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We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution. [ABSTRACT FROM AUTHOR]

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]

This paper studies the finite‐time stability (FTS) for stochastic impulsive systems (SIS). The notions of FTS including GKL$$ \mathcal{GKL} $$‐FTS and event‐ GKL$$ \mathcal{GKL} $$‐FTS are proposed for SIS. The comparison principles for GKL$$ \mathcal{GKL} $$‐FTS and event‐ GKL$$ \mathcal{GKL} $$‐FTS are established, respectively. Then the FTS criteria with the settling time estimates are derived. The criteria of GKL$$ \mathcal{GKL} $$‐FTS and event‐ GKL$$ \mathcal{GKL} $$‐FTS relax the requirements on stabilizing flow and jump for FTS in the literature. Finally, two examples are presented to demonstrate the obtained results. [ABSTRACT FROM AUTHOR]

This study investigates a mathematical model encompassing both HIV-1 and HIV-2 epidemics, with a focus on drug resistance and dual saturated incidence rates. We begin by analyzing the local stability of equilibrium points in the deterministic system using Routh-Hurwitz and Jacobian matrix approaches. Furthermore, we establish the existence of a global positive solution for our model. We identify specific parameter conditions that lead to disease extinction. Additionally, we explore the persistence of HIV-1 and HIV-2 infections by examining their mean values. To assess the practical implications of our theoretical analysis, we conduct graphical evaluations under various noise disturbances, employing Milstein's Higher Order Method. This research contributes to a deeper understanding of the dynamics of HIV-1 and HIV-2 co-infection, with potential implications for treatment strategies and disease management.

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]

This study investigates a mathematical model encompassing both HIV-1 and HIV-2 epidemics, with a focus on drug resistance and dual saturated incidence rates. We begin by analyzing the local stability of equilibrium points in the deterministic system using Routh-Hurwitz and Jacobian matrix approaches. Furthermore, we establish the existence of a global positive solution for our model. We identify specific parameter conditions that lead to disease extinction. Additionally, we explore the persistence of HIV-1 and HIV-2 infections by examining their mean values. To assess the practical implications of our theoretical analysis, we conduct graphical evaluations under various noise disturbances, employing Milstein's Higher Order Method. This research contributes to a deeper understanding of the dynamics of HIV-1 and HIV-2 co-infection, with potential implications for treatment strategies and disease management. [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]

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the n th exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô's formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.

The two- and three-dimensional incompressible backward stochastic convective Brinkman–Forchheimer (BSCBF) equations on a torus driven by Lévy noise are considered in this paper. A-priori estimates for adapted solutions of the finite-dimensional approximation of 2D and 3D BSCBF equations are obtained. For a given terminal data, the existence and uniqueness of pathwise adapted strong solutions is proved by using a standard Galerkin (or spectral) approximation technique and exploiting the monotonicity arguments. We also establish the continuity of the adapted solutions with respect to the terminal data. The above results are obtained for the absorption exponent r ∈ [ 1 , ∞) for d = 2 and r ∈ [ 3 , ∞) for d = 3 , and any Brinkman coefficient μ > 0 , Forchheimer coefficient β > 0 , and hence the 3D critical case ( r = 3 ) is also handled successfully. We deduce analogous results for 2D backward stochastic Navier–Stokes equations perturbed by Lévy noise also. [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]

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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]

In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts. [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 investigate the stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs) with general decay rate. By applying the method of the Lyapunov function, the pth moment and almost sure stability with general decay rate of solution for NPSDEs are derived. Finally, an example is presented to show the effectiveness of the proposed methods.

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Campanato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space ℝd) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions belong to the space Cγ (DT; Lp(Ω)) with the optimal Hölder continuity index (which is given explicitly), where DT := [0, T] × D for T > 0, and D ⊂ ℝdd being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in Lp(Ω; Cγ* (DT)). What's more, we give an explicit formula between the two indexes γ and γ*. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criterion to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method is suitable to the case of space-time white noise. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs) with general decay rate. By applying the method of the Lyapunov function, the pth moment and almost sure stability with general decay rate of solution for NPSDEs are derived. Finally, an example is presented to show the effectiveness of the proposed methods. [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]

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]

Abstract 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.

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]

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices. [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]

In this paper, a modified Leslie-Gower Holling-type Ⅱ two-predator one-prey stochastic model in polluted environments with impulsive toxicant input is proposed where we use an Ornstein-Uhlenbeck process to improve the stochasticity of the environment. The sharp sufficient conditions for persistence in the mean and extinction are established. The results reveal that the persistence and extinction of the species have close relationships with the toxicant and environmental stochasticity. In addition, the theoretical results are verified by numerical simulation.

Abstract 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.

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]

In this paper, we aim to study a stochastic process from a macro point of view, and thus, periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations. [ABSTRACT FROM AUTHOR]

This paper is related to the stochastic smoking model for the purpose of creating the effects of smoking that are not observed in deterministic form. First, formulation of the stochastic model is presented. Then the sufficient conditions for extinction and persistence are determined. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, the numerical simulations are shown graphically with the software MATLAB.

Abstract Acknowledging many effects on humans, which are ignored in deterministic models for COVID-19, in this paper, we consider stochastic mathematical model for COVID-19. Firstly, the formulation of a stochastic susceptible–infected–recovered model is presented. Secondly, we devote with full strength our concentrated attention to sufficient conditions for extinction and persistence. Thirdly, we examine the threshold of the proposed stochastic COVID-19 model, when noise is small or large. Finally, we show the numerical simulations graphically using MATLAB.

Abstract 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.

Diabetes mellituse has been one of the major diseases in the world due to the high percentage of diabetics in the global population and the increasing growth rate of its onset. Identifying individual physiological characteristics, e.g., insulin sensitivity and glucose effectiveness and others, is extremely important in developing effective drugs and investigating genetic pathways causing the defects in these physiological responses. Intravenous glucose tolerance test (IVGTT) is such a protocol to determine an individual insulin sensitivity and glucose effectiveness indices. In this paper, we propose a stochastic delay differential equation model for the IVGTT protocol attempting to develop a method to increase the accuracy of parameter estimation. We first study the existence and uniqueness of the global positive solution and its asymptotic behavior of the stochastic path close to the steady state of the corresponding deterministic model. Then we develop a maximum likelihood estimation method to estimate the parameters involved in the proposed model. Our simulation studies numerically confirm our theoretical findings and demonstrate that the proposed model with estimated parameters can improve the fitness of clinical data.

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]

Abstract This paper is concerned with almost sure exponential stabilization of neural networks by intermittent control based on delay observations. By the stochastic comparison principle and Itô’s formula, a sufficient criterion is derived, under which unstable neural networks can be stabilized by stochastic intermittent control based on delay observations. The range of intermittent rate is given, and the upper bound of time delay can be solved from a transcend equation. Finally, two examples are provided to demonstrate the feasibility and validity of our proposed methods.

In this paper, we propose and study the deterministic and stochastic lake ecosystem models to investigate the impact of terrestrial organic matter upon persistence of the plankton populations. By constructing Lyapunov function and using the LaSalle's Invariance Principle, we establish global properties of the deterministic model. The dynamical behavior of solutions fits well with some experimental results. It is concluded that the terrestrial organic matter plays an important role in influencing interactions between phytoplankton and zooplankton. Based on the fluctuations of lake ecosystem, we further develop a stochastically perturbed model. Theoretic analysis implies that the stochastic model exists a stationary distribution which is ergodic. The key point of our analysis is to enhance our knowledge of the factors governing the dynamics of plankton population models.

Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the e ects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.

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]