Stationary Distribution Research Articles

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Sampling random graphs with specified degree sequences

The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network’s structure can be explained by its degree structure alone. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model. However, accurately and efficiently detecting when this Markov chain is sufficiently close to its stationary distribution remains an unsolved problem. Here, we provide a solution to sample from the configuration model using this standard MCMC algorithm. We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived from analyzing a corpus of 509 empirical networks. We provide a convergence detection method based on the Dickey-Fuller Generalized Least Squares test, which we show is more accurate and efficient than three alternative Markov chain convergence tests. Supplementary materials for the proposed methods can be found here.

Long time behavior of a rumor model with Ornstein-Uhlenbeck process

In order to study the propagation of rumors under the influence of media, this paper analyzes a random rumor propagation system with Ornstein-Uhlenbeck process. By constructing the Lyapunov function, we get that the established model has a stationary distribution, which means that rumors will persist under the side effects of the media. In addition, we solve the corresponding matrix and get the exact expression of the probability density near the positive equilibrium. At the end of this paper, numerical simulations verify our results.

Analytic computation for stationary distributions of D-BMAP/D -MSP(a,b)/1 queueing system

ABSTRACT This paper analyses single server correlated batch arrival and correlated batch service queue with innumerable waiting space in discrete-time. The arrival and service processes are governed by discrete batch Markovian arrival process and discrete Markovian service process with general bulk service rule, respectively. Observing two consecutive random epochs, we construct general structure transition probability matrix and it is then reblocked to the desired M/G/1 structure. The UL-type RG-factorization approach combined with spectral expansion method is adopted to unravel the queue length distribution at random epoch. The relationships are undertaken to obtain the queue length distributions at pre-arrival, outside observer's, intermediate and post-departure epochs. Our most challenging effort is to derive the probability mass functions of waiting time and service batch size for arbitrary customer in an arriving batch. We address the warehouse management process for finished products from dealer to retailer as a potential practical use of our batch queue with correlation in both arrival and service, and provide the effect of our queueing behaviour in this application based on our analytical and computational investigation. Based on the implementation of parametrized numeric experiments with numerous categories, we impart our computational results to validate the analytical findings reported in this investigation.

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Assessing extreme significant wave height in China’s coastal waters under climate change

Accurately estimating the return values of significant wave height is essential for marine and coastal infrastructure, particularly as climate change intensifies the frequency and intensity of extreme wave events. Traditional models, which assume stationarity in wave data, often underestimate future risks by neglecting the impacts of climate change on wave dynamics. Combining time series decomposition and recurrence analysis, the research develops a nonstationary framework to predict significant wave height. The stochastic component is modelled using a stationary probability distribution, while the deterministic component is predicted based on sea surface temperature projections from CMIP6 climate scenarios. The model evaluation demonstrates strong predictive capability for both stochastic and deterministic components. Application of the model to China’s coastal waters reveals significant discrepancies between stationary and nonstationary return value estimates. Compared to conventional distribution models, the nonstationary model predicts substantial increases in extreme wave heights. These findings underscore the importance of adopting nonstationary models to more accurately assess future risks posed by extreme wave events in a changing climate.

Nonlinear Dynamics for the Ising Model

We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann’s model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely spin systems at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} and O(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log n)$$\\end{document} respectively, where n is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erdős-Rényi random graphs, and non-traditional branching processes augmented by a “fragmentation” process. Our results extend immediately to any spin system with a finite number of spins and bounded interactions.

Influence of electron cyclotron current drive on the pressure crashes caused by the 2/1 double tearing modes

Abstract A module with self-consistent evolution of driven current is developed and coupled with the resistive-MHD equations in the three-dimensional, toroidal, and nonlinear simulation code (CLT). The driven current equation is solved with a second-order accuracy symmetric scheme, which exhibits good conservation properties. With the new module, we find that the driven current can self-consistently concentrate inside the magnetic island when the parallel diffusion of the driven current is sufficiently large. The efficiency of the driven current on tearing mode suppression will then be much higher than those with stationary distributions. With the new module, the influence of electron cyclotron current drive (ECCD) on the nonlinear evolution of the 2/1 double tearing modes (DTMs) is investigated. When co-ECCD deposits on the outer resonant surface, the local magnetic shear is reduced, and the growth rates of the DTMs decrease; if ctr-ECCD deposits on the outer resonant surface, the local magnetic shear increases, and the DTMs become more unstable. However, things will be different if ECCD deposits on the inner resonant surface since the local magnetic shear is negative. The co-ECCD deposited on the inner resonant surface increases the negative shear and then promotes the growth of the DTMs; while the ctr-ECCD suppresses the DTMs. It is also found that the off-axis and central pressure crashes associated with the 2/1 DTMs can be converted to each other by properly depositing the driven current. To convert a central crash to an off-axis crash, the co-ECCD should be deposited on the outer resonant surface, or the ctr-ECCD deposited on the inner resonant surface. While, the co-ECCD should be deposited on the inner rational surface, or the ctr-ECCD deposited on the outer rational surface to convert an off-axis crash to a central crash. The co- or ctr-ECCD should be larger than a threshold for such transitions, and the threshold value is mainly determined by the location of the inner resonant surface.

Dual process in the two-parameter Poisson–Dirichlet diffusion

The two-parameter Poisson–Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson–Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman’s coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson–Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman’s Pólya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.

Stochastic Optimal Control Analysis for HBV Epidemic Model with Vaccination

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV.

Extinction Dynamics and Equilibrium Patterns in Stochastic Epidemic Model for Norovirus: Role of Temporal Immunity and Generalized Incidence Rates

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic SEIQR epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when R^s>1. Extinction occurs when R0E<1. Computer simulations confirm that noise level significantly influences epidemic spread.

Asymmetric autocatalytic reactions and their stationary distribution

We consider a general class of autocatalytic reactions, which has been shown to display stochastic switching behaviour (discreteness-induced transitions (DITs)) in some parameter regimes. This behaviour was shown to occur either when the overall species count is low or when the rate of inflow and outflow of species is relatively much smaller than the rate of autocatalytic reactions. The long-term behaviour of this class was analysed in Bibbona et al. (Bibbona et al. 2020 J. R. Soc. Interface 17 , 20200243 (doi:10.1098/rsif.2020.0243)) with an analytic formula for the stationary distribution in the symmetric case. We focus on the case of asymmetric autocatalytic reactions and provide a formula for an approximate stationary distribution of the model. We show this distribution has different properties corresponding to the distinct behaviour of the process in the three parameter regimes; in the DIT regime, the formula provides the fraction of time spent at each of the stable points.

A new consecutive high-performance counter-current chromatography approach based on co-current and overlap mode for isolating active compounds from olive leaves

Conventional consecutive separation methods based on overlapping injection modes produce interactions between samples as the separation time increases. This will lead to a reduction in the purity of the target due to the gradual saturation of the stationary phase distribution. To enhance the purity of the sample preparation and separation efficiency, we devised a consecutive separation method under high-performance counter-current chromatography (HPCCC) utilizing co-current elution and overlapping injection. The method was used for the separation and preparation of olive leaves for their main bioactive polyphenolic constituents. Six HPCCC separations were conducted in reversed-phase mode using a two-phase solvent system consisting of petroleum ether/ethyl acetate/water (1:40:40, v/v). Finally, 302.7 mg of luteolin-4’-O-β-D-glucoside and oleuropein were isolated from 720 mg of crude sample with purities of 93.3 % and 96.2 %, respectively. The optimization strategy outlined in this study can be used to enhance the extraction of bioactive components from olive leaves.

Meta-Reinforcement Learning in Nonstationary and Nonparametric Environments.

Recent state-of-the-art artificial agents lack the ability to adapt rapidly to new tasks, as they are trained exclusively for specific objectives and require massive amounts of interaction to learn new skills. Meta-reinforcement learning (meta-RL) addresses this challenge by leveraging knowledge learned from training tasks to perform well in previously unseen tasks. However, current meta-RL approaches limit themselves to narrow parametric and stationary task distributions, ignoring qualitative differences and nonstationary changes between tasks that occur in the real world. In this article, we introduce a Task-Inference-based meta-RL algorithm using explicitly parameterized Gaussian variational autoencoders (VAEs) and gated Recurrent units (TIGR), designed for nonparametric and nonstationary environments. We employ a generative model involving a VAE to capture the multimodality of the tasks. We decouple the policy training from the task-inference learning and efficiently train the inference mechanism on the basis of an unsupervised reconstruction objective. We establish a zero-shot adaptation procedure to enable the agent to adapt to nonstationary task changes. We provide a benchmark with qualitatively distinct tasks based on the half-cheetah environment and demonstrate the superior performance of TIGR compared with state-of-the-art meta-RL approaches in terms of sample efficiency (three to ten times faster), asymptotic performance, and applicability in nonparametric and nonstationary environments with zero-shot adaptation. Videos can be viewed at https://videoviewsite.wixsite.com/tigr.

Dynamics of a stochastic impulsive vegetation system with regime switching

This paper describes an analytical and numerical investigation of a novel stochastic impulsive vegetation system with regime switching. Impulsive control, white noise, regime switching, and rainfall are modeled as crucial factors to simulate natural ecological phenomena, with the aim to explore the effects of those factors on the dynamics of vegetation system. Dynamics of the system, including the existence and uniqueness of global positive solutions, extinction, non-persistence in the mean, weak persistence and stochastic persistence, are investigated firstly under the effects of impulsive control and regime switching. For system without impulsive perturbations, we investigate the existence and uniqueness of the stationary distribution for the system, demonstrating that the vegetation can survive forever. Using a sophisticated sensitivity analysis technique, it is found that the vegetation biomass is highly sensitive to the rainfall but least sensitive to the saturation constant of the vegetation consumption. Our numerical results reveal that either the reduced rainfall or increased environmental disturbance can tip the balance of vegetation system, but this relationship can be effectively regulated by implementing impulsive control schemes. Additionally, it is observed that system at high-level rainfall may be detrimental to maintain the balance of vegetation, but regime switching in rainfall patterns can balance different states of vegetation and provide higher survival chance for vegetation. Significantly, it emphasizes that the persistence-extinction behaviors of the vegetation are more sensitive to the change of regime switching, indicating that the proper and general environmental disturbances is beneficial to maintain dynamic balance of vegetation system. These findings may provide new insights into the complex vegetation system dynamics.

Stationary distribution of a stochastic generalized SIRI epidemic model with reinfection and relapse

In this paper, we propose and investigate the stochastic SIRI epidemic model with two generalized incidence rate functions. We firstly study the existence and uniqueness of the globally positive solution to the stochastic SIRI model with positive initial value. Then we obtain sufficient conditions for the extinction of the disease in the stochastic epidemic model, and find that the large noise can make the disease die out exponentially. Meanwhile, we obtain that the solution to the stochastic model has a unique stationary distribution when R0s˜ is greater than one. Our results show that the intensity of white noise can affect the dynamical behaviors of the model. Finally, we use numerical simulation to illustrate theoretical results, and apply both the stochastic and deterministic models to analyze the outbreak of COVID-19 epidemic in Serbia.

Gelation in input-driven aggregation.

We investigate irreversible aggregation processes driven by a source of small mass clusters. In the spatially homogeneous situation, a well-mixed system consists of clusters of various masses whose concentrations evolve according to an infinite system of nonlinear ordinary differential equations. We focus on the cluster mass distribution in the long-time limit. An input-driven aggregation with rates proportional to the product of merging partners undergoes a percolation transition. We examine this process analytically and numerically. There are two theoretical schemes and two natural ways of numerical integration on the level of a truncated system with a finite number of equations. After the percolation transition, the behavior depends on the adopted approach: The giant component quickly engulfs the entire system (Flory approach), or a nontrivial stationary mass distribution emerges (Stockmayer approach). We also outline ageneralization to ternary aggregation.

The Dynamic Behavior of a Stochastic SEIRM Model of COVID-19 with Standard Incidence Rate

This paper studies the dynamic behavior of a stochastic SEIRM model of COVID-19 with a standard incidence rate. The existence of global solutions for dynamic system models is proven by integrating stochastic process theory and the concept of stopping times, together with the contradiction method. Moreover, we construct appropriate Lyapunov functions to analyze system stability and apply Dynkin’s formula and Fatou’s lemma to handle stopping times and expectations of stochastic processes. Notably, the extinction study provides mathematical proof that under the given system dynamics, the total population does not grow indefinitely but tends to stabilize over time. The properties of the diffusion matrix are harnessed to guarantee the system’s stationary distribution. Conclusively, numerical simulations confirm the model’s extinction outcomes.

Nonequilibrium cluster-cluster aggregation in the presence of anchoring sites.

Nonequilibrium cluster-cluster aggregation of particles diffusing in or at the cell membrane has been hypothesized to lead to domains of finite size in different biological contexts, such as lipid rafts, cell adhesion complexes, or postsynaptic domains in neurons. In this scenario, the desorption of particles balances a continuous flux to the membrane, imposing a cutoff on possible aggregate sizes and giving rise to a stationary size distribution. Here, we investigate the case of nonequilibrium cluster-cluster aggregation in two dimensions where diffusing particles and/or clusters remain fixed in space at specific anchoring sites, which should be particularly relevant for synapses but may also be present in other biological or physical systems. Using an effective mean-field description of the concentration field around anchored clusters, we derive an expression for their average size as a function of parameters such as the anchoring site density. We furthermore propose and solve appropriate rate equationsthat allow us to predict the size distributions of both diffusing and fixed clusters. We confirm our results with particle-based simulations and discuss potential implications for biological and physical systems.

Interactive multi-agent convolutional broad learning system for EEG emotion recognition

Electroencephalogram (EEG) emotion recognition is gaining significance in intelligent human–computer interaction. Multi-agent learning can capture more complete and reliable features, compensating for the lack of a single agent’s knowledge. However, previous EEG emotion recognition only focused on a single agent. Therefore, to extract effective features from high-dimensional EEG data, a novel interactive multi-agent (IMA) learning framework is proposed, and introduced into convolutional broad learning system (CNNBLS), then the interactive multi-agent convolutional broad learning system (IMA-CNNBLS) is proposed for EEG emotion recognition. It can effectively model high-dimensional EEG data, automatically extract EEG features related to emotions through convolutional neural network (CNN), and then extend the above features to a vast space to quickly extract generalized features through broad learning system (BLS), consider a CNNBLS as an agent, and multi-agent interact in the IMA part. As the theoretical basis of IMA-CNNBLS, the consistency is mathematically proved through the stationary distribution theory of Markov processes. To demonstrate the superiority of our proposed model, sufficient experiments are conducted on the DEAP, DREAMER and SEED datasets. The experimental results show the model can effectively improve the EEG emotion recognition accuracy, the best recognition performance is achieved on all datasets. In addition, our proposed model also shows the multi-agent interaction significantly affects EEG emotion recognition.