4,561 results on '"Stationary distribution"'
Search Results
2. Dynamic Properties for a Second-Order Stochastic SEIR Model with Infectivity in Incubation Period and Homestead-Isolation of the Susceptible Population
- Author
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Chun Lu, Honghui Liu, and Junhua Zhou
- Subjects
Statistics and Probability ,Statistical and Nonlinear Physics ,stochastic SEIR model ,stationary distribution ,extinction ,density function ,Analysis - Abstract
In this article, we analyze a second-order stochastic SEIR epidemic model with latent infectious and susceptible populations isolated at home. Firstly, by putting forward a novel inequality, we provide a criterion for the presence of an ergodic stationary distribution of the model. Secondly, we establish sufficient conditions for extinction. Thirdly, by solving the corresponding Fokker–Plank equation, we derive the probability density function around the quasi-endemic equilibrium of the stochastic model. Finally, by using the epidemic data of the corresponding deterministic model, two numerical tests are presented to illustrate the validity of the theoretical results. Our conclusions demonstrate that nations should persevere in their quarantine policies to curb viral transmission when the COVID-19 pandemic proceeds to spread internationally.
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- 2023
- Full Text
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3. Threshold Analysis of a Stochastic SIRS Epidemic Model with Logistic Birth and Nonlinear Incidence
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Huyi Wang, Ge Zhang, Tao Chen, and Zhiming Li
- Subjects
General Mathematics ,Computer Science (miscellaneous) ,stochastic SIRS epidemic model ,Logistic birth ,nonlinear incidence ,global stability ,stationary distribution ,Engineering (miscellaneous) - Abstract
The paper mainly investigates a stochastic SIRS epidemic model with Logistic birth and nonlinear incidence. We obtain a new threshold value (R0m) through the Stratonovich stochastic differential equation, different from the usual basic reproduction number. If R0m1, the disease is permanent in the mean with probability one and has an endemic stationary distribution. Numerical simulations are given to illustrate the theoretical results. Interestingly, we discovered that random fluctuations can suppress outbreaks and control the disease.
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- 2023
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4. Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations
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Hui Chen, Xuewen Tan, Jun Wang, Wenjie Qin, and Wenhui Luo
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epidemic model ,vaccine inoculation ,extinction ,stationary distribution ,General Mathematics ,Computer Science (miscellaneous) ,Engineering (miscellaneous) - Abstract
In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold R0* can be used to control the stochastic dynamics of the model. If R0*0, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model.
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- 2023
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5. Stochastic resonance of an asymmetric tristable system driven by cross-correlated Ornstein–Uhlenbeck noise
- Author
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Yilin Liu, Lifang He, and Gang Zhang
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Physics ,Stationary distribution ,Stochastic resonance ,General Physics and Astronomy ,Probability density function ,Ornstein–Uhlenbeck process ,Statistical physics ,First-hitting-time model ,Adiabatic process ,Noise (electronics) ,Stability (probability) - Abstract
Stochastic resonance of an asymmetric tristable system driven by a periodic forcing and cross-correlated Ornstein-Uhlenbeck (O-U) noise is investigated. An approximate Fokker-Planck Equation for an asymmetric tristable system driven by a periodic forcing and cross-correlated O-U noise is derived in the adiabatic limit by using the unified colored-noise approximation method and the stochastic equivalent rules. Afterwards, the expression for spectral amplification (SA), the stationary probability density function (PDF) and the mean first passage time (MFPT) are derived. By considering the influence of asymmetric potential function and cross-correlated noise, the phenomenon of noise enhancing stability (NES) and stochastic resonance (SR) can be found. It can also be found that the response of the tristable system can be effectively improved by properly selecting the relevant noise intensity and the asymmetric constant. This proves the excellent performance of the system, and provides a good theoretical support for practical engineering application.
- Published
- 2022
6. Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects
- Author
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Humera Batool, Weiyu Li, and Zhonggui Sun
- Subjects
numerical simulations ,Physics and Astronomy (miscellaneous) ,Chemistry (miscellaneous) ,extinction ,General Mathematics ,Computer Science (miscellaneous) ,stochastic COVID-19 epidemic model ,stationary distribution - Abstract
Human society always wants a safe environment from pollution and infectious diseases, such as COVID-19, etc. To control COVID-19, we have started the big effort for the discovery of a vaccination of COVID-19. Several biological problems have the aspects of symmetry, and this theory has many applications in explaining the dynamics of biological models. In this research article, we developed the stochastic COVID-19 mathematical model, along with the inclusion of a vaccination term, and studied the dynamics of the disease through the theory of symmetric dynamics and ergodic stationary distribution. The basic reproduction number is evaluated using the equilibrium points of the proposed model. For well-posedness, we also test the given problem for the existence and uniqueness of a non-negative solution. The necessary conditions for eradicating the disease are also analyzed along with the stationary distribution of the proposed model. For the verification of the obtained result, simulations of the model are performed.
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- 2023
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7. Modeling evidence accumulation decision processes using integral equations: Urgency-gating and collapsing boundaries
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Philip L. Smith and Roger Ratcliff
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Flexibility (engineering) ,Stationary distribution ,Computer science ,media_common.quotation_subject ,Decision Making ,Integral equation ,Article ,Stimulus (psychology) ,Encoding (memory) ,Perception ,Reaction Time ,Humans ,Diffusion (business) ,Constant (mathematics) ,Algorithm ,General Psychology ,media_common - Abstract
Diffusion models of evidence accumulation have successfully accounted for the distributions of response times and choice probabilities from many experimental tasks, but recently their assumption that evidence is accumulated at a constant rate to constant decision boundaries has been challenged. One model assumes that decision-makers seek to optimize their performance by using decision boundaries that collapse over time. Another model assumes that evidence does not accumulate and is represented by a stationary distribution that is gated by an urgency signal to make a response. We present explicit, integral-equation expressions for the first-passage time distributions of the urgency-gating and collapsing-bounds models and use them to identify conditions under which the models are equivalent. We combine these expressions with a dynamic model of stimulus encoding that allows the effects of perceptual and decisional integration to be distinguished. We compare the resulting models to the standard diffusion model with variability in drift rates on data from three experimental paradigms in which stimulus information was either constant or changed over time. The standard diffusion model was the best model for tasks with constant stimulus information; the models with time-varying urgency or decision bounds performed similarly to the standard diffusion model on tasks with changing stimulus information. We found little support for the claim that evidence does not accumulate and attribute the good performance of the time-varying models on changing-stimulus tasks to their increased flexibility and not to their ability to account for systematic experimental effects. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
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- 2022
8. EXTINCTION AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE AND STAGE STRUCTURE OF PREY
- Author
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Rongyan Wang and Wencai Zhao
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Extinction ,Stationary distribution ,Ecology ,General Mathematics ,Functional response ,Stage (hydrology) ,Biology ,Predation - Published
- 2022
9. Stationary distribution and density function analysis of a stochastic epidemic HBV model
- Author
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Junyan Ge, Daqing Jiang, and Wenjie Zuo
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Lyapunov function ,Numerical Analysis ,Stationary distribution ,General Computer Science ,Stochastic modelling ,Applied Mathematics ,Ergodicity ,Ode ,Probability density function ,Critical value ,Theoretical Computer Science ,symbols.namesake ,Modeling and Simulation ,symbols ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Basic reproduction number ,Mathematics - Abstract
In this paper, we present a stochastic hepatitis B virus (HBV) infection model and the dynamic behaviors of the model are investigated. When the fraction of vertical transmission μ ω ν C is not considered to be new infections, the existence and ergodicity of the stationary distribution of the model are obtained by constructing a suitable Lyapunov function, which determines a critical value ρ 0 s corresponding to the basic reproduction number of ODE system. This implies the persistence of the diseases when ρ 0 s > 1 . Meanwhile, the sufficient conditions for the extinction of the diseases are derived when ρ 0 T 0 . What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. Finally, the numerical simulations are illustrated to verify the theoretical results and match the HBV epidemic data in China.
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- 2022
10. STATIONARY DISTRIBUTION AND CONTROL STRATEGY OF A STOCHASTIC DENGUE MODEL WITH SPATIAL DIFFUSION
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Kangkang Chang, Huaimin Yuan, and Qimin Zhang
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Stationary distribution ,General Mathematics ,medicine ,Statistical physics ,medicine.disease ,Spatial diffusion ,Dengue fever ,Mathematics - Published
- 2022
11. Analog Solutions of Discrete Markov Chains via Memristor Crossbars
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Fernando Corinto, Anil Korkmaz, Gianluca Zoppo, R. Stanley Williams, Samuel Palermo, and Francesco Marrone
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Stationary distribution ,Markov chain ,Computer science ,Markov process ,Memristor ,law.invention ,Accuracy ,analog computation ,crossbars ,discrete Markov chains ,memristor ,pagerank ,precision ,symbols.namesake ,law ,symbols ,Multiplication ,Electrical and Electronic Engineering ,Crossbar switch ,Algorithm ,Eigenvalues and eigenvectors ,Electronic circuit - Abstract
Problems involving discrete Markov Chains are solved mathematically using matrix methods. Recently, several research groups have demonstrated that matrix-vector multiplication can be performed analytically in a single time step with an electronic circuit that incorporates an open-loop memristor crossbar that is effectively a resistive random-access memory. Ielmini and co-workers have taken this a step further by demonstrating that linear algebraic systems can also be solved in a single time step using similar hardware with feedback. These two approaches can both be applied to Markov chains, in the first case using matrix-vector multiplication to compute successive updates to a discrete Markov process and in the second directly calculating the stationary distribution by solving a constrained eigenvector problem. We present circuit models for open-loop and feedback configurations, and perform detailed analyses that include memristor programming errors, thermal noise sources and element nonidealities in realistic circuit simulations to determine both the precision and accuracy of the analog solutions. We provide mathematical tools to formally describe the trade-offs in the circuit model between power consumption and the magnitude of errors. We compare the two approaches by analyzing Markov chains that lead to two different types of matrices, essentially random and ill-conditioned, and observe that ill-conditioned matrices suffer from significantly larger errors. We compare our analog results to those from digital computations and find a significant power efficiency advantage for the crossbar approach for similar precision results.
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- 2021
12. Non-standard limits for a family of autoregressive stochastic sequences
- Author
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Sergey Foss and Matthias Schulte
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Statistics and Probability ,Normal distribution ,Distribution (mathematics) ,Stationary distribution ,Markov chain ,Autoregressive model ,Characteristic function (probability theory) ,Applied Mathematics ,Modeling and Simulation ,Asymptotic distribution ,Applied mathematics ,Random variable ,Mathematics - Abstract
We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.
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- 2021
13. Impact of information intervention on stochastic dengue epidemic model
- Author
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Peijiang Liu, Anwarud Din, and Zenab
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Lyapunov function ,Dengue stochastic epidemic model ,Stochastic modelling ,Computer science ,020209 energy ,02 engineering and technology ,Numerical simulation ,01 natural sciences ,Quantitative Biology::Other ,010305 fluids & plasmas ,Dengue fever ,Stochastic differential equation ,symbols.namesake ,Stationary distribution ,Exponential stability ,Intervention (counseling) ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,medicine ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,Extinction analysis ,General Engineering ,Environmental noise ,medicine.disease ,Engineering (General). Civil engineering (General) ,humanities ,symbols ,Information intervention ,TA1-2040 ,Epidemic model ,Basic reproduction number - Abstract
The investigated manuscript is devoted to the study of the dengue epidemic model. Generally, the infection of dengue occurs in extremely hot and humid conditions where the transmission and control depend on many factors like information intervention, humid weather, etc. With the help of the deterministic dengue model, we have taken the stochastic perturbed dengue model along with the factor of information intervention. The qualitative analysis for the positive solution of the stochastic differential equation based model is built. The scheme for basic reproduction number introduced for sure exponential stability by constructing the Lyapunov function. For the validity of our obtained results, we have simulated the proposed scheme and compare the stochastic model with its corresponding deterministic version.
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- 2021
14. Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion
- Author
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Zhouchao Wei and Yuxi Li
- Subjects
Original Paper ,Stationary distribution ,Computer simulation ,Reaction–diffusion ,Turing instability ,Applied Mathematics ,Mechanical Engineering ,COVID-19 ,Aerospace Engineering ,Ocean Engineering ,Optimal control ,Nonlinear system ,symbols.namesake ,Stochastic epidemic model ,Control and Systems Engineering ,Reaction–diffusion system ,Taylor series ,symbols ,Applied mathematics ,Uniqueness ,Electrical and Electronic Engineering ,Epidemic model ,Amplitude equations ,Mathematics - Abstract
In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.
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- 2021
15. The stationary distribution of a stochastic rumor spreading model
- Author
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Dapeng Gao, Peng Guo, and Chaodong Chen
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Lyapunov function ,Stationary distribution ,Stochastic modelling ,General Mathematics ,lcsh:Mathematics ,White noise ,Rumor ,lcsh:QA1-939 ,stationary distribution ,symbols.namesake ,rumor spreading ,symbols ,threshold ,Applied mathematics ,Ergodic theory ,Uniqueness ,Persistence (discontinuity) ,Mathematics - Abstract
In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.
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- 2021
16. Mean square exponential stability of discrete-time Markov switched stochastic neural networks with partially unstable subsystems and mixed delays
- Author
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Quanxin Zhu and Lina Fan
- Subjects
Mean square ,Information Systems and Management ,Stationary distribution ,Markov chain ,Computer Science Applications ,Theoretical Computer Science ,Term (time) ,Discrete time and continuous time ,Exponential stability ,Artificial Intelligence ,Control and Systems Engineering ,Bernoulli distribution ,Applied mathematics ,Stochastic neural network ,Software ,Mathematics - Abstract
In this paper, we study the mean square exponential stability of discrete-time stochastic neural networks with partially unstable subsystems and mixed delays. The mixed delays under consideration involve discrete delay and distributed delay. Moreover, the discrete delay term satisfies the Bernoulli distribution . Different from the deterministic switching, we consider Markov switching and our system has partially unstable subsystems. By constructing a novel Lyapunov–Krasovskii functional and using the stationary distribution of Markov chain , we give sufficient conditions for the mean square exponential stability of the suggested system. Finally, two numerical examples are given to check the theory results.
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- 2021
17. Energy saving strategy and Nash equilibrium of hybrid P2P networks
- Author
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Liyuan Zhang, Shunzhi Wang, Changzhen Zhang, and Zhanyou Ma
- Subjects
Consumption (economics) ,Queueing theory ,Mathematical optimization ,Stationary distribution ,Computer Networks and Communications ,Computer science ,Energy consumption ,Differentiated service ,Theoretical Computer Science ,symbols.namesake ,Artificial Intelligence ,Hardware and Architecture ,Asynchronous communication ,Nash equilibrium ,Computer Science::Networking and Internet Architecture ,symbols ,Software ,Energy (signal processing) - Abstract
This paper proposes a penalty strategy with differentiated service rate based on the free riding phenomenon in P2P networks, and establishes an M/M/c+d queueing model. Based on this model, a sleep/wakeup mechanism is introduced for the peers at the service end, and a single asynchronous vacation strategy is adopted to reduce the energy consumption of the system. In addition, the energy consumption of peers in each state is quantified, and the relationship between the energy consumption and parameters of the system is analyzed. In order to avoid excessive requests for unnecessary services from requesting nodes and increasing energy consumption of the system, this paper analyzes the Nash equilibrium between the arrival rate and the net profit of a single node, and then studies the optimization of social profit. The stationary distribution of queueing model is obtained by the method of matrix geometric solution, the performance indicators of the system are constructed, and the system performance is analyzed by numerical experiments. Experimental results show that the model developed in this paper has a significant penalty effect on free riding behavior, and that the single asynchronous vacation strategy not only saves more than 10% of the total energy consumption compared with the single synchronous vacation strategy, but also makes the hybrid P2P networks more flexible and efficient.
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- 2021
18. A heavy-traffic-limit formula for the moments of the stationary distribution in GI/G/1-type Markov chains
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Tatsuaki Kimura and Hiroyuki Masuyama
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Stationary distribution ,Markov chain ,Characteristic function (probability theory) ,Applied Mathematics ,Mathematical analysis ,Limit (mathematics) ,Management Science and Operations Research ,Type (model theory) ,Heavy traffic ,Industrial and Manufacturing Engineering ,Software ,Mathematics - Abstract
This paper studies the heavy-traffic limit of the moments of the stationary distribution in GI/G/1-type Markov chains. For these Markov chains, several researchers have derived heavy-traffic-limit formulas for the stationary distribution itself. However, for its moments, no such formulas have been reported in the literature. This paper presents a heavy-traffic-limit formula for the moments of the stationary distribution and a sufficient condition for the formula to hold, by using a characteristic function approach.
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- 2021
19. Stochastic analysis of a SIRI epidemic model with double saturated rates and relapse
- Author
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Yan Zhang, Shujing Gao, and Shihua Chen
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Computational Mathematics ,Stationary distribution ,Extinction ,Stochastic process ,Applied Mathematics ,Econometrics ,Quantitative Biology::Populations and Evolution ,Ergodic theory ,Relapse rate ,Uniqueness ,Epidemic model ,Mathematics - Abstract
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.
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- 2021
20. The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission
- Author
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Guihua Li and Yuanhang Liu
- Subjects
Lyapunov function ,State variable ,Stationary distribution ,Article Subject ,Computer simulation ,Stochastic modelling ,symbols.namesake ,Modeling and Simulation ,QA1-939 ,symbols ,Initial value problem ,Applied mathematics ,Ergodic theory ,Epidemic model ,Mathematics - Abstract
In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold R 0 s is given. When R 0 s < 1 , the diseases can be extincted with probability one. When R 0 s > 1 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.
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- 2021
21. A stochastic turbidostat model coupled with distributed delay and degenerate diffusion: dynamics analysis
- Author
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Daqing Jiang, Xiaojie Mu, Ahmed Alsaedi, and Bashir Ahmad
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Computational Mathematics ,Current (mathematics) ,Stationary distribution ,Applied Mathematics ,Theory of computation ,Turbidostat ,Ergodic theory ,White noise ,Uniqueness ,State (functional analysis) ,Statistical physics ,Mathematics - Abstract
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.
- Published
- 2021
22. Uniqueness of Stationary Distributions in Random Access Poisson Networks
- Author
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Plinio S. Dester, Paulo Cardieri, and Pedro H. J. Nardelli
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Queueing theory ,Stationary distribution ,Asymptotic distribution ,Poisson distribution ,Computer Science Applications ,symbols.namesake ,Modeling and Simulation ,symbols ,Applied mathematics ,Uniqueness ,Electrical and Electronic Engineering ,Random access ,Mathematics ,Counterexample ,Rayleigh fading - Abstract
This letter presents sufficient conditions for the existence and uniqueness of the limiting stationary distribution in random access Poisson networks with packet queueing and under Rayleigh fading and a general path loss model. This system model is traditionally used in the literature assuming uniqueness of the stationary distribution. We demonstrate here through a counterexample that this assumption might not always hold. From the sufficient conditions, an interesting and perhaps counterintuitive result emerged, that is, the arrival rate of packets per node greater than 1/e guarantees uniqueness of the limiting distribution. When that is the case, then setting the medium access probability to 1 minimizes the proportion of unstable nodes.
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- 2021
23. Continuity properties and the support of killed exponential functionals
- Author
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Alexander Lindner, Jana Reker, Victor Rivero, and Anita Behme
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Statistics and Probability ,Stationary distribution ,Exponential distribution ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Absolute continuity ,01 natural sciences ,Lévy process ,Exponential function ,010104 statistics & probability ,Stochastic differential equation ,Mathematics::Probability ,Modeling and Simulation ,FOS: Mathematics ,High Energy Physics::Experiment ,0101 mathematics ,Connection (algebraic framework) ,Random variable ,Mathematics - Probability ,Mathematical physics ,Mathematics - Abstract
For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.
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- 2021
24. Dynamics Analysis of a Class of Stochastic SEIR Models with Saturation Incidence Rate
- Author
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Pengpeng Liu and Xuewen Tan
- Subjects
Physics and Astronomy (miscellaneous) ,saturation incidence ,Lyapunov functions ,gradual properties ,stochastic persistence ,extinction ,stationary distribution ,numerical simulation ,Chemistry (miscellaneous) ,General Mathematics ,Computer Science (miscellaneous) - Abstract
In this article, a class of stochastic SEIR models with saturation incidence is studied. The model is a symmetric and compatible distribution family. This paper studies various properties of the system by constructing Lyapunov functions. First, the gradual properties of the systematic solution near the disease-free equilibrium of the deterministic model is studied, followed by the final behavior of the model, including stochastic persistence and final extinction. Finally, the existence conditions of the stationary distribution of the model are given, and then it is proved that it is traversed, and the corresponding conclusions are verified through numerical simulation.
- Published
- 2022
- Full Text
- View/download PDF
25. Estimation of stationary probability of semi-Markov Chains
- Author
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Bei Wu and Nikolaos Limnios
- Subjects
Statistics and Probability ,Stationary distribution ,Markov chain ,Rate of convergence ,Consistency (statistics) ,Ergodic theory ,Applied mathematics ,Asymptotic distribution ,Estimator ,Law of the iterated logarithm ,Mathematics - Abstract
This paper concerns the estimation of stationary probability of ergodic semi-Markov chains based on an observation over a time interval. We derive asymptotic properties of the proposed estimator, when the time of observation goes to infinity, as consistency, asymptotic normality, law of iterated logarithm and rate of convergence in a functional setting. The proofs are based on asymptotic results on discrete-time semi-Markov random evolutions.
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- 2021
26. Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine‐induced immunity under nonlinear regime switching
- Author
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Daqing Jiang, Bingtao Han, Tasawar Hayat, and Baoquan Zhou
- Subjects
Nonlinear system ,Stationary distribution ,Transmission (telecommunications) ,General Mathematics ,Ergodicity ,General Engineering ,Ergodic theory ,Regime switching ,Statistical physics ,Mathematics - Published
- 2021
27. Rich dynamics in a stochastic predator-prey model with protection zone for the prey and multiplicative noise applied on both species
- Author
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Mustapha Belabbas, Fethi Souna, and Abdelghani Ouahab
- Subjects
Lyapunov function ,education.field_of_study ,Stationary distribution ,Extinction ,Applied Mathematics ,Mechanical Engineering ,Population ,Aerospace Engineering ,Ocean Engineering ,Predation ,symbols.namesake ,Stochastic differential equation ,Nonlinear Sciences::Adaptation and Self-Organizing Systems ,Control and Systems Engineering ,symbols ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Ergodic theory ,Uniqueness ,Electrical and Electronic Engineering ,education ,Mathematics - Abstract
In this manuscript, a new approach of a stochastic predator-prey interaction with protection zone for the prey is developed and studied. The considered mathematical model consists of a system of two stochastic differential equations, SDEs, describing the interaction between the prey and predator populations where the prey exhibits a social behavior called also by “herd behavior.” First, according to the theory of the SDEs, some properties of the solution are obtained, including: the existence and uniqueness of the global positive solution and the stochastic boundedness of the solutions. Then, the sufficient conditions for the persistence in the mean and the extinction of the species are established, where the extinction criteria are discussed in two different cases, namely, the first case is the survival of the prey population, while the predator population goes extinct; the second case is the extinction of all prey and predator populations. Next, by constructing a suitable stochastic Lyapunov function and under certain parametric restrictions, it has been proved that the system has a unique stationary distribution which is ergodic. Finally, some numerical simulations based on the Milstein’s higher-order scheme are performed to illustrate the theoretical predictions.
- Published
- 2021
28. Conversion of a stable fixed point into a transient peak by stochastic fluctuation in a gene regulatory network
- Author
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Julian Lee
- Subjects
Maxima and minima ,Physics ,Stationary distribution ,Exponential growth ,Gene regulatory network ,General Physics and Astronomy ,Transient (oscillation) ,Statistical physics ,Limit (mathematics) ,Fixed point ,Numerical integration - Abstract
The deterministic rate equation has been considered as an approximate description of the dynamics of the gene regulatory network, derived in the limit of vanishing stochastic fluctuations. When stochastic fluctuations are introduced, the stable fixed points of the deterministic equation usually turn into local maxima of the stationary distribution. Here, I consider a simple genetic regulatory network with a positive feedback and a cooperative binding, where the stationary distribution does not have a local minimum at a stable fixed point. I show that a transient peak appears at the stable fixed point instead, whose life-time grows exponentially as the stochastic fluctuation decreases. The result shows that the order of the limits is important for retrieving the deterministic dynamics.
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- 2021
29. Continuous Random Process Modeling of AGC Signals Based on Stochastic Differential Equations
- Author
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Feng Liu, Yiwei Qiu, Jin Lin, Yonghua Song, and Ning-Yi Dai
- Subjects
Stochastic differential equation ,Electric power system ,Stationary distribution ,Automatic Generation Control ,Control theory ,Stochastic process ,Computer science ,Energy Engineering and Power Technology ,Probability distribution ,Probability density function ,Electrical and Electronic Engineering ,Signal - Abstract
Reflecting the uncertainty of renewable energy generations and loads, power system AGC signals are essentially random processes. For the sake of the optimal operation and control of the AGC participant, such as energy storage systems (ESSs) in a performance/mileage-based regulation market, taking the uncertain nature, especially the temporal correlation, of the AGC signals into consideration can be beneficial; hence, random process models of the AGC signals are needed. However, a continuous random process model of the AGC signal that jointly considers the probability distribution and the temporal correlation is still lacking. To fill this gap, this paper first presents a systematic methodology for modeling the continuous random processes of AGC signals based on stochastic differential equations (SDEs). It is shown that AGC signals may have a saturated stationary probability density function and a biexponential temporal correlation, which are very different from the renewable generations. To capture these special characteristics, SDEs are then carefully constructed, which are easy to use in optimization and control. Using the PJM traditional and dynamic regulation (RegD and RegA) signals and the signal received from a battery ESS (BESS) plant in Jiangsu, China for example, simulation shows that the SDE is able to simultaneously capture the probability distribution and temporal correlation accurately.
- Published
- 2021
30. Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers
- Author
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Yang Song and Huijun Lu
- Subjects
Computer Science::Performance ,Service (business) ,Stationary distribution ,Applied Mathematics ,Applied mathematics ,Management Science and Operations Research ,Orbit (control theory) ,Queue ,Rate function ,Industrial and Manufacturing Engineering ,Software ,Mathematics - Abstract
In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.
- Published
- 2021
31. Long-time behaviors of two stochastic mussel-algae models
- Author
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Dengxia Zhou, Meng Liu, Zhijun Liu, and Ke Qi
- Subjects
Lyapunov function ,Models, Biological ,symbols.namesake ,QA1-939 ,Ergodic theory ,Animals ,Periodic model ,Mathematics ,Stochastic Processes ,Extinction ,Stationary distribution ,random perturbations ,Applied Mathematics ,periodic solution ,General Medicine ,Mussel ,Plants ,Bivalvia ,stationary distribution ,Computational Mathematics ,Modeling and Simulation ,symbols ,mussel-algae models ,General Agricultural and Biological Sciences ,Biological system ,TP248.13-248.65 ,Biotechnology - Abstract
In this paper, we develop two stochastic mussel-algae models: one is autonomous and the other is periodic. For the autonomous model, we provide sufficient conditions for the extinction, nonpersistent in the mean and weak persistence, and demonstrate that the model possesses a unique ergodic stationary distribution by constructing some suitable Lyapunov functions. For the periodic model, we testify that it has a periodic solution. The theoretical findings are also applied to practice to dissect the effects of environmental perturbations on the growth of mussel.
- Published
- 2021
32. Periodicity and stationary distribution of two novel stochastic epidemic models with infectivity in the latent period and household quarantine
- Author
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Ronghua Tan, Zhijun Liu, Lianwen Wang, and Dongchen Shangguan
- Subjects
Lyapunov function ,Periodicity ,Stationary distribution ,Markov chain ,Series (mathematics) ,Applied Mathematics ,Regime switching ,Computational Mathematics ,symbols.namesake ,SEIR epidemic model ,Theory of computation ,92D30 ,symbols ,Ergodic theory ,Applied mathematics ,60H10 ,Ergodic stationary distribution ,Mathematics ,Original Research - Abstract
Two types of stochastic epidemic models are formulated, in which both infectivity in the latent period and household quarantine on the susceptible are incorporated. With the help of Lyapunov functions and Has’minskii’s theory, we derive that, for the nonautonomous periodic version with white noises, it owns a positive periodic solution. For the other version with white and telephone noises, we construct stochastic Lyapunov function with regime switching to present easily verifiable sufficient criteria for the existence of ergodic stationary distribution. Also, we introduce a series of numerical simulations to support our analytical findings. At last, a brief discussion of our theoretical results shows that the stochastic perturbations and household quarantine measures can significantly affect both periodicity and stationary distribution.
- Published
- 2021
33. Permanence of a stochastic prey–predator model with a general functional response
- Author
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Shangjiang Guo and Shangzhi Li
- Subjects
Numerical Analysis ,education.field_of_study ,Extinction ,Stationary distribution ,General Computer Science ,Weak convergence ,Applied Mathematics ,Population ,Functional response ,Boundary (topology) ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,Predation ,Modeling and Simulation ,0202 electrical engineering, electronic engineering, information engineering ,Quantitative Biology::Populations and Evolution ,020201 artificial intelligence & image processing ,Statistical physics ,0101 mathematics ,education ,Predator ,Mathematics - Abstract
Different from the existing methods, a new method is introduced to analyze the stochastic permanence and extinction of a stochastic predator–prey model with a general functional response and the random factors acting on both the intrinsic growth rates and the intra-specific interaction rates. In particular, the existence of a stationary distribution and weak convergence to a boundary process are investigated as well. Some numerical simulations are performed to illustrate our theoretical results and to show that the stochastic noises play an essential role in determining the permanence and extinction. To be more specific, appropriate intensities of white noises may make the predator and prey population fluctuate around their deterministic steady-state values; but too large intensities of white noises may make the predator and/or prey population go to extinction.
- Published
- 2021
34. Analysis of a stochastic mathematical model for tuberculosis with case detection
- Author
-
D. Okuonghae
- Subjects
Lyapunov function ,education.field_of_study ,Control and Optimization ,Stationary distribution ,Case detection ,Mechanical Engineering ,Population ,symbols.namesake ,Control and Systems Engineering ,Modeling and Simulation ,symbols ,Applied mathematics ,Ergodic theory ,Uniqueness ,Electrical and Electronic Engineering ,Persistence (discontinuity) ,education ,Civil and Structural Engineering ,Mathematics - Abstract
In this work, we investigate some qualitative properties of a stochastic dynamical model for tuberculosis with case detection. Using appropriately formulated stochastic Lyapunov functions, we derive sufficient conditions for the existence (and uniqueness) of an ergodic stationary distribution of the positive solutions of the model, guaranteeing persistence of the disease in the presence of case detection. We also obtained conditions that will allow for the eradication of the disease from the population. Using numerical simulations, we were able to illustrate the analytical results obtained herein.
- Published
- 2021
35. Stationary distribution and extinction of a stochastic multigroup DS-DI-a model for the transmission of HIV
- Author
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Qun Liu and Daqing Jiang
- Subjects
Statistics and Probability ,Stationary distribution ,Extinction ,Applied Mathematics ,Ergodicity ,Human immunodeficiency virus (HIV) ,medicine.disease_cause ,Transmission (telecommunications) ,medicine ,Ergodic theory ,Statistical physics ,Uniqueness ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, we analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distributio...
- Published
- 2021
36. Synchronization of Discrete-Time Switched 2-D Systems with Markovian Topology via Fault Quantized Output Control
- Author
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Lei Shi, Zhengwen Tu, Xue Qin, Yi Zou, Xinsong Yang, and Xiaolin Xiong
- Subjects
Stationary distribution ,Markov chain ,Computer Networks and Communications ,Computer science ,General Neuroscience ,Markov process ,Topology (electrical circuits) ,Topology ,Synchronization ,symbols.namesake ,Discrete time and continuous time ,Artificial Intelligence ,Control theory ,symbols ,Almost surely ,Software - Abstract
This paper considers the global exponential synchronization almost surely (GES a.s.) in an array of two-dimensional (2-D) discrete-time systems with Markovian jump topology. Considering the fact that external disturbances are inevitable and communication resources are limited, mode-dependent quantized output control with actuator fault (AF) is designed. Sufficient conditions formulated by linear matrix inequalities (LMIs) are given to ensure the GES a.s. Control gains of the controller without AF is designed by solving the LMIs. It is shown that the stationary distribution of the transition probability of the Markov chain plays an important role in our study, which makes it possible that some of the modes are not controlled. Numerical examples are given to illustrate the effectiveness of theoretical analysis.
- Published
- 2021
37. Estimating drift and minorization coefficients for Gibbs sampling algorithms
- Author
-
David A. Spade
- Subjects
Statistics and Probability ,Stationary distribution ,Computer science ,Applied Mathematics ,Sampling (statistics) ,Markov chain Monte Carlo ,Conditional probability distribution ,Statistics::Computation ,symbols.namesake ,Mixing (mathematics) ,symbols ,Probability distribution ,Algorithm ,Gibbs sampling ,Curse of dimensionality - Abstract
Gibbs samplers are common Markov chain Monte Carlo (MCMC) algorithms that are used to sample from intractable probability distributions when sampling directly from full conditional distributions is possible. These types of MCMC algorithms come up frequently in many applications, and because of their popularity it is important to have a sense of how long it takes for the Gibbs sampler to become close to its stationary distribution. To this end, it is common to rely on the values of drift and minorization coefficients to bound the mixing time of the Gibbs sampler. This manuscript provides a computational method for estimating these coefficients. Herein, we detail the several advantages of the proposed methods, as well as the limitations of this approach. These limitations are primarily related to the “curse of dimensionality”, which for these methods is caused by necessary increases in the numbers of initial states from which chains need be run and the need for an exponentially increasing number of grid points for estimation of minorization coefficients.
- Published
- 2021
38. Long-Time Behavior and Density Function of a Stochastic Chemostat Model with Degenerate Diffusion
- Author
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Xiangdan Wen, Miaomiao Gao, and Daqing Jiang
- Subjects
Stationary distribution ,Markov chain ,Semigroup ,Computer Science (miscellaneous) ,Complex system ,Applied mathematics ,Probability density function ,Chemostat ,Expression (computer science) ,Information Systems ,Deterministic system ,Mathematics - Abstract
This paper considers a stochastic chemostat model with degenerate diffusion. Firstly, the Markov semigroup theory is used to establish sufficient criteria for the existence of a unique stable stationary distribution. The authors show that the densities of the distributions of the solutions can converge in L1 to an invariant density. Then, conditions are obtained to guarantee the washout of the microorganism. Furthermore, through solving the corresponding Fokker-Planck equation, the authors give the exact expression of density function around the positive equilibrium of deterministic system. Finally, numerical simulations are performed to illustrate the theoretical results.
- Published
- 2021
39. Stationary probability density of a stochastic chemostat model with Monod growth response function
- Author
-
Chaoqun Xu
- Subjects
Stationary distribution ,General Mathematics ,General Engineering ,Applied mathematics ,Function (mathematics) ,Chemostat ,Mathematics - Published
- 2021
40. Asymptotic behavior for Markovian iterated function systems
- Author
-
Cheng-Der Fuh
- Subjects
Statistics and Probability ,Combinatorics ,Stationary distribution ,Distribution (mathematics) ,Rate of convergence ,Applied Mathematics ,Modeling and Simulation ,Random function ,Almost surely ,Composition (combinatorics) ,Random variable ,Separable space ,Mathematics - Abstract
Let ( U , d ) be a complete separable metric space and ( F n ) n ≥ 0 a sequence of random functions from U to U . Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function ( F n ) n ≥ 0 is driven by a Markov chain X = { X n , n ≥ 0 } . Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations M n u = F n ∘ ⋯ ∘ F 1 ( u ) , n ≥ 0 , converge weakly to a unique stationary distribution Π for each u ∈ U , where ∘ denotes composition of two maps. The associated backward iterations M n u = F 1 ∘ ⋯ ∘ F n ( u ) are almost surely convergent to a random variable M ∞ which does not depend on u and has distribution Π . Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d ( M ∞ , M n u ) and d ( M n u , M n v ) , u , v ∈ U . The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.
- Published
- 2021
41. Pass-and-Swap Queues
- Author
-
Céline Comte, J.L. Dorsman, Eindhoven University of Technology [Eindhoven] (TU/e), University of Amsterdam [Amsterdam] (UvA), Stochastic Operations Research, Stochastics (KDV, FNWI), and KdV Other Research (FNWI)
- Subjects
FOS: Computer and information sciences ,Computer science ,G.3 ,Quasi-reversibility ,Order-independent queue ,02 engineering and technology ,Management Science and Operations Research ,Assign-to-the-longest-idle-server ,01 natural sciences ,Stability (probability) ,Scheduling (computing) ,[INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI] ,010104 statistics & probability ,First-come-first-served ,Swap (finance) ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Queue ,Discrete mathematics ,Queueing theory ,Supply chain management ,Stationary distribution ,Computer Science - Performance ,Network of queues ,Product-form stationary distribution ,020206 networking & telecommunications ,Computer Science Applications ,Performance (cs.PF) ,Computational Theory and Mathematics ,Irreducibility ,60K25 (Primary), 60K30 (Secondary) - Abstract
Order-independent (OI) queues, introduced by Berezner, Kriel, and Krzesinski in 1995, expanded the family of multi-class queues that are known to have a product-form stationary distribution by allowing for intricate class-dependent service rates. This paper further broadens this family by introducing pass-and-swap (P&S) queues, an extension of OI queues where, upon a service completion, the customer that completes service is not necessarily the one that leaves the system. More precisely, we supplement the OI queue model with an undirected graph on the customer classes, which we call a swapping graph, such that there is an edge between two classes if customers of these classes can be swapped with one another. When a customer completes service, it passes over customers in the remainder of the queue until it finds a customer it can swap positions with, that is, a customer whose class is a neighbor in the graph. In its turn, the customer that is ejected from its position takes the position of the next customer it can be swapped with, and so on. This is repeated until a customer can no longer find another customer to be swapped with; this customer is the one that leaves the queue. After proving that P&S queues have a product-form stationary distribution, we derive a necessary and sufficient stability condition for (open networks of) P&S queues that also applies to OI queues. We then study irreducibility properties of closed networks of P&S queues and derive the corresponding product-form stationary distribution. Lastly, we demonstrate that closed networks of P&S queues can be applied to describe the dynamics of new and existing load-distribution and scheduling protocols in clusters of machines in which jobs have assignment constraints., 44 pages, 15 figures
- Published
- 2021
42. Diffusion Approximation for Fair Resource Control—Interchange of Limits Under a Moment Condition
- Author
-
David D. Yao and Heng-Qing Ye
- Subjects
Class (set theory) ,050208 finance ,Stationary distribution ,General Mathematics ,05 social sciences ,Control (management) ,Management Science and Operations Research ,Heavy traffic approximation ,01 natural sciences ,Computer Science Applications ,Moment (mathematics) ,010104 statistics & probability ,Resource (project management) ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Diffusion limit ,Mathematics - Abstract
In a prior study [Ye HQ, Yao DD (2016) Diffusion limit of fair resource control–Stationary and interchange of limits. Math. Oper. Res. 41(4):1161–1207.] focusing on a class of stochastic processing network with fair resource control, we justified the diffusion approximation (in the context of the interchange of limits) provided that the pth moment of the workloads are bounded. To this end, we introduced the so-called bounded workload condition, which requires the workload process be bounded by a free process plus the initial workload. This condition is for a derived process, the workload, as opposed to primitives such as arrival processes and service requirements; as such, it could be difficult to verify. In this paper, we establish the interchange of limits under a moment condition of suitable order on the primitives directly: the required order is [Formula: see text] on the moments of the primitive processes so as to bound the pth moment of the workload. This moment condition is trivial to verify, and indeed automatically holds in networks where the primitives have moments of all orders, for instance, renewal arrivals with phase-type interarrival times and independent and identically distributed phase-type service times.
- Published
- 2021
43. Dynamics analysis of stochastic modified Leslie–Gower model with time-delay and Michaelis–Menten type prey harvest
- Author
-
Ming Liu, Xiaofeng Xu, and Yu Liu
- Subjects
Computational Mathematics ,Correctness ,Stationary distribution ,Applied Mathematics ,Theory of computation ,Structure (category theory) ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Ergodic theory ,Perturbation (astronomy) ,Type (model theory) ,Michaelis–Menten kinetics ,Mathematics - Abstract
In this paper, we study the dynamics of stochastic time-delayed predator–prey model with modified Leslie–Gower and ratio-dependent schemes including additional food for predator and prey with Michaelis–Menten type harvest. We introduce the effects of time delay, Michaelis–Menten type harvest and stochastic perturbation under the structure of the original model to make the model more consistent with the actual system. We first prove that the system has a globally unique positive solution. Secondly we obtain conditions for the persistence in mean and extinction of the system. Besides, we verify that the system is stochastic permanence under certain conditions. In addition to that, we prove that the system has an ergodic stationary distribution when the parameters satisfy certain conditions. Finally, some numerical simulations were performed to verify the correctness and validity of the theoretical results.
- Published
- 2021
44. Stationary distribution and periodic solution of a stochastic Nicholson's blowflies model with distributed delay
- Author
-
Ahmed Alsaedi, Tasawar Hayat, Daqing Jiang, Xiaojie Mu, and Bashir Ahmad
- Subjects
Stationary distribution ,Extinction ,General Mathematics ,Mathematical analysis ,General Engineering ,Positive recurrence ,Mathematics - Published
- 2021
45. Impact of environmental noises on the stability of a waterborne pathogen model
- Author
-
Daqing Jiang and Xinhong Zhang
- Subjects
Lyapunov function ,Stationary distribution ,Markov chain ,Stochastic modelling ,Applied Mathematics ,White noise ,Stability (probability) ,Computational Mathematics ,Noise ,symbols.namesake ,symbols ,Ergodic theory ,Statistical physics ,Mathematics - Abstract
A stochastic waterborne disease model is analyzed to reveal the effects of the white noise and telegraph noise. For this stochastic model, a critical quantity $$R_0^s$$ is derived, which depends on not only model parameters but also environmental noises. If $$R_0^s>1$$ , then this model admits an ergodic stationary distribution. One of the most important contributions is the suitable Lyapunov function depending on the state of Markov chains is constructed. Theoretical results and numerical simulations provide a clear insight of the impact of environmental noises on the dynamical behavior of the model.
- Published
- 2021
46. Dynamics of a stochastic multigroup SEI epidemic model
- Author
-
Qun Liu and Daqing Jiang
- Subjects
Statistics and Probability ,Extinction ,Stationary distribution ,Salient ,Applied Mathematics ,Dynamics (mechanics) ,Ergodicity ,Ergodic theory ,Statistical physics ,Uniqueness ,Statistics, Probability and Uncertainty ,Epidemic model ,Mathematics - Abstract
In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of...
- Published
- 2021
47. A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey
- Author
-
Wanying Shi, Chunjin Wei, Youlin Huang, and Shuwen Zhang
- Subjects
Lyapunov function ,education.field_of_study ,Extinction ,Stationary distribution ,Article Subject ,Physics ,QC1-999 ,Applied Mathematics ,010102 general mathematics ,Population ,Structure (category theory) ,General Physics and Astronomy ,01 natural sciences ,Predation ,symbols.namesake ,0103 physical sciences ,symbols ,Quantitative Biology::Populations and Evolution ,Ergodic theory ,Applied mathematics ,Uniqueness ,0101 mathematics ,education ,010301 acoustics ,Mathematics - Abstract
In this paper, we study a stochastic Holling-type II predator-prey model with stage structure and refuge for prey. Firstly, the existence and uniqueness of the global positive solution of the system are proved. Secondly, the stochastically ultimate boundedness of the solution is discussed. Next, sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the positive solution are established by constructing a suitable stochastic Lyapunov function. Then, sufficient conditions for the extinction of predator population in two cases and that of prey population in one case are obtained. Finally, some numerical simulations are presented to verify our results.
- Published
- 2021
48. Dynamics of a Stochastic HIV Infection Model with Logistic Growth and CTLs Immune Response under Regime Switching
- Author
-
Lin Hu and Linfei Nie
- Subjects
General Mathematics ,Computer Science (miscellaneous) ,HIV model ,logistic growth ,Brownian motion and Markovian switching ,stationary distribution ,extinction ,Engineering (miscellaneous) - Abstract
Considering the influences of uncertain factors on the reproduction of virus in vivo, a stochastic HIV model with CTLs’ immune response and logistic growth was developed to research the dynamics of HIV, where uncertain factors are white noise and telegraph noise. which are described by Brownian motion and Markovian switching, respectively. We show, firstly, the existence of global positive solutions of this model. Further, by constructing suitable stochastic Lyapunov functions with regime switching, some sufficient conditions for the existence and uniqueness of the stationary distribution and the conditions for extinction are obtained. Finally, the main results are explained by some numerical examples. Theoretical analysis and numerical simulation show that low-intensity white noise can maintain the persistence of the virus, and high intensity white noise can make the virus extinct after a period of time with multi-states.
- Published
- 2022
- Full Text
- View/download PDF
49. Analysis of a Class of Predation-Predation Model Dynamics with Random Perturbations
- Author
-
Xuewen Tan, Pengpeng Liu, Wenhui Luo, and Hui Chen
- Subjects
random perturbation ,Lipschitz conditions ,stationary distribution ,gradual properties ,population extinction ,numerical simulation ,General Mathematics ,Computer Science (miscellaneous) ,Engineering (miscellaneous) - Abstract
In this paper, we study a class of predation–prey biological models with random perturbation. Firstly, the existence and uniqueness of systematic solutions can be proven according to Lipschitz conditions, and then we prove that the systematic solution exists globally. Moreover, the article discusses the long-term dynamical behavior of the model, which studies the stationary distribution and gradual properties of the system. Next, we use two different methods to give the conditions of population extinction. From what has been discussed above, we can safely draw the conclusion that our results are reasonable by using numerical simulation.
- Published
- 2022
- Full Text
- View/download PDF
50. Dynamics Analysis of a Predator–Prey Model with Hunting Cooperative and Nonlinear Stochastic Disturbance
- Author
-
Yuke Zhang and Xinzhu Meng
- Subjects
hunting cooperation ,stochastic prey–predator model ,nonlinear perturbation ,stationary distribution ,persistence in mean ,General Mathematics ,Computer Science (miscellaneous) ,Engineering (miscellaneous) - Abstract
This paper proposes a stochastic predator–prey model with hunting cooperation and nonlinear stochastic disturbance, and focuses on the effects of nonlinear white noise and hunting cooperation on the populations. First, we present the thresholds R1 and R2 for extinction and persistence in mean of the predator. When R1 is less than 0, the predator population is extinct; when R2 is greater than 0, the predator population is persistent in mean. Moreover, by establishing suitable Lyapunov functions, we investigate the threshold R0 for the existence of a unique ergodic stationary distribution. At last, we carry out the numerical simulations. The results show that white noise is harmful to the populations and hunting cooperation is beneficial to the predator population.
- Published
- 2022
- Full Text
- View/download PDF
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