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To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein–Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus. [ABSTRACT FROM AUTHOR]

Influenza remains one of the most widespread epidemics, characterized by serious pathogenicity and high lethality, posing a significant threat to public health. This paper focuses on an influenza A infection model that includes vaccination and asymptomatic patients. The deterministic model examines the existence and local asymptotic stability of equilibria. In light of the influence of environmental disruption on the spread of disease, we develop a stochastic model in which the transmission rate follows a log-normal Ornstein–Uhlenbeck process. To demonstrate the dynamic behavior of the stochastic model, we verify the existence and uniqueness of the global positive solution. The establishment of suitable Lyapunov functions allows for the determination of sufficient conditions for the stationary distribution and extinction of the disease. Furthermore, the expression of the local density function around the quasi-endemic equilibrium is represented. Eventually, numerical simulations are conducted to support theoretical results and explore the effect of environmental noise. Our findings indicate that high noise intensity can expedite the extinction of the disease, while low noise intensity can facilitate the disease reaching a stationary distribution. This information may be valuable in developing strategies for disease prevention and control. [ABSTRACT FROM AUTHOR]

The density of a population in a natural state often fluctuates greatly, but it is not an unlimited change. Throughout the full text, we consider a stochastic predator–prey system with fear, Holling-II response function, distributed delay and mean-reverting Ornstein–Uhlenbeck process, which can better reflect the actual situation and provide theoretical basis for species research in the actual environment. And we drew some significant conclusions. The first step is to obtain the stability of equilibrium point. Then, the existence and uniqueness of positive solution for stochastic system is got and we also reach the stationary distribution of the stochastic system. In addition, we present an exact local expression for the density function of the random system near the unique positive equilibrium. Then we obtain the pth moment boundedness and the asymptotic behavior of the solution. At last, we give the extinction condition of the population system by the asymptotic behavior of the solution. In particular, we confirm the theoretical results by numerical simulation in the corresponding section. [ABSTRACT FROM AUTHOR]

Motivated by the notion of isotropic α -stable Lévy processes confined, by reflections, to a bounded open Lipschitz set D ⊂ R d , we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time. [ABSTRACT FROM AUTHOR]

In this article, a SIRS epidemic model with a general incidence rate is proposed and investigated. We briefly verify the global existence of a unique positive solution for the proposed system. Moreover, and unlike other works, we were able to find the stochastic threshold R s of the proposed model which was used for the discussion of the persistence in mean and extinction of the disease. Moreover, we utilize stochastic Lyapunov functions to show under sufficient conditions the existence and uniqueness of stationary distributions of the solution. Lastly, numerical simulation is executed to conform our analytical results. [ABSTRACT FROM AUTHOR]

With the vigorous development of information technology, online rumors are spreading recklessly. In this article, by comprehensive consideration of the stochastic disturbance, generalized nonlinear incidence, education age and punishment age, the dynamical behaviors of a novel stochastic SEIFR (susceptible-educated-infected-forced silence-removed) rumor spreading model with class-age structure under the generalized nonlinear incidence is analyzed and discussed. Firstly, given that the ubiquity of stochastic disturbance in the process of rumor spreading, a stochastic rumor spreading model is established. Secondly, the existence of the unique global positive solution for the stochastic model is manifested. Thirdly, the sufficient condition of rumor extinction is attained by virtue of Itô's formula and strong law of large numbers. Additionally, the existence of a unique stationary distribution implying the rumor persistence is investigated based on the method of Khasminskii. Finally, some simulations and a practical application are carried to validate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study an SICA model with a standard incidence rate, where the contact rate β is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when R 0 s > 1 , the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value R 0 e for disease extinction, and when R 0 e < 1 , the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate a novel stochastic SIHR epidemic model, which includes a general incidence rate and mean-reversion Ornstein–Uhlenbeck process. Firstly, the existence of global positivity of the solution is testified by Lyapunov function. Secondly, this disease will be eradicated if the reproduction number R 0 s < 1 . Otherwise, if the reproduction number R 0 ∗ > 1 , then the system has a stationary distribution, which means that the pandemic will persist. In addition, an explicit expression of the probability density function for a linear system near quasi-endemic equilibrium is obtained under certain conditions. Finally, a series of numerical simulations is carried out to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this article, an S V I R age structure model is formulated that describes the dynamics of the Hepatitis B virus (HBV), keeping in view the vaccinated population and the standard incidence rate. After the formulation of the deterministic system, the model is extended to a stochastic system, taking into account the noises in the transmission coefficient. Utilizing the well-known formula of Itô's and the Lyapunov function theory, first, demonstrate that the perturbed system has a unique global positive solution subject to positive initial conditions. Analyze the model for the extinction of HBV, and the required condition is derived. The threshold parameter for the model is calculated, and by using this quantity, it is proved that for small values of the white noises and R s < 1 , the disease will eventually go extinct at a negative exponential rate. It is also proved that for R s 0 > 1 , the weak permanence of the disease is observed. The criterion R s 0 > 1 confirms the stationary distribution of mean infected persons, suggesting that the illness will remain throughout time. To reinforce the primary analytical findings of this paper, various illustrative examples are conducted separately through numerical simulations. [ABSTRACT FROM AUTHOR]

Vaccination is an important tool in disease control to suppress disease, and vaccine-influenced diseases no longer conform to the general pattern of transmission. In this paper, by assuming that the infection rate is affected by the Ornstein–Uhlenbeck process, we obtained a stochastic SIRV model. First, we prove the existence and uniqueness of the global positive solution. Sufficient conditions for the extinction and persistence of the disease are then obtained. Next, by creating an appropriate Lyapunov function, the existence of the stationary distribution for the model is proved. Further, the explicit expression for the probability density function of the model around the quasi-equilibrium point is obtained. Finally, the analytical outcomes are examined by numerical simulations. [ABSTRACT FROM AUTHOR]

A stochastic three species predator–prey model with intermediate predator and anti-predator behavior is established and studied in this paper. By constructing suitable Lyapunov functions and combining knowledge of stochastic differential equations, the behavior of global positive solutions to the model are investigated. Firstly, we prove that there is unique global positive solution to the model, and establish the stochastic boundedness of the positive solution as well. Secondly, the sufficient condition for the existence of a unique ergodic stationary distribution is provided. Thirdly, the persistence and extinction of the populations are discussed. Finally, some numerical simulations demonstrate that the results obtained in this paper are true, moreover, the influence of white noise on the populations is revealed. [ABSTRACT FROM AUTHOR]

Multi-strain infectious diseases, which are usually prevented from spreading widely by vaccination, have two main transmission mechanisms: competitive exclusion and co-existence. In this paper, a stochastic multi-strain coinfection model with amplification and vaccination is developed. For the deterministic model, the basic reproduction number R 0 and fixed points are provided. For the stochastic model, we first prove the existence and uniqueness of the positive solution under any initial value. Then, a portion of those infected with the common strain will always become patients with the amplified strain, which increases the risk of death from the disease. Therefore, we verified that patients with common strains would become extinct if R 1 s < 1 . Furthermore, by constructing the Lyapunov function, we find that model (3) has a unique ergodic stationary distribution if R 0 S > 1 . Particularly, we get a concrete form of the probability density of the distribution κ (·) near equilibrium E ∗ , where E ∗ is the quasi-local equilibrium of the stochastic model. Finally, the results are verified by numerical simulation. The results show that vaccination can control disease outbreaks or even eliminate them. [ABSTRACT FROM AUTHOR]

In this paper, assuming the certain variable satisfies the Ornstein–Uhlenbeck process, we formulate a stochastic within-host dengue model with immune response to obtain further understanding of the transmission dynamics of dengue fever. Then we analyze the dynamical properties of the stochastic system in detail, including the existence and uniqueness of the global solution, the existence of a stationary distribution, and the extinction of infected monocytes and free viruses. In particular, it is worth revealing that we get the specific form of covariance matrix in its probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the dynamic behavior of a novel stochastic epidemic model (SEM) that incorporates distributed delay and time switching. We prove the existence and uniqueness of a global positive solution and establish the presence of ergodic stationary distribution (ESD) through the construction of appropriate Lyapunov functions. The threshold R 0 S derived from our analysis plays a vital role in this procedure. Additionally, we conduct computer simulations to validate our theoretical discoveries, demonstrating that the incorporation of white noise can induce random fluctuations in the system variables under time switching. [ABSTRACT FROM AUTHOR]

Vaccination and social media play pivotal roles in affecting disease transmission. Research has shown that disease transmission can be subject to random events. Consequently, we develop a stochastic infectious disease dynamical model that incorporates saturated media coverage and vaccination age. The Itô's formula and the Lyapunov function method are applied to study the extinction behavior of the disease and the existence of a unique ergodic stationary distribution. The findings suggest that the media effect is delayed and cannot eliminate the disease completely. To directly control disease transmission, a combination of high-intensity noise disturbance and low vaccine wane rate is required. Furthermore, the shorter the disease incubation period, the more difficult it is to control. [ABSTRACT FROM AUTHOR]

Savannas are widely distributed on the earth, and the growth of plants in savannas often encounters the environmental perturbations. In this research, based on a deterministic tree–grass savanna model proposed by Tamen et al. (Biomath 3: 1407191, 2014), we use white noise and a continuous-time Markov chain to model the slight environmental perturbations and environmental regime switching, respectively, and formulate a stochastic tree–grass savanna model under regime switching. We first give conditions under which the model is permanent (i.e., the tree species and the grass species coexist) in the sense that the model admits a unique ergodic non-boundary stationary distribution and the transition probability of the solution of the model converges to this stationary distribution exponentially under total variation norm. Then, we provide conditions under which the grass species or the tree species die out. Finally, we discuss the impacts of random environmental perturbations on the coexistence and disintegration of the model with the help of exemplary data and numerical simulations. [ABSTRACT FROM AUTHOR]

Queueing systems where certain inventory items are required to provide service to a customer have become popular in the literature from early 1990th. Such systems are similar to those models analysed in the literature models with paired customers, assembly-like queues, passenger–taxi models, etc. During the last few years they are considered in the context of modelling operation of the nodes of a wireless sensor network with energy harvesting. Distinguishing feature of the model considered in this paper, besides the suggestion that arrival flow of customers is described by the Markovian arrival process, is the assumption about a general distribution of the service time while only exponential or phase-type distribution was previously assumed in the existing literature. We apply the well-known technique of M/G/1 type Markov chains and semi-regenerative processes to obtain the ergodicity criterion in a transparent form, stationary distribution of the system under study and the Laplace–Stieltjes transform of the sojourn time distribution. This creates an opportunity to formulate and solve various optimization problems. A number of numerical examples illustrate the computational tractability of the theoretical results and illustrate the behavior of the system performance measures depending on its parameters. [ABSTRACT FROM AUTHOR]

We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities v 1 and v 2 . We assume that the sequences of random times concerning the motions along each velocity follow two independent geometric counting processes of intensity λ , and that the resetting times are Poissonian with rate ξ > 0 . Under these assumptions we obtain the probability laws of the modified telegraph process describing the position and the velocity of the running particle. Our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. We obtain also the asymptotic distribution of the particle position when (i) λ tends to infinity, and (ii) the time goes to infinity. In the latter case the asymptotic distribution arises properly as an effect of the resetting mechanism. A quite different behavior is observed in the two cases when v 2 < 0 < v 1 and 0 < v 2 < v 1 . Furthermore, we focus on the determination of the moment-generating function and on the main moments of the process describing the particle position under reset. Finally, we analyse the mean-square distance between the process subject to resets and the same process in absence of resets. Quite surprisingly, the lowest mean-square distance can be found for ξ = 0 , for a positive ξ , or for ξ → + ∞ depending on the choice of the other parameters. [ABSTRACT FROM AUTHOR]

In this paper, considering the inevitable effects of environmental perturbations on disease transmission, we mainly study a stochastic SVEIR epidemic model in which the transmission rate satisfies the log-normal Ornstein–Uhlenbeck process and the incidence rate is general. To analyze the dynamic properties of the stochastic model, we firstly verify that there is a unique positive global solution. By constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. The sufficient condition for disease extinction is also given. Next, as a special case, we investigate the asymptotic stability of equilibria for the deterministic model and establish the exact expression of the probability density function of stationary distribution for the stochastic model. Finally, we calculate the mean first passage time from the initial value to the stationary state or extinction state to study the influence of environmental perturbations; meanwhile, some numerical simulations are carried out to demonstrate theoretical conclusions. [ABSTRACT FROM AUTHOR]

Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on Z ≥ 0 d , and simulated via a version of the “Gillespie algorithm,” have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in Duso and Zechner (Proc Nat Acad Sci 117(37):22674–22683 , Duso and Zechner (2020)), where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in Duso and Zechner (Proc Nat Acad Sci 117(37):22674–22683 , Duso and Zechner (2020)). [ABSTRACT FROM AUTHOR]

In this contribution, a higher-order stochastically perturbed predator–prey system involving fear effect, Crowley–Martin functional response and a modified Leslie–Gower functional response is investigated. Based on the preliminary lemma as regards global positive solutions, the sufficient criteria ensuring the existence and uniqueness of an ergodic stationary distribution are established by virtue of constructing suitable Lyapounov functions. Later, the exponential extinction of every species indicates that sufficiently large noise intensities are unfavorable to populations' growth. It is worth mentioning that we study the probability density function of this system near the quasi-positive equilibrium, and further show the approximate expression. Finally, four numerical examples are supplied to support and illustrate our theoretical results. Analytical results clearly reveal that white noise plays a crucial role in the dynamic behaviors of predator and prey. [ABSTRACT FROM AUTHOR]

Two different approaches to incorporate environmental perturbations in stochastic systems are compared analytically and computationally. Then we present a stochastic model for COVID-19 that considers susceptible, exposed, infected, and recovered individuals, in which the contact rate between susceptible and infected individuals is governed by the Ornstein–Uhlenbeck process. We establish criteria for the existence of a stationary distribution of the system by constructing a suitable Lyapunov function. Next, we derive the analytical expression of the probability density function of the model near the quasi-equilibrium. Additionally, we establish sufficient conditions for the extinction of disease. Finally, we analyze the effect of the Ornstein–Uhlenbeck process on the dynamic behavior of the stochastic model in the numerical simulation section. Overall, our findings shed light on the underlying mechanisms of COVID-19 dynamics and the influence of environmental factors on the spread of the disease, which can inform policy decisions and public health interventions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a stochastic SIRC model with Ornstein–Uhlenbeck process. Firstly, we give the existence and uniqueness of global solution of stochastic SIRC model and prove it. In addition, the existence of ergodic stationary distributions for stochastic SIRC system is proved by constructing a suitable series of Lyapunov functions. A quasi-endemic equilibrium related to endemic equilibrium of deterministic systems is defined by considering randomness. And we obtain the probability density function of the linearized system near the equilibrium point. After the proof of probability density function, the sufficient condition of disease extinction is given and proved. We prove the theoretical results in the paper by numerical simulation at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper uses the stationary distribution of Markov chains to obtain new criteria for the almost sure exponential stability of nonlinear hybrid stochastic functional differential equations with Lévy noise (HSFDEs-LN) under weaker conditions and an upper bound τ ∗ of time delay is obtained. That is, if corresponding HSDEs-LN is almost surely exponentially stable, then the HSFDEs-LN is also almost surely exponentially stable as long as time delay τ < τ ∗ . Furthermore, the new theory enabled us to design discrete-time state observations feedback control by using Lévy noise as the control source to stabilize a class of unstable hybrid differential equations. [ABSTRACT FROM AUTHOR]

Due to many uncertain factors, parameter values in many microorganism cultivation systems are affected to a greater or lesser extent by environmental fluctuation. In this paper, the authors develop a stochastic turbidostat model that considers white noise, formulate and analyze dynamical behavior for the stochastic model. The authors obtain the existence and uniqueness of globally positive equilibrium. Sufficient conditions of the extinction for the system are established. Since the uniformly elliptic condition can fail to satisfy, the existence of the stationary distribution of the system is proved by using Markov semigroups theory. Biologically speaking, the existence of stationary distribution suggests that microorganism will exist for a long time. The analytical results are tested and verified by numerical simulation, which evaluate the influence of white noise on the dynamics of microorganism. [ABSTRACT FROM AUTHOR]

In this paper, a generalized n-species Gilpin–Ayala competition system with saturation effect and nonlinear perturbations is proposed and examined. We first develop a new mathematical technique called “stochastic ϵ -threshold method”, to tackle the nonlinear perturbations and study the competitive coexistence, which includes the existence of ergodic stationary distribution as well as stochastic positive periodic solution. It should be mentioned that the method can be successfully applied to the species coexistence of other biological models. Then, we establish sufficient conditions for exponential extinction of competing species. Finally, several numerical examples are provided to support our theoretical results and analyze the impact of key parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic differential equation model which is used to explore how the environmental noise affects the spread of hepatitis B virus. Firstly, we show that there exists a unique global positive solution of the stochastic system with any positive initial value. Then, we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model. Especially, under the same conditions as the existence of a stationary distribution, it is worth noting that we obtain the specific form of probability density around the quasi-infected steady state of the stochastic system. Thirdly, we obtain sufficient criteria for extinction of the infected hepatocytes. Finally, numerical simulations are introduced to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we examine a stochastic avian influenza model with a nonlinear incidence rate within avian populations and the psychological effect within the human population, where susceptible humans reduce their contact with infected avians as the number of infected humans increases. For the deterministic model, the basic reproduction number R 0 , possible equilibria, and related asymptotic stability are first studied. Then, for the stochastic model, we obtain a critical value R 0 S , which can determine the persistence and extinction of avian influenza. It is theoretically proved that the stochastic model has a unique stationary distribution ϖ (·) if R 0 S > 1 , but the disease will go to extinction when R 0 S < 1 . Taking stochasticity into account, a quasi-endemic equilibrium T ¯ ∗ related to the endemic equilibrium of the deterministic model is defined. We develop an important lemma for solving the special Fokker–Planck equation and derive the explicit expression of the density function of the distribution ϖ (·) around the equilibrium T ¯ ∗ . Numerical simulations verify our theoretical results, and we study the impact of noise and the psychological effect on the transmission dynamics of avian influenza. [ABSTRACT FROM AUTHOR]

In this paper, we extend a social obesity epidemic model from a deterministic system to a stochastic differential equation by introducing white noise. We first present some preliminaries needed for later proof. Then, we demonstrate several theorems and their proofs. The existence of the global positive solution of the model is proved by using stochastic Lyapunov function. Furthermore, the solutions of model are stochastically ultimately bounded and permanent are also obtained and the sufficient condition for the existence of a unique ergodic stationary distribution is established by using Khasminskii's theorem. Finally, the theoretical results are applied to study the target population of 24- to 65-year-old adult residents in the region of Valencia, Spain, and the influence of relevant parameters on the proportion of the three populations is further analyzed and the control strategy is explored. [ABSTRACT FROM AUTHOR]

A stochastic SIQS model via isolation with regime-switching is studied in this paper. The range of positive solution of the model is presented. Threshold to determine extinction and invariant measure is obtained by a new technique, which can be seen as the sufficient and almost necessary condition. Meantime, a value to judge the existence of stationary distribution is acquired by constructing the suitable hybrid Lyapunov function. Two values are proved to be consistent. Several examples are enumerated to test the theoretical results. [ABSTRACT FROM AUTHOR]

A two-species stochastic competitive model with distributed delays and feedback controls is constructed to reveal the effect of both environmental noises and feedback controls on the long-time behaviors of two species, and the saturation effect is also incorporated by considering the boundedness of inter-specific competition. Through theoretical analysis, sufficient conditions for partial permanence are obtained. Moreover, the existence and uniqueness of stationary distribution to the positive solution is acquired. Finally, plentiful numerical experiments are performed as the authentication to support the theoretical results, and biological interpretations on the main results are also given. [ABSTRACT FROM AUTHOR]

To explore the relationship between hunting of predator and anti-predation behavior of prey in the natural environment, considering a stochastic predator-prey model with fear effect and hunting cooperation, which is perturbed by the nonlinear white noise. Firstly, the relevant properties of solutions for the model are studied. Then, we present sufficient conditions for extinction, persistence, and stationary distribution. Especially, the density function near positive equilibrium of model without white noise is established. Finally, based on the results of numerical simulation and theoretical analysis, it is revealed that fear effect, hunting cooperation, and white noise have indispensable influence on the dynamical behaviors of this model. [ABSTRACT FROM AUTHOR]

In this paper, the authors develop and study an HIV infection system with two distinct cell subsets and nonlinear stochastic perturbation. Firstly, the authors obtain that the solution of the system is positive and global. Secondly, for the corresponding linear case, the authors derive a critical condition R0S similar to deterministic system. When R0S > 1, the authors establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution to the stochastic system, respectively. Finally, the authors give sufficient criterions for extinction of the diseases. The proposed work provides a new method in overcoming difficulty conduced by nonlinear stochastic perturbation. [ABSTRACT FROM AUTHOR]

Dependency network (DN) aims at using a collection of conditional distributions to identify a joint pdf. When the DN is compatible (self-consistent), the Gibbs sampler (GS) has been the algorithm to approximate the joint pdf. Without compatibility, GS will have multiple stationary distributions, named pseudo-Gibbs distributions (PGD), associated with different updating orders. To increase the computational efficiency and stability, we propose computing the marginal distributions. Closed-form marginal transition matrix is unearthed from DN. Thus, it becomes possible to compute the marginal distribution of PGD, which will be paired with a conditional distribution to obtain a PGD. We also show that multiple PGDs can be derived from one PGD. When the support is a union of disjoint regions, GS could not converge because the stationary pdf is a mixture of several joint distributions. Examples here show that our approach can obtain correct PGDs even for partitioned support. A new way to verify compatibility, under such circumstances, will also be proposed. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in stochastic fluid flow model with upward jumps at level zero and instantaneous phase transitions at jumps epochs. This mathematical model is governed by a non-homogeneous differential system with specific boundary conditions. We derive three algorithms for computing explicitly the fluid level distribution. We study the general case of the model, the case where the phase process conserves its state when the fluid jumps, and a special case where the phase process transits to a deterministic fixed state at jumps epochs. Numerical results are carried out to illustrate the numerical stability and the advantages in computational time for the proposed approaches. [ABSTRACT FROM AUTHOR]

The present article formulates and dissects a single-species model with intraspecific cooperation and stochastic perturbations. Sufficient criteria for extinction and permanence are provided. In addition, the existence of a unique ergodic stationary distribution (UESD) is dissected, and the explicit form of the density function of the UESD is obtained under some conditions. Finally, the theoretical outcomes are used to study the growth of gypsy moth (Lymantria dispar) in North America, and some critical impacts of stochastic perturbations are uncovered and numerically illustrated. [ABSTRACT FROM AUTHOR]

The stochastic fluid-fluid model (SFFM) is a Markov process { (X t , Y t , φ t) , t ≥ 0 } , where { φ t , t ≥ 0 } is a continuous-time Markov chain, the first fluid, { X t , t ≥ 0 } , is a classical stochastic fluid process driven by { φ t , t ≥ 0 } , and the second fluid, { Y t , t ≥ 0 } , is driven by the pair { (X t , φ t) , t ≥ 0 } . Operator-analytic expressions for the stationary distribution of the SFFM, in terms of the infinitesimal generator of the process { (X t , φ t) , t ≥ 0 } , are known. However, these operator-analytic expressions do not lend themselves to direct computation. In this paper the discontinuous Galerkin (DG) method is used to construct approximations to these operators, in the form of finite dimensional matrices, to enable computation. The DG approximations are used to construct approximations to the stationary distribution of the SFFM, and results are verified by simulation. The numerics demonstrate that the DG scheme can have a superior rate of convergence compared to other methods. [ABSTRACT FROM AUTHOR]

Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniqueness of a unique globally positive solution. The disease will be extinct under the threshold criterion. We analyze the asymptotic behavior around the disease-free equilibrium of a deterministic SIRS model. By using the Khasminskii method, we prove the existence of a unique stationary distribution. Further, solutions of the stochastic model fluctuate around endemic equilibrium under certain conditions. Some numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

A generalized stepping stone model with Ξ-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by Ξ-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued Ξ-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) { (X n , J n) } on Z + 2 × S 0 , where X n = (X 1 , n , X 2 , n) is the level state, J n the phase state (background state) and S 0 a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in an arbitrary direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques and the singularity analysis methods. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results. [ABSTRACT FROM AUTHOR]

In this paper, a bi-dimensional continuous-time stochastic differential system of Leslie–Grower types is considered. The proposed predator-prey model with white noises is theoretically and numerically investigated. The global existence of the positive unique solution of the model is obtained. The boundedness in the sense of moments of the solution is shown. Some asymptotic properties of the stochastic system are also elaborated. New methodological approaches based on the statistical linearisation method as well as on the coupled and weighted procedures are elaborated. Numerical results of the density function for the stationary state are obtained. [ABSTRACT FROM AUTHOR]

Infectious diseases have for centuries been the leading causes of death and disability worldwide and the environmental fluctuation is a crucial part of an ecosystem in the natural world. In this paper, we proposed and discussed a stochastic SIRI epidemic model incorporating double saturated incidence rates and relapse. The dynamical properties of the model were analyzed. The existence and uniqueness of a global positive solution were proven. Sufficient conditions were derived to guarantee the extinction and persistence in mean of the epidemic model. Additionally, ergodic stationary distribution of the stochastic SIRI model was discussed. Our results indicated that the intensity of relapse and stochastic perturbations greatly affected the dynamics of epidemic systems and if the random fluctuations were large enough, the disease could be accelerated to extinction while the stronger relapse rate were detrimental to the control of the disease. [ABSTRACT FROM AUTHOR]

By taking full consideration of contact heterogeneity of individuals, quarantine measures, demographics, information transmission and random environments, we present a stochastic SIQR epidemic model with demographics and non-monotone incidence rate on scale-free networks, which introduces stochastic perturbations to death rate. The formula of the basic reproduction number of the deterministic model is obtained by utilizing the existence of the endemic equilibrium. Next, we define a stopping time, then the existence of a unique global positive solution for the stochastic model is proved by constructing appropriate Lyapunov function to demonstrate the stopping time is infinite. In addition, we also manifest sufficient conditions for diseases extinction and the existence of ergodic stationary distribution by constructing appropriate stochastic Lyapunov functions. At last, numerical simulations illustrate the analytical results. [ABSTRACT FROM AUTHOR]

We considered multi-server retrial queue with heterogeneous servers. Servers are distinguished by the rates of service, breakdowns and repairs. Arrivals of customers are described by the Markov arrival process. Arriving customer occupies the available server with the minimal number. Customers who meet all servers busy move to the orbit to retry access to the servers later on. The orbit capacity is infinite. The total retrial rate is assumed to be an infinitely increasing function of the number of customers staying in the orbit. Stationary behavior of the system is analyzed, numerical illustrations are presented. [ABSTRACT FROM AUTHOR]

In this paper, we develop a stochastic multi-molecule chemical reaction model with reaction rate perturbed by log-normal Ornstein-Uhlenbeck\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Ornstein-Uhlenbeck$$\end{document} process in order to consider the effects of random factors on chemical reaction dynamics. Firstly, we prove the existence and uniqueness of the global positive solution for the stochastic model. In addition, we obtain the conditions under which the corresponding stochastic system exist a stationary distribution. Then, we derive a sufficient condition to end the reaction. Furthermore, the stochastic system has been transformed into a linearized system, by solving Fokker-Planck\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Fokker-Planck$$\end{document} equation, we obtain the exact expression of the density function around the quasi-equilibrium of this system. Finally, we draw a conclusion that the dynamical behaviors of the stochastic system will be affected by random factor, Ornstein-Uhlenbeck\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Ornstein-Uhlenbeck$$\end{document} process respectively [ABSTRACT FROM AUTHOR]

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Motivated by a BAR approach studied in the recent paper (Braverman et al. in the BAR approach for multi-class queueing networks with SBP service policies, 2023), we are interested in two problems; when this Markov process inherits the same Markov structure under the Palm distributions, and how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We affirmatively answer the first problem, and propose a framework for resolving the second problem, which is applicable to a general stationary process, which is not needed to be Markov. We also discuss how those results can be applied in deriving BAR’s for the diffusion approximation of queueing models in heavy traffic. In particular, as their new application, the heavy traffic limit of the stationary distribution is derived for a single-server queue with a finite waiting room. [ABSTRACT FROM AUTHOR]

In this paper, we consider a high-dimensional stochastic HIV/AIDS model that incorporates both multiple stages treatment and higher order perturbation. Firstly, we establish sufficient criteria for the existence of a unique ergodic stationary distribution by making use of stochastic Lyapunov analysis method. Stationary distribution shows that the disease will be persistent in the long term. Then, conditions for extinction of the disease are obtained. Theoretical analysis indicates that large noise intensity can suppress the prevalence of HIV/AIDS epidemic. Finally, we provide some numerical simulations to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the features of a stochastically perturbed two-species predator-prey patch-system with ratio-dependent functional response. We first prove that the system which we investigate has a unique global positive solution. Then, the sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system are presented by establishing a series of suitable Lyapunov functions. [ABSTRACT FROM AUTHOR]

Time delay, where it depends on the current state and on the past situation, is often occurred in biological activities, for example, the process by which microorganism consume nutrients into their available biomass is not instantaneous. This investigation inspects the dynamic behavior of stochastic turbidostat model coupled with distributed delay and degenerate diffusion, including sufficient conditions of the extinction and the existence of a unique stationary distribution. What's more, the existence and uniqueness of globally positive equilibrium of the exploited model are studied. The findings manifest that the turbidostat system is ergodic only when the intensity of white noise is very small. Finally, some numerical examples are proposed to indicate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Phytoplankton is an important indicator organism to evaluate the quality of water environment, which may reflect the nutritional level of the sea area. Conversely, environmental conditions can directly affect the community structure of phytoplankton. A stochastic phytoplankton–zooplankton model considering non-degenerate and degenerate diffusions is formulated in this paper. What’s more, in both systems, we obtain sufficient conditions of extinction and ergodicity. Our results demonstrate that the weaker white noise can ensure the permanence of zooplankton and the stronger white noise can lead to the disappearance of zooplankton. Moreover, the threshold value of extinction and persistence can serve as a theoretical basis for controlling phytoplankton and zooplankton. Numerical examples are performed on the analysis results of the two cases to confirm our theoretical results. In addition, we also provide a real-life case study, the validity of the model is verified based on experimental data, and it is shown that fluctuation of external environment and the consumption of phytoplankton by zooplankton will affect the growth and number of phytoplankton. Effectively controlling the quantity of phytoplankton can delay the occurrence of water bloom or red tide. [ABSTRACT FROM AUTHOR]