Unique Stationary Distribution Research Articles

The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence

In this paper, a stochastic SIS type epidemic model with general nonlinear incidence is investigated. The diffusion process in the model is indicated to be degenerate. A new technique is proposed to investigate the existence of stationary distribution. By using the theory of integral Markov semigroup, the threshold criterion is established to ensure the existence of unique stationary distribution for the model. The numerical examples are carried out to illustrate our theoretical results.

Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate

In this paper, we study a stochastic SIS epidemic model with nonlinear incidence rate. By employing the Markov semigroups theory, we verify that the reproduction number R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α) can be used to govern the threshold dynamics of the studied system. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)>1, we show that there is a unique stable stationary distribution and the densities of the distributions of the solutions can converge in L1 to an invariant density. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)<1, under mild extra conditions, we establish sufficient conditions for extinction of the epidemic. Our results show that larger white noise can lead to the extinction of the epidemic while smaller white noise can ensure the existence of a stable stationary distribution which leads to the stochastic persistence of the epidemic.

A [formula omitted]-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains

In this paper, we study a new tensor eigenvalue problem, which involves E- and S-eigenvalues as its special cases. Some theoretical results such as existence of an eigenvalue and the number of eigenvalues are given. For an application of the proposed eigenvalue problem, we establish a tensor model for a higher-order multivariate Markov chain. The core issue of this problem is to study a stationary probability distribution of a higher-order multivariate Markov chain. A sufficient condition of the unique stationary positive distribution is given. An algorithm for computing stationary probability distribution is also developed. Numerical examples of applications in stock market modeling, sales demand prediction and biological sequence analysis are given to illustrate the proposed tensor model and the computed stationary probability distribution can provide a better prediction in these Markov chain applications.

Dynamics of a stochastic regime-switching predator–prey model with modified Leslie–Gower Holling-type II schemes and prey harvesting

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

A stochastic differential equation SIS epidemic model with two independent Brownian motions

In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ + γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.

Stationary distribution and extinction of a stochastic one-prey two-predator model with Holling type II functional response

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations.

Threshold of a regime-switching SIRS epidemic model with a ratio-dependent incidence rate

In this paper, we study the dynamical behavior of a regime-switching SIRS epidemic model with a ratio-dependent incidence rate. We propose a stochastic reproduction number R0S which can be regarded as a threshold to use in identifying the stochastic extinction and persistence: if R0S<1, the disease dies out exponentially with probability one; while if R0S>1, there exists a unique ergodic stationary distribution of the positive solutions to the system which implies the stochastic persistence of the infectious disease.

Global analysis of a vector-host epidemic model in stochastic environments

In this paper, a vector-host epidemic model is formulated from a classical deterministic framework to a stochastic differential equation. The model incorporates the potential impact of the free pathogens as well as the random environment on disease dynamics. The dynamics of the vector-host epidemic model is analyzed for the deterministic and stochastic cases, respectively. In the deterministic case, the global dynamics of the system is determined by the criteria R0, i.e., if R0≤1 the disease-free equilibrium is globally asymptotically stable; if R0>1 the unique endemic equilibrium is globally asymptotically stable. In the stochastic case, the stochastic system admits a unique ergodic stationary distribution if R˜0>1, which indicates that the disease can be persistent in vivo. Finally, numerical simulations are conducted to verify these analytical results.

Dynamical behaviors of a stochastic malaria model: A case study for Yunnan, China

In this paper, we investigate dynamical behaviors for a stochastic malaria model by introducing the effect of environmental white noise on transmission dynamics of malaria. Firstly, the distance between stochastic solutions and the disease-free equilibrium of the corresponding deterministic model is estimated in the time mean sense by constructing stochastic Lyapunov functions. Particularly, by the comparison theorem of stochastic differential equations and the law of large numbers, sufficient conditions for the extinction of the disease are obtained in the special case. Next, the existence of the unique ergodic stationary distribution is proved by constructing a suitable Lyapunov function. Finally, the model is used to simulate the human malaria data in Yunnan province and predict the trend of malaria in Yunnan.

Dynamics analysis of stochastic epidemic models with standard incidence

In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

Dynamics of a stochastic hepatitis C virus system with host immunity

In this paper, stochastic differential equations that model the dynamics of a hepatitis C virus are derived from a system of ordinary differential equations. The stochastic model incorporates the host immunity. Firstly, the existence of a unique ergodic stationary distribution is derived by using the theory of Hasminskii. Secondly, sufficient conditions are obtained for the destruction of hepatocytes and the convergence of target cells. Moreover based on realistic parameters, numerical simulations are carried out to show the analytical results. These results highlight the role of environmental noise in the spread of hepatitis C viruses. The theoretical work extend the results of the corresponding deterministic system.

DYNAMICAL BEHAVIOR OF A STOCHASTIC FOOD CHAIN CHEMOSTAT MODEL WITH MONOD RESPONSE FUNCTIONS

This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results.

Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation.

In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as R0 < 1. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.

Analysis of a stochastic HIV-1 infection model with degenerate diffusion

This paper studies a stochastic HIV-1 infection model with degenerate diffusion. The asymptotic dynamics of the stochastic model are shown to be governed by a threshold parameter. When the parameter is negative, the infection is predicted to go extinct exponentially while the level of healthy cells converges weakly to a unique invariant measure. When the threshold parameter is positive, the solution of the stochastic model converges polynomially to a unique invariant probability measure, indicating that the system admits a unique ergodic stationary distribution. Numerical simulations are conducted to show the analytical results. These results highlight the role of environmental noise in the spread of HIV-1. The method can also be applied to the non-degenerate systems.

Convergence rates for semistochastic processes

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

Classification with label noise: a Markov chain sampling framework

The effectiveness of classification methods relies largely on the correctness of instance labels. In real applications, however, the labels of instances are often not highly reliable due to the presence of label noise. Training effective classifiers in the presence of label noise is a challenging task that enjoys many real-world applications. In this paper, we propose a Markov chain sampling (MCS) framework that accurately identifies mislabeled instances and robustly learns effective classifiers. MCS builds a Markov chain where each state uniquely represents a set of randomly sampled instances. We show that the Markov chain has a unique stationary distribution, which puts much larger probability weights on the states dominated by correctly labeled instances than the states dominated by mislabeled instances. We propose a Markov Chain Monte Carlo sampling algorithm to approximate the stationary distribution, which is further used to compute the mislabeling probability for each instance, and train noise-resistant classifiers. The MCS framework is highly compatible with a wide spectrum of classifiers that produce probabilistic classification results. Extensive experiments on both real and synthetic data sets demonstrate the superior effectiveness and efficiency of the proposed MCS framework. [ABSTRACT FROM AUTHOR]

Diffusion approximations for double-ended queues with reneging in heavy traffic

We study a double-ended queue consisting of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system. The matching is instantaneous following the first-come–first-match principle. If a customer cannot be matched immediately, he/she will stay in a queue. We also assume customers are impatient with generally distributed patience times. Under suitable heavy traffic conditions, we establish simple linear asymptotic relationships between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time processes and show that the diffusion-scaled queue length process converges weakly to a diffusion process that admits a unique stationary distribution.

Stochastic mutualism model under regime switching with Lévy jumps

This paper is concerned with a stochastic mutualism model under regime switching with Lévy jumps. To begin with, the existence and uniqueness of the global positive solution is proved with any given positive initial value. Then, the sufficient conditions for stochastic permanence are established. The critical value between extinction and persistence in mean is also obtained. In addition, under some suitable conditions, we proved that there is a unique stationary distribution for the system without Lévy jumps. Our method relies on the Lyapunov function analysis and the Fredholm alternative. The results demonstrate that regime switching may contribute to the permanence but jump noise may suppress the permanence.