On numerical stationary distribution of overdamped Langevin equation in harmonic system

Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time.In this work we study the highly accurate numerical algorithm for the overdamped Langevin equation.In particular,our interest is in the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system.Based on the large friction limit of the underdamped Langevin dynamic scheme,three algorithms for overdamped Langevin equation are obtained.We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory for both one-dimensional case and multi-dimensional case.The accuracy of the stationary distribution of each algorithm is illustrated by comparing with the exact Boltzmann distribution.Our results demonstrate that the“BAOA-limit”algorithm generates an accurate distribution of the harmonic system in a canonical ensemble,within a stable range of time interval.The other algorithms do not produce the exact distribution of the harmonic system.

Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives,when emergencies occur,rumors will spread widely on the internet.However,it is quite difficult for the netizens to distinguish the truth of the information.The main reasons are the uncertainty of netizens’behavior and attitude,which make the transmission rates of these information among social network groups be not fixed.In this paper,we propose a stochastic rumor propagation model with general incidence function.The model can be described by a stochastic differential equation.Applying the Khasminskii method via a suitable construction of Lyapunov function,we first prove the existence of a unique solution for the stochastic model with probability one.Then we show the existence of a unique ergodic stationary distribution of the rumor model,which exhibits the ergodicity.We also provide some numerical simulations to support our theoretical results.The numerical results give us some possible methods to control rumor propagation.Firstly,increasing noise intensity can effectively reduce rumor propagation when R_(0)>1That is,after rumors spread widely on social network platforms,government intervention and authoritative media coverage will interfere with netizens’opinions,thus reducing the degree of rumor propagation.Secondly,speed up the rumor refutation,intensify efforts to refute rumors,and improve the scientific quality of netizen(i.e.,increase the value ofβand decrease the value ofαandγ),which can effectively curb the rumor propagation.

Extinction and Stationary Distribution of a Stochastic SIR Epidemic Model with Jumps

A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a unique global positive solution;(2) the results reveal that the solution of this epidemic model will be stochastically ultimately bounded,and the non-linear SDE admits a unique stationary distribution under certain parametric conditions;(3) the coefficients play an important role in the extinction of the diseases.

Ergodic stationary distribution of two stochastic tuberculosis models with imperfect vaccination and early and late latency

This contribution probes into ergodic stationary distribution for two stochastic SVELIT(susceptible-vaccinated-early latent-late latent-infective-treated)tuberculosis(TB)models to observe the impact of white noises and color noises on TB control in random environments.We first investigate the existence and uniqueness of ergodic stationary distribution(EUESD)for the autonomous SVELIT model subject to white noises via the proper Lyapunov functions,and suficient conditions on the extinction of disease are acquired.Next,sufficient conditions for the EUESD and the extinction of disease for the SVELIT model with Markov switching are also established.Eventually,some numerical examples validate the theoretical findings.What's more,it has been observed that higher amplitude noises may lead to the eradication of TB,which is conducive to TB control.

Stationary distribution and long-time behavior of COVID-19 model with stochastic effect

The coronavirus disease(COVID-19)is a dangerous pandemic and it spreads to many people in most of the world.In this paper,we propose a COVID-19 model with the assumption that it is affected by randomness.For positivity,we prove the global existence of positive solution and the system exhibits extinction under certain parametric restrictions.Moreover,we establish the stability region for the stochastic model under the behavior of stationary distribution.The stationary distribution gives the guarantee of the appearance of infection in the population,Besides that,we find the reproduction ratio R for prevail and disappear of infection within the human population.From the graphical representation,we have validated the threshold conditions that define in our theoretical findings.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.

The stationary distribution and stochastic persistence for a class of disease models:Case study of malarial

This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates,and also from the random delays of the incubation and immunity periods.Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state,and also for the disease to remain permanently in the system over time.Moreover,the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined,and the statistical characteristics of the distribution are given mumerically.The results of this study show that the disease will persist and become permanent in the system,regardless of(1)whether the noises are from the discase transmission rate and/or from the natural death rates or(2)whether the delays in the system are constant or random for individuals in the system.Furthermore,it is shown that"weak"noise is associated with the existence of an endemic stationary distribution for the disease,while"strong"noise is associated with extinction of the population over time.Numerical simulation examples for Plasnodiurr vitar malaria are given.

Stationary distribution and extinction for a stochastic two-compartment model of B-cell chronic lymphocytic leukemia

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

Dynamical Analysis of the Stochastic COVID-19 Model Using Piecewise Differential Equation Technique

Various data sets showing the prevalence of numerous viral diseases have demonstrated that the transmission is not truly homogeneous.Two examples are the spread of Spanish flu and COVID-19.The aimof this research is to develop a comprehensive nonlinear stochastic model having six cohorts relying on ordinary differential equations via piecewise fractional differential operators.Firstly,the strength number of the deterministic case is carried out.Then,for the stochastic model,we show that there is a critical number RS0 that can predict virus persistence and infection eradication.Because of the peculiarity of this notion,an interesting way to ensure the existence and uniqueness of the global positive solution characterized by the stochastic COVID-19 model is established by creating a sequence of appropriate Lyapunov candidates.Adetailed ergodic stationary distribution for the stochastic COVID-19 model is provided.Our findings demonstrate a piecewise numerical technique to generate simulation studies for these frameworks.The collected outcomes leave no doubt that this conception is a revolutionary doorway that will assist mankind in good perspective nature.

DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE

In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function.Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system,which greatly improves upon previous results.

Semi-Markovian Model of Two-Line Queuing System with Losses

In the present paper, to build model of two-line queuing system with losses GI/G/2/0, the approach introduced by V.S. Korolyuk and A.F. Turbin, is used. It is based on application of the theory of semi-Markov processes with arbitrary phase space of states. This approach allows us to omit some restrictions. The stationary characteristics of the system have been defined, assuming that the incoming flow of requests and their service times have distributions of general form. The particular cases of the system were considered. The used approach can be useful for modeling systems of various purposes.

On the Dynamics of a Stochastic Ratio-Dependent Predator-Prey System with Infection for the Prey

In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold <img alt="" src="Edit_0a62b9be-7934-457b-aca3-af3420f5b5ee.png" /> for the infected prey population, that is, the disease will tend to extinction if <img alt="" src="Edit_e6cd63f6-de07-42be-a22a-8750d6c8aac9.png" />< 1, and it will exist in the long time if <img alt="" src="Edit_5964fdd8-a9fe-4dc2-b897-f4206f046f65.png" />> 1. Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

Time-Delayed Feedback Control in a Single-Mode Laser System

The effects of time-delayed feedback control in a single-mode laser system is investigated.Using the smalltime delay approximation,the analytic expression of the stationary probability distribution function of the laser field isobtained.The mean,normalized variance and skewness of the steady-state laser intensity are calculated.It is found thatthe time-delayed feedback control can suppress the intensity fluctuation of the laser system.The numerical simulationsare in good agreement with the approximate analytic results.

Exact distributions for stochastic models of gene expression with arbitrary regulation

Stochasticity in gene expression can result in fluctuations in gene product levels. Recent experiments indicated that feedback regulation plays an important role in controlling the noise in gene expression.A quantitative understanding of the feedback effect on gene expression requires analysis of the corresponding stochastic model. However, for stochastic models of gene expression with general regulation functions, exact analytical results for gene product distributions have not been given so far. Here, we propose a technique to solve a generalized ON-OFF model of stochastic gene expression with arbitrary(positive or negative, linear or nonlinear) feedbacks including posttranscriptional or posttranslational regulation. The obtained results, which generalize results obtained previously, provide new insights into the role of feedback in regulating gene expression. The proposed analytical framework can easily be extended to analysis of more complex models of stochastic gene expression.

Long-Time Behavior of a Stochastic Turbidostat Model Under Degenerate Diffusion

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 simulations,which evaluate the influence of white noise on the dynamics of microorganism.

The Impact of Nonlinear Stochastic Perturbation on Dynamical Behavior of an HIV Infection System

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.

Dynamics of a stochastic multi-stage sheep brucellosis model with incomplete immunity

This paper considered a multi-stage sheep brucellosis model with incomplete immunity.First,we established a deterministic model,calculated the basic reproduction number R_(0),set out the conditions for the global stability of the disease-free equilibrium and endemic equilibrium.Second,considering the influence of environmental white noise on brucellosis infection,we further established the stochastic version of the model.By constructing a suitable Lyapunov function,we proved the existence and uniqueness of the global positive solution.Further,we got the sufficient conditions for disease extinction and the existence of ergodic stationary distribution.Finally,we carried out some numerical simulations to verify the theoretical results.

Analysis of an avian influenza model with Allee effect and stochasticity

In this paper,we investigate a two-dimensional avian influenza model with Allee effect and stochasticity.We first show that a unique global positive solution always exists to the stochastic system for any positive initial value.Then,under certain conditions,this solution is proved to be stochastically ultimately bounded.Furthermore,by constructing a suitable Lyapunov function,we obtain sufficient conditions for the existence of stationary distribution with ergodicity.The conditions for the extinction of infected avian population are also analytically studied.These theoretical results are conformed by computational simulations.We numerically show that the environmental noise can bring different dynamical outcomes to the stochastic model.By scanning different noise intensities,we observe that large noise can cause extinction of infected avian population,which suggests the repression of noise on the spread of avian virus.

Analysis and simulation of a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures: A case study of the outbreak in Urumqi, China in August 2022

In this paper,a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures is proposed.Firstly,the existence and uniqueness of the global positive solution is obtained.Secondly,threshold criteria for the stochastic extinction and persistence in the mean with probability one are established.Moreover,a sufficient condition for the existence of unique ergodic stationary distribution for any positive solution is also established.Finally,numerical simulations are carried out in combination with real COVID-19 data from Urumqi,China and the theoretical results are verified.