölder continuous in time with α ∈ (1/3, 1/2 ]. Also, we study the Kunita-type equation Yt =ξ + ∫0t W(dr, Ys), obtaining the local and global existence and uniqueness of the solution under suitable sufficient conditions. As an application, we study transport equations with rough vector fields and observe that the classical solution formula for smooth and Young's cases does not provide a solution to the rough equation. Indeed this formula satisfies a transport equation with additional compensator terms (see (1.7)). [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair (u , ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random regular measure satisfying minimal Skohorod condition. [ABSTRACT FROM AUTHOR]

In this paper, we aim to obtain the Hölder continuous of solutions to stochastic nonlocal heat equations. By using Campanato estimates and Sobolev embedding theorem, we first prove the Hölder continuous of the mild solution of stochastic nonlocal diffusion equations locally in R d in the sense that the solution u belongs to the space C γ (D T ; L p (Ω)). Then by using tail estimates, we obtain the estimates of the mild solution in L p (Ω ; C γ (D T)). Lastly, we prove the Hölder continuous of the mild solution on bounded domain. [ABSTRACT FROM AUTHOR]

Abstract After introducing an entropy formulation for stochastic nonlocal conservation laws via an integral formula, we develop a uniqueness theory by using a doubling variables technique and an existence theory by using a vanishing viscosity method. We further consider the continuous dependence of the entropy solution on initial data. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t (x) = Δ α 2 u t (x) d t + g (t , x) d η t , u 0 = 0 , t ∈ (0 , T ] , x ∈ G , for a random field u : (t , x) ∈ [ 0 , T ] × G ↦ u (t , x) = : u t (x) ∈ R , where Δ α 2 : = − (− Δ) α 2 , α ∈ (0 , 2 ] , is the fractional Laplacian, T ∈ (0 , ∞) is arbitrarily fixed, G ⊂ R d is a bounded domain, g : [ 0 , T ] × G × Ω → R is a joint measurable coefficient, and η t , t ∈ [ 0 , ∞) , is either a Brownian motion or a Lévy process on a given filtered probability space (Ω , F , P ; { F t } t ∈ [ 0 , T ]). To this end, we derive the BMO estimates and Morrey–Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilizing the embedding theory between the Campanato space and the Hölder space, we establish the controllability of the norm of the space C θ , θ / 2 (D ¯) , where θ ≥ 0 , D ¯ = [ 0 , T ] × G ¯. With all these in hand, we are able to show that the q -th order BMO quasi-norm of the α q 0 -order derivative of the solution u is controlled by the norm of g under the condition that η t is a Lévy process. Finally, we derive the Schauder estimate for the p -moments of the solution of the above stochastic fractional heat equations driven by Lévy noise. [ABSTRACT FROM AUTHOR]

We find a pathwise decomposition of a certain class of Brownian semistationary processes (BSS) in terms of fractional Brownian motions. To do this, we specialize in the case when the kernel of the BSS is given by ϕα(x) = L(x)xα with α ∈ (-1/2, 0) ∪ (0, 1/2) and L a continuous function slowly varying at zero. We use this decomposition to study some path properties and derive Itˆo's formula for this subclass of BSS processes. [ABSTRACT FROM AUTHOR]

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution ( Y t , ξ , Z t , ξ , H t , ξ ) , ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) of the split equations are proved. The first and the second order derivatives of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to x , the derivative of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to the measure P ξ , and the derivative of the process ( ∂ μ Y t , x , P ξ ( y ) , ∂ μ Z t , x , P ξ ( y ) , ∂ μ H t , x , P ξ ( y ) ) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L 2 . The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V ( t , x , P ξ ) ≔ Y t t , x , P ξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type. [ABSTRACT FROM AUTHOR]

The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function. [ABSTRACT FROM AUTHOR]

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512-5524 2011), and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in Lp) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370-384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. [ABSTRACT FROM AUTHOR]

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the G -expectation theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a rumor propagation model with L évy noise. We first prove that there exists a positive global solution. Then, the asymptotic behaviors around the rumor‐free equilibrium and rumor‐epidemic equilibrium are obtained. Lastly, simulations verify our results. [ABSTRACT FROM AUTHOR]

The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation. [ABSTRACT FROM AUTHOR]

With the methods of H2/H∞-control theory, in the presence of noise we solve the optimization problem for a multiplicative stochastic system with several external disturbances (the multiperturbation problem) and vector Wiener processes with arbitrary intensity matrices. We obtain matrix differential equations of Riccati type, reducing the original optimization problem to solving these equations. [ABSTRACT FROM AUTHOR]

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. In this paper, we are concerned with a stochastic rumor propagation model. Sufficient conditions for extinction and persistence in the mean of the rumor are established. The threshold between persistence in the mean and extinction of the rumor is obtained. Compared with the corresponding deterministic model, the threshold affected by the white noise is smaller than the basic reproduction number R 0 of the deterministic system. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the local time associated with symmetric α -stable processes is of bounded p -variation for any p > 2 α − 1 partly based on Barlow’s estimation of the modulus of the local time of such processes. The fact that the local time is of bounded p -variation for any p > 2 α − 1 enables us to define the integral of the local time ∫ − ∞ ∞ ▿ − α − 1 f ( x ) d x L t x as a Young integral for less smooth functions being of bounded q -variation with 1 ≤ q < 2 3 − α . When q ≥ 2 3 − α , Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric α -stable processes for 2 3 − α ≤ q < 4 . [ABSTRACT FROM AUTHOR]

We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications. [ABSTRACT FROM AUTHOR]

In this paper, we present the threshold of a stochastic SIQS epidemic model which determines the extinction and persistence of the disease. Furthermore, we find that noise can suppress the disease outbreak. Numerical simulations are also carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper we introduce randomness into a model of AIDS and condom use through the technique of parameter perturbation which is standard in stochastic population modeling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system. [ABSTRACT FROM AUTHOR]

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results. [ABSTRACT FROM AUTHOR]