677 results on '"Stationary distribution"'
Search Results
152. Infinite-step stationarity of rotor walk and the wired spanning forest.
- Author
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Chan, Swee Hong
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ROTORS , *PEDOMETERS , *HARDWOODS , *RANDOM walks - Abstract
We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk (after the walker escapes to infinity) follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018). [ABSTRACT FROM AUTHOR]
- Published
- 2021
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153. Analysis on stochastic predator-prey model with distributed delay.
- Author
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Gokila, C. and Sambath, M.
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STOCHASTIC analysis , *STOCHASTIC models , *PREDATION , *LOTKA-Volterra equations , *LYAPUNOV functions , *COMPUTER simulation - Abstract
In the present work, we consider a stochastic predator-prey model with disease in prey and distributed delay. Firstly, we establish sufficient conditions for the extinction of the disease and also permanence of healthy prey and predator. Besides, we obtain the condition for the existence of an ergodic stationary distribution through the stochastic Lyapunov function. Finally, we provide some numerical simulations to validate our theoretical findings. [ABSTRACT FROM AUTHOR]
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- 2021
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154. A note on ergodicity for CIR model with Markov switching.
- Author
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Tong, Jinying, Meng, Qingting, Zhang, Zhenzhong, and Lu, Yunfang
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MARKOV processes , *WASTE products - Abstract
Recently, Zhang et al. show that if ∑ i = 1 N π i β (i) ≠ 0 , then the Cox-Ingersoll-Ross (CIR) model with Markov switching (see below, the SDE (1.2)) is ergodic in the Wasserstein distance if and only if ∑ i = 1 N π i β (i) > 0. In this article, we will show that if ∑ i = 1 N π i β (i) = 0 , the Cox-Ingersoll-Ross (CIR) model with Markov switching is non-ergodic. Explicit expressions for the mean and variance of the CIR model with Markov switching are obtained. As a byproduct, the explicit expressions for mean of stationary distribution and second-order moments for such model are presented. Besides, we find the necessary and sufficient conditions of weak stationarity for CIR model with Markov switching. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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155. Dynamic characterization of a stochastic SIR infectious disease model with dual perturbation.
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Kiouach, Driss and Sabbar, Yassine
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COMMUNICABLE diseases , *MEDICAL model , *WHITE noise , *RANDOM noise theory , *INFECTIOUS disease transmission - Abstract
Environmental perturbations are unavoidable in the propagation of infectious diseases. In this paper, we introduce the stochasticity into the susceptible–infected–recovered (SIR) model via the parameter perturbation method. The stochastic disturbances associated with the disease transmission coefficient and the mortality rate are presented with two perturbations: Gaussian white noise and Lévy jumps, respectively. This idea provides an overview of disease dynamics under different random perturbation scenarios. By using new techniques and methods, we study certain interesting asymptotic properties of our perturbed model, namely: persistence in the mean, ergodicity and extinction of the disease. For illustrative purposes, numerical examples are presented for checking the theoretical study. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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156. Dynamic behaviors of a stochastic virus infection model with Beddington–DeAngelis incidence function, eclipse-stage and Ornstein–Uhlenbeck process.
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Liu, Yuncong, Wang, Yan, and Jiang, Daqing
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ORNSTEIN-Uhlenbeck process , *VIRUS diseases , *PROBABILITY density function , *LYAPUNOV functions , *SET functions - Abstract
In this paper, we present a virus infection model that incorporates eclipse-stage and Beddington–DeAngelis function, along with perturbation in infection rate using logarithmic Ornstein–Uhlenbeck process. Rigorous analysis demonstrates that the stochastic model has a unique global solution. Through construction of appropriate Lyapunov functions and a compact set, combined with the strong law of numbers and Fatou's lemma, we obtain the existence of the stationary distribution under a critical condition, which indicates the long-term persistence of T-cells and virions. Moreover, a precise probability density function is derived around the quasi-equilibrium of the model, and spectral radius analysis is employed to identify critical condition for elimination of the virus. Finally, numerical simulations are presented to validate theoretical results, and the impact of some key parameters such as the speed of reversion, volatility intensity and mean infection rate are investigated. • A stochastic virus model with Beddington–DeAngelis incidence function and Ornstein–Uhlenbeck process is built. • A novel method is proposed to prove the existence of stationary distribution. • The probability density function near the quasi-equilibrium is obtained. • Investigate the effect of some key parameters on virus dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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157. Existence of stationary distribution for stochastic coupled nonlinear strict-feedback systems with Markovian switching.
- Author
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Guo, Wanying, Meng, Shuyu, Qi, Ruotong, Li, Wenxue, and Wu, Yongbao
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NONLINEAR systems , *BACKSTEPPING control method , *LYAPUNOV functions , *GRAPH theory , *VIRTUAL design - Abstract
In this paper, a class of stochastic coupled nonlinear strict-feedback systems with Markovian switching (SCNSSMs) are introduced, and the existence of stationary distribution for SCNSSMs under pinning control is studied for the first time. In particular, an appropriate technique for handling the impact of Markovian switching is proposed in the process of designing virtual controllers through the back-stepping method. Regarding the global Lyapunov function for an SCNSSM, we first construct a quartic Lyapunov function for each stochastic nonlinear strict-feedback system with Markovian switching of the SCNSSM, and then combine graph theory to construct the global Lyapunov function. After that, the theoretical results are applied to stochastic coupled oscillator systems with Markovian switching. Finally, the effectiveness of proposed results is verified by some numerical simulations. • This article represents the innovative effort to address the existence of stationary distribution for a novel class of SCNSSMs. • It is worthwhile noting that we incorporate stochastic disturbances, Markovian switching, and coupling factors in nonlinear strict-feedback systems for the first time. • We employ pinning control to investigate existence of stationary distribution for SCNSSMs , and a strategy of selecting vertices for pinning control is developed. • By adopting back-stepping method and the design of virtual controllers, the problem of designing actual controllers is successfully addressed. • In this article, the combination of Lyapunov method and graph theory is ingeniously utilized to formulate a global Lyapunov function. [ABSTRACT FROM AUTHOR]
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- 2024
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158. Dynamics caused by the mean-reverting Ornstein–Uhlenbeck process in a stochastic predator–prey model with stage structure.
- Author
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Mu, Xiaojie and Jiang, Daqing
- Subjects
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ORNSTEIN-Uhlenbeck process , *STOCHASTIC processes , *STOCHASTIC models , *STOCHASTIC systems , *STATIONARY processes , *MARKOV processes - Abstract
There are many species that go through different life stages in nature, they have different ecological characteristics in different stages, besides, stochastic variation exists in the natural environment. Based on this biological phenomenon, we develop and investigate dynamics in a predator–prey system with stage structure and the mean-reverting Ornstein–Uhlenbeck processes. We prove that this system satisfies the existence and uniqueness of globally positive equilibrium. The sufficient conditions for the existence of a stationary Markov process in stochastic system are attained. We deal with explicit expression and the existence of density function of the system which make a great contribution to biological intrinsic essence for the system. In addition, the sufficient criteria for extinction of the predator populations is derived. Finally, numerical simulations are furnished to illustrate the analysis results. [ABSTRACT FROM AUTHOR]
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- 2024
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159. Stochastic multi-group epidemic SVIR models: Degenerate case.
- Author
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Tuong, Tran D., Nguyen, Dang H., and Nguyen, Nhu N.
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WHITE noise , *EPIDEMICS , *EPIDEMIOLOGICAL models , *STOCHASTIC models , *PREVENTIVE medicine - Abstract
This work considers a multi-group epidemic SIR models with vaccination. The fluctuation of the environment is taken into account by introducing both color noise and white noise to a compartmental model. Unlike existing results on stochastic multigroup models, which were not successful in finding the reproduction numbers and fully classify the longtime behaviors of the models, we will provide a formula for the reproduction number R 0 of our model and will show that the disease is persistent if R 0 > 1 while the disease will be eradicated if R 0 ≤ 1. We also provide the explicit formulae for R 0 in some special cases. The formulae will be useful to determine the herb immunity threshold of the disease, which can be used to make right and timely policy to control a disease. • Fully characterize the long-term behaviors of a multi-group epidemic SIR models with vaccination with degenerate noise. • Provide a formula for the reproduction number. • Complement and generalize some existing work on epidemiological models. [ABSTRACT FROM AUTHOR]
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- 2024
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160. Mean square exponential stability of discrete-time Markov switched stochastic neural networks with partially unstable subsystems and mixed delays.
- Author
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Fan, Lina and Zhu, Quanxin
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EXPONENTIAL stability , *BINOMIAL distribution , *MARKOV processes - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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161. Virus dynamic behavior of a stochastic HIV/AIDS infection model including two kinds of target cell infections and CTL immune responses.
- Author
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Qi, Kai, Jiang, Daqing, Hayat, Tasawar, and Alsaedi, Ahmed
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CYTOTOXIC T cells , *IMMUNE response , *SIMIAN immunodeficiency virus , *WHITE noise , *T cells , *AIDS , *VIRUSES , *HIV - Abstract
This study investigated the impact of white noise on the HIV/AIDS model with a cytotoxic T lymphocyte (CTL) immune response. The model introduced the interactions between the virus and two kinds of target cells, CD4+ T cells and macrophages. It was theoretically proved that the solution of the stochastic model is positive and global, as well as the existence of an ergodic stationary distribution. The sufficient conditions were established for viral eradication. By comparing these new results to those of a deterministic model, it is determined that white noise can promote the extinction of the virus. Theoretical results have been verified by numerical simulation of several examples. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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162. Permanence, Extinction and Periodicity to a Stochastic Competitive Model with Infinite Distributed Delays.
- Author
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Ji, Chunyan, Yang, Xue, and Li, Yong
- Subjects
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STOCHASTIC models , *STATIONARY processes , *MARKOV processes , *NOISE , *BIOLOGICAL extinction , *LOTKA-Volterra equations - Abstract
In this paper, we study a stochastic Lotka–Volterra competitive model with infinite distributed delays. We obtain sufficient but almost necessary conditions for permanence and extinction of species. Competitive exclusion also takes place in some cases. Besides, criteria are established for the existence and uniqueness of a stationary Markov process. Our results not only cover the traditional Lotka–Volterra competition model without environmental noise, but also reflect the influence of noise to model's dynamics. Moreover there are some challenges to give the existence of periodic solutions for system with periodic coefficients. Via a stochastic comparison approach, we obtain the existence and uniqueness of periodic solutions in distribution. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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163. Dynamics of an autonomous Gilpin–Ayala competition model with random perturbation.
- Author
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Fu, Jing, Han, Qixing, Jiang, Daqing, and Yang, Yanyan
- Subjects
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WHITE noise , *POSITIVE systems , *COMPUTER simulation - Abstract
This paper discusses the dynamics of a Gilpin–Ayala competition model of two interacting species perturbed by white noise. We obtain the existence of a unique global positive solution of the system and the solution is bounded in p th moment. Then, we establish sufficient and necessary conditions for persistence and the existence of an ergodic stationary distribution of the model. We also establish sufficient conditions for extinction of the model. Moreover, numerical simulations are carried out for further support of present research. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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164. Analysis on stochastic dynamics of two-consumers-one-resource competing systems with Beddington–DeAngelis functional response.
- Author
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Zhao, Shufen
- Subjects
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STOCHASTIC analysis , *BIOLOGICAL extinction , *STOCHASTIC resonance , *COMPUTER simulation , *NOISE - Abstract
In this paper, stochastic dynamics with Lévy noise of two-consumers-one-resource competing systems with Beddington–DeAngelis functional response are considered. We first show the existence of the global positive solution, then discuss the effects of noises on the extinction of the species and the stochastic persistence of the species. In the meanwhile, numerical simulations are carried to support results. Finally, we show the existence of the stationary distribution for a special case. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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165. Stationary distribution and extinction for a food chain chemostat model with random perturbation.
- Author
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Gao, Miaomiao, Jiang, Daqing, Hayat, Tasawar, and Alsaedi, Ahmed
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FOOD chains , *LYAPUNOV functions , *PREDATION , *WHITE noise , *FOOD habits - Abstract
In this paper, we study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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166. Exponential stability of the stationary distribution of a mean field of spiking neural network.
- Author
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Drogoul, Audric and Veltz, Romain
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EXPONENTIAL stability , *SYSTEMS theory , *DYNAMICAL systems , *TRANSPORT equation , *TIME management , *STOCHASTIC convergence , *KERNEL (Mathematics) - Abstract
In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in [1,2]. We complement the convergence result proved in [2] using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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167. Markov modeling of traffic flow in Smart Cities.
- Author
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Bátfai, Norbert, Besenczi, Renátó, Jeszenszky, Péter, Szabó, Máté, and Ispány, Márton
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SMART cities , *MARKOV processes , *CITY traffic , *INNER cities , *TRAFFIC flow , *STOCHASTIC models - Abstract
Modeling and simulating the traffic flow in large urban road networks are important tasks. A mathematically rigorous stochastic model proposed in [8] is based on the synthesis of the graph and Markov chain theories. In this model, the transition probability matrix describes the traffic dynamics and its unique stationary distribution approximates the proportion of the vehicles at the segments of the road network. In this paper various Markov models are studied and a simulation method is presented for generating random traffic trajectories on a road network based on the two-dimensional stationary distribution of the models. In a case study we apply our method to the central region of the city of Debrecen by using the road network data from the OpenStreetMap project which is available publicly. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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168. Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence.
- Author
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Gao, Shuaibin, Li, Xiaotong, and Liu, Zhuoqi
- Subjects
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STOCHASTIC differential equations , *MARKOV processes , *FUZZY neural networks - Abstract
• We are very grateful to the editor and the reviewer for evaluating our manuscript and for the comments. In accordance with the comments, we have carefully and thoroughly revised and rewritten our manuscript. To our best knowledge, this work is the first paper to consider the stationary distribution of the Milstein scheme for stochastic differential delay equations. We reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Then the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, we have demonstrated the reliability of the theoretical results through abundant numerical experiments. This paper focuses on the stationary distribution of the Milstein scheme for stochastic differential delay equations. The numerical segment process is constructed, which is proved to be a time homogeneous Markov process. We show that this numerical segment process admits a unique numerical stationary distribution. Then we reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Moreover, the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, abundant numerical experiments confirm the reliability of theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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169. The effect of stochasticity with respect to reinfection and nonlinear transition states for some diseases with relapse.
- Author
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El Fatini, Mohamed, El Khalifi, Mohamed, Lahrouz, Aadil, Pettersson, Roger, and Settati, Adel
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DISEASE relapse , *DISEASE progression , *STOCHASTIC models , *COMPUTER simulation , *REINFECTION - Abstract
In this paper, we consider a stochastic epidemic model with relapse, reinfection, and a general incidence function. Using stochastic tools, we establish a stochastic threshold Rs0 and prove the extinction of the disease when its value is equal or less than unity. We also show the persistence in mean of the disease when Rs0>1. Moreover, we prove the existence and uniqueness of a stationary distribution. Finally, numerical simulations are presented to show the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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170. Survival analysis of a stochastic predator–prey model with prey refuge and fear effect.
- Author
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Xia, Yixiu and Yuan, Sanling
- Subjects
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STOCHASTIC analysis , *STOCHASTIC models , *SURVIVAL analysis (Biometry) , *FEAR , *WHITE noise , *LOTKA-Volterra equations , *PREY availability - Abstract
In this paper, we propose and investigate a stochastic Holling type-II predator–prey model with prey refuge and fear effect. We first prove the existence and uniqueness of the global positive solution. Then we perform the survival analysis of the model, including the existence of a unique ergodic stationary distribution and the extinction of the model. Numerical simulations are carried out to validate our analytical results. Our findings indicate that the white noise is adverse to the growth of predator and prey populations, and the increase of fear effect will lead to the decrease of predator density, but with no obvious effect on prey density. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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171. Tier structure of strongly endotactic reaction networks.
- Author
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Anderson, David F., Cappelletti, Daniele, Kim, Jinsu, and Nguyen, Tung D.
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CHEMICAL species , *STOCHASTIC models , *MOLECULAR models , *ORDINARY differential equations , *LARGE deviations (Mathematics) - Abstract
Reaction networks are mainly used to model the time-evolution of molecules of interacting chemical species. Stochastic models are typically used when the counts of the molecules are low, whereas deterministic models are often used when the counts are in high abundance. The mathematical study of reaction networks has increased dramatically over the last two decades as these models are now routinely used to investigate cellular behavior. In 2011, the notion of "tiers" was introduced to study the long time behavior of deterministically modeled reaction networks that are weakly reversible and have a single linkage class. This "tier" based argument was analytical in nature. Later, in 2014, the notion of a strongly endotactic network was introduced in order to generalize the previous results from weakly reversible networks with a single linkage class to this wider family of networks. The point of view of this later work was more geometric and algebraic in nature. The notion of strongly endotactic networks was later used in 2018 to prove a large deviation principle for a class of stochastically modeled reaction networks. In the current paper we provide an analytical characterization of strongly endotactic networks in terms of tier structures. By doing so, we not only shed light on the connection between the two points of view, but also make available a new proof technique for the study of strongly endotactic networks. We show the power of this new technique in two distinct ways. First, we demonstrate how the main previous results related to strongly endotactic networks, both for the deterministic and stochastic modeling choices, can be quickly obtained from our characterization. Second, we demonstrate how new results can be obtained by proving that a sub-class of strongly endotactic networks, when modeled stochastically, is positive recurrent. Finally, and similarly to recent independent work by Agazzi and Mattingly, we provide an example which closes a conjecture in the negative by showing that stochastically modeled strongly endotactic networks can be transient (and even explosive). [ABSTRACT FROM AUTHOR]
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- 2020
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172. Stationary Distribution of Telomere Lengths in Cells with Telomere Length Maintenance and its Parametric Inference.
- Author
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Lee, Kyung Hyun and Kimmel, Marek
- Abstract
Telomeres are nucleotide caps located at the ends of each eukaryotic chromosome. Under normal physiological conditions as well as in culture, they shorten during each DNA replication round. Short telomeres initiate a proliferative arrest of cells termed ‘replicative senescence’. However, cancer cells possessing limitless replication potential can avoid senescence by the telomere maintenance mechanism, which offsets telomeric loss. Therefore, cancer cells have sufficiently long telomeres even though their lengths are significantly shorter than their normal counterparts. This implies that the attrition and elongation rates play crucial roles in deciding whether and when cells ultimately become carcinogenic. In this research, we propose a concise mathematical model that shows the shortest telomere length at each cell division and prove mathematical conditions related to the attrition and elongation rates, which are necessary and sufficient for the existence of stationary distribution of telomere lengths. Moreover, we estimate the parameters of the telomere length maintenance process based on frequentist and Bayesian approaches. This study expands our knowledge of the mathematical relationship between the telomere attrition and elongation rates in cancer cells, which is important because the telomere length dynamics is a useful biomarker of cancer diagnosis and prognosis. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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173. Stationary distribution of a stochastic Alzheimer's disease model.
- Author
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Hu, Jing, Zhang, Qimin, Meyer‐Baese, Anke, and Ye, Ming
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ALZHEIMER'S disease , *MEDICAL model , *STOCHASTIC models - Abstract
In this paper, based on the pathogenesis of Alzheimer's disease, we investigate a stochastic mathematical model, focusing on the dynamics of β‐amyloid (Aβ) plaques, Aβ oligomers, PrPC proteins, and the Aβ‐x‐PrPC complex. Within the framework of the Lyapunov method, we first show existence and uniqueness of global positive solution of the model and then establish the sufficient conditions for existence of an ergodic stationary distribution of the positive solution. Ultimately, numerical examples are presented to illustrate the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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174. Permanence and extinction for the stochastic SIR epidemic model.
- Author
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Du, N.H. and Nhu, N.N.
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WHITE noise - Abstract
The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for SIR epidemic system with multi noises { d S (t) = [ a 1 − b 1 S (t) − I (t) f (S (t) , I (t)) ] d t + σ 1 S (t) d B 1 (t) − I (t) g (S (t) , I (t)) d B 3 (t) , d I (t) = [ − b 2 I (t) + I (t) f (S (t) , I (t)) ] d t + σ 2 I (t) d B 2 (t) + I (t) g (S (t) , I (t)) d B 3 (t). Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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175. Asymptotic behaviors of stochastic epidemic models with jump-diffusion.
- Author
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Thanh Dieu, Nguyen, Fugo, Takasu, and Huu Du, Nguyen
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STOCHASTIC models , *INVARIANT measures , *BROWNIAN motion , *STOCHASTIC systems , *NONLINEAR equations , *NONLINEAR difference equations , *STOCHASTIC analysis - Abstract
• Classify the asymptotic behavior SIR models with noise perturbing to linear terms. • Construct a threshold value based on coefficients. • The negativity of threshold implies the extinction of the model at exponential rate. • If it is positive, the solution converges to unique stationary measure in total variation. • We can use this technique to classify models with noise perturbing to non linear terms. In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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176. Modeling a stochastic avian influenza model under regime switching and with human-to-human transmission.
- Author
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Shi, Zhenfeng and Zhang, Xinhong
- Subjects
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STOCHASTIC models , *STOCHASTIC systems , *LYAPUNOV functions , *HUMAN-to-human transmission , *WHITE noise , *AVIAN influenza , *POSITIVE systems - Abstract
In this paper, we investigate the stochastic avian influenza model with human-to-human transmission, which is disturbed by both white and telegraph noises. First, we show that the solution of the stochastic system is positive and global. Furthermore, by using stochastic Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then we obtain the conditions for extinction. Finally, numerical simulations are employed to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
177. Time to extinction and stationary distribution of a stochastic susceptible-infected-recovered-susceptible model with vaccination under Markov switching.
- Author
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Li, Xiaoni, Li, Xining, and Zhang, Qimin
- Subjects
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STOCHASTIC models , *VACCINATION - Abstract
A stochastic susceptible-infected-recovered-susceptible model with vaccination includes stochastic variation in its parameters. Sufficient conditions for the extinction and the existence of the stationary distribution of the population are proved. [ABSTRACT FROM AUTHOR]
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- 2020
- Full Text
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178. Tail asymptotics for the M1,M2/G1,G2/1 retrial queue with non-preemptive priority.
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Liu, Bin and Zhao, Yiqiang Q.
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QUEUING theory , *TELECOMMUNICATION network management , *TAILS , *NEW trials , *QUEUEING networks , *KEY performance indicators (Management) - Abstract
Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. The system considered here receives two types of customers, of which Type-1 customers (in a queue) have non-pre-emptive priority to receive service over Type-2 customers (in an orbit). For this type of system, we propose an exhaustive version of the stochastic decomposition approach, which is one of the main contributions made in this paper, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this retrial queue with two types of customers. Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the numbers of customers in the queue and in the orbit, respectively, conditioning on the server's status, in terms of the exhaustive stochastic decomposition results. These tail asymptotic results are new, which is another main contribution made in this paper. Tail asymptotic properties are very important, not only on their own merits but also often as key tools for approximating performance metrics and constructing numerical algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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179. Dynamics of stochastic single‐species models.
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Zhang, Yan, Lv, Jingliang, and Zou, Xiaoling
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STOCHASTIC models , *LOTKA-Volterra equations , *COMPUTER simulation - Abstract
A type of stochastic single‐species model is proposed and studied. The sufficient conditions of the existence of a unique solution, the existence of its stationary distribution, and stochastic permanence are obtained. Besides, the threshold conditions for its strong stochastic persistence and extinction are found. Finally, some examples and numerical simulations are introduced to support our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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180. The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion.
- Author
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Tuerxun, Nafeisha, Wen, Buyu, and Teng, Zhidong
- Subjects
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DIFFUSION , *EPIDEMICS , *FOKKER-Planck equation , *LYAPUNOV functions , *BASIC reproduction number - Abstract
A class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion is investigated. By using the Lyapunov function method, the existence of global positive solutions and the ultimate boundedness with probability one are obtained. By using the Markov semigroups theory, Fokker–Planck equation and Khasminskiǐ functions, the existence of unique stationary distribution for the model is established. That is, when the stochastic basic reproduction number R 0 S > 1 and some extra conditions are satisfied then the distribution density of any positive solutions of the model converges to a unique invariant density as t → + ∞. Finally, the main conclusions and open problems are illustrated and verified by the numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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181. The impact of virus carrier screening and actively seeking treatment on dynamical behavior of a stochastic HIV/AIDS infection model.
- Author
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Qi, Kai and Jiang, Daqing
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HIV , *AIDS , *STOCHASTIC systems , *DISEASE eradication , *WHITE noise , *VIRUSES , *INFECTION - Abstract
• A HIV/AIDS model is established in which the model is disturbed by white noises. • The model includes the screening for virus carriers and the infected individuals seeking treatment actively. • We establish sufficient conditions for the existence of ergodic stationary distribution and disease eradication, respectively. To screen for virus carriers and for infected individuals to actively seek treatment are the key factors to curb the spread of HIV/AIDS. We propose a stochastic HIV/AIDS model to evaluate their effect. First, we theoretically prove that the solution of the stochastic model is positive and global. Second, we obtain the existence of an ergodic stationary distribution if R 0 S > 1 , and we establish sufficient conditions R 0 ′ < 1 for disease extinction. We provide examples and numerical simulations to verify our theoretical results. Finally, we analyze the impact of the above two factors on the dynamical behavior of the stochastic system, and draw conclusions regrading disease prevention and elimination. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
182. A new family of positive recurrent semimartingale reflecting Brownian motions in an orthant.
- Author
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Yaacoubi, Abdelhak
- Subjects
- *
BROWNIAN motion , *WIENER processes , *QUEUEING networks , *COVARIANCE matrices , *DIFFUSION processes - Abstract
Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular d × d {d\times d} covariance matrix Δ, and a d × d {d\times d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift θ ∈ Γ ̊ {\theta\in\mathring{\Gamma}} , where Γ ̊ {\mathring{\Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
183. Dynamical behavior of a stochastic predator-prey model with stage structure for prey.
- Author
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Liu, Qun, Jiang, Daqing, Hayat, Tasawar, Alsaedi, Ahmed, and Ahmad, Bashir
- Subjects
- *
LOTKA-Volterra equations , *STOCHASTIC models , *STOCHASTIC programming , *LYAPUNOV functions , *COMPUTER simulation , *BIOLOGICAL extinction , *BEHAVIOR - Abstract
In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
184. Ergodicity of CIR type SDEs driven by stable processes with random switching.
- Author
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Zhang, Zhenzhong, Cao, Jingwen, Tong, Jinying, and Zhu, Enwen
- Subjects
- *
STOCHASTIC processes , *INTEREST rates - Abstract
In this paper, we focus on ergodicity and transience of the Cox–Ingersoll–Ross interest rate model driven by stable processes with random switching. Under some assumptions, we prove that the interest rate process and switching process has a unique stationary distribution. In addition, some sufficient conditions for transience of such interest rate model are given. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
185. Dynamics of a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps.
- Author
-
Lu, Chun
- Subjects
- *
LOTKA-Volterra equations , *WHITE noise , *INTEGRAL transforms , *PHASE space , *MEMORY - Abstract
This paper investigates a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps. Firstly, we transfer a classic infinite memory predator–prey model with weak kernel case into an equivalent model through integral transform. Then, for the corresponding stochastic Markovian switching model, we establish the sufficient conditions for permanence in time average and the threshold between stability in time average and extinction. Finally, sufficient criteria for a unique ergodic stationary distribution of the model are derived. Our results show that, firstly, both white noise and infinite memory are unfavorable to the existence of the stationary distribution; secondly, the general Lévy jumps could make the stationary distribution vanish as well as happen; finally, the Markovian switching could make the stationary distribution appear. • The method is combining asymptotic approach with phase space Cg. • Stationary distribution is investigated as well as permanence in time average. • Both white noises and infinite memory count against stationary distribution. • General Lévy jumps could make stationary distribution vanish as well as happen. • The Markovian switching is propitious to appearance of stationary distribution. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
186. Study on a susceptible–exposed–infected–recovered model with nonlinear incidence rate.
- Author
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Chen, Lijun and Wei, Fengying
- Subjects
- *
LOTKA-Volterra equations , *COMPUTER simulation , *BIOLOGICAL extinction , *RATES - Abstract
A stochastic susceptible–exposed–infected–recovered (SEIR) model with nonlinear incidence rate is investigated. Under suitable conditions, existence and uniqueness of a global solution, stationary distribution with ergodicity, persistence in the mean, and extinction of the disease are obtained. Numerical simulations and conclusions are separately carried out at the end of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
187. Analysis of a Stochastic Holling Type II Predator–Prey Model Under Regime Switching.
- Author
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Jiang, Xiaobo, Zu, Li, Jiang, Daqing, and O'Regan, Donal
- Subjects
- *
STOCHASTIC analysis , *PREDATION , *WHITE noise , *BIOLOGICAL extinction , *STOCHASTIC models - Abstract
The goal of this paper is to introduce and to initiate a study of a predator–prey system with Holling type II functional response, which is disturbed by both Markov switching and white noise. Our main aim is to investigate the dynamical properties of this model. First, we establish sufficient conditions of the strong persistence in the mean and the extinction of the predator population and obtain the threshold between them. Then, we show the existence of a unique ergodic stationary distribution under some natural conditions. Finally, a numerical example and figures are presented to illustrate the results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
188. The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria.
- Author
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Wanduku, Divine
- Subjects
- *
DISTRIBUTION (Probability theory) , *INFECTIOUS disease transmission , *STOCHASTIC processes , *RANDOM noise theory , *PLASMODIUM vivax - Abstract
This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates, and also from the random delays of the incubation and immunity periods. Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state, and also for the disease to remain permanently in the system over time. Moreover, the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined, and the statistical characteristics of the distribution are given numerically. The results of this study show that the disease will persist and become permanent in the system, regardless of (1) whether the noises are from the disease transmission rate and/or from the natural death rates or (2) whether the delays in the system are constant or random for individuals in the system. Furthermore, it is shown that "weak" noise is associated with the existence of an endemic stationary distribution for the disease, while "strong" noise is associated with extinction of the population over time. Numerical simulation examples for Plasmodium vivax malaria are given. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
189. A stochastic switched SIRS epidemic model with nonlinear incidence and vaccination: Stationary distribution and extinction.
- Author
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Xin Zhao, Xin He, Tao Feng, and Zhipeng Qiu
- Subjects
- *
STOCHASTIC systems , *STOCHASTIC analysis , *WHITE noise , *VACCINATION , *BIOLOGICAL extinction - Abstract
In this paper, a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases. Nonlinear incidence and vaccination strategies are also considered in the proposed model. By using the method of stochastic analysis, we point out the key parameters that determine the persistence and extinction of the diseases. Specifically, if Rs 0 is greater than 0, the stochastic system has a unique ergodic stationary distribution; while if R* 0 is less than 0, the diseases will be extinct at an exponential rate. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
190. LONG-TERM ANALYSIS OF A STOCHASTIC SIRS MODEL WITH GENERAL INCIDENCE RATES.
- Author
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DANG HAI NGUYEN, YIN, GEORGE, and CHAO ZHU
- Subjects
- *
STOCHASTIC models , *INVARIANT measures , *LYAPUNOV exponents , *EXPONENTIAL functions , *RATES , *STOCHASTIC analysis , *INTERVAL analysis - Abstract
This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing, a real-valued threshold. We show that if < 0, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value > 0, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all t 2 [0;1), approximation in a long but finite interval [0; T] can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
191. Stationary distributions and convergence for M/M/1 queues in interactive random environment.
- Author
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Pang, Guodong, Sarantsev, Andrey, Belopolskaya, Yana, and Suhov, Yuri
- Subjects
- *
JUMP processes , *EMPLOYEE reviews , *ECOLOGY - Abstract
A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depend on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump diffusion. In both cases, the joint dynamics are constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for the exponential rate of convergence to the stationary distribution via coupling. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
192. ASYMPTOTIC BEHAVIOUR OF THE STOCHASTIC MAKI–THOMPSON MODEL WITH A FORGETTING MECHANISM ON OPEN POPULATIONS.
- Author
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LI, HAIJIAO and YANG, KUAN
- Abstract
Rumours have become part of our daily lives, and their spread has a negative impact on a variety of human affairs. Therefore, how to control the spread of rumours is an important topic. In this paper, we extend the classic Maki–Thompson model from a deterministic framework to a stochastic framework with a forgetting mechanism, because real-world person-to-person communications are inevitably affected by random factors. By constructing suitable stochastic Lyapunov functions, we show that the asymptotic behaviour of the stochastic rumour model is governed by the basic reproductive number. If this number is less than one, then the solution of the stochastic rumour model oscillates around the rumour-free equilibrium under extra mild conditions, indicating the extinction of the rumour with a probability of one. Otherwise, the solution always fluctuates around the endemic equilibrium under certain parametric restrictions, implying that the rumour will continually persist. In addition, we discuss a possible intervention strategy that stops the spread of rumours by strengthening the intensity of white noise, which is very different from the deterministic rumour model without white noise. Also, numerical simulations are conducted to support our analytical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
193. Influence of stochastic perturbation on an SIRI epidemic model with relapse.
- Author
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Liu, Qun, Jiang, Daqing, Hayat, Tasawar, and Alsaedi, Ahmed
- Subjects
- *
WHITE noise , *BIOLOGICAL extinction - Abstract
In this paper, we investigate a stochastic susceptible-infective-removed-infective (SIRI) epidemic model with relapse. We show that the densities of the distributions of the solutions can converge in L 1 to an invariant density or can converge weakly to a singular measure under certain condition. We also find the support of the invariant density. Moreover, we establish sharp sufficient criteria for the extinction of the disease in two cases. The results show that the smaller white noise can assure the existence of a stationary distribution which implies the persistence of the disease while the larger white noise can lead to the extinction of the disease. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
194. Asymptotic properties of a stochastic SIRS epidemic model with nonlinear incidence and varying population sizes.
- Author
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Rifhat, Ramziya, Muhammadhaji, Ahmadjan, and Teng, Zhidong
- Subjects
- *
GLOBAL analysis (Mathematics) , *DISEASE incidence , *STABILITY criterion , *EQUILIBRIUM - Abstract
This paper studies a class of stochastic SIRS epidemic models with general nonlinear incidence f (S , I , R) and varying population sizes. A new threshold value is obtained. The sufficient conditions for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution in probability meaning are established. As applications of the main results, the stochastic SIRS epidemic models with standard incidence, Beddington–DeAngelis incidence and nonlinear incidence rate h (S) g (I) are discussed, and a series of new criteria for the global stability of the disease-free equilibrium, the permanence in the mean of the disease and the existence of a unique stationary distribution are established. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
195. Analysis of a stochastic predator-prey system with foraging arena scheme.
- Author
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Cai, Yongmei, Cai, Siyang, and Mao, Xuerong
- Subjects
- *
PREDATION , *STOCHASTIC systems , *STOCHASTIC analysis , *ARENAS , *COMPUTER simulation , *BROWNIAN motion - Abstract
This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
196. Markov-modulated Brownian motions perturbed by catastrophes.
- Author
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Simon, Matthieu
- Subjects
- *
MATRIX analytic methods , *DISASTERS , *WIENER processes - Abstract
We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
197. Stationary Distribution and Extinction of a Stochastic HIV-1 Infection Model with Distributed Delay and Logistic Growth.
- Author
-
Liu, Qun, Jiang, Daqing, Hayat, Tasawar, and Alsaedi, Ahmed
- Subjects
- *
BIOLOGICAL extinction , *STOCHASTIC systems , *WHITE noise , *BASIC reproduction number , *LINEAR systems , *LYAPUNOV functions - Abstract
In this paper, we propose a stochastic HIV-1 infection model with distributed delay and logistic growth. Firstly, we transfer the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Then, we establish sufficient conditions for the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Moreover, we obtain sufficient criteria for extinction of the infected cells; that is, the uninfected cells are survival and the infected cells are extinct. Our results show that the smaller white noise can ensure the existence of a stationary distribution when the basic reproduction number R 0 S of the stochastic system is bigger than one, while the larger white noise can lead to the extinction of the infected cells when the basic reproduction number R 0 of the deterministic system is smaller than one. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
198. Dynamical analysis of a stochastic epidemic HBV model with log-normal Ornstein–Uhlenbeck process and vertical transmission term.
- Author
-
Wang, Haile, Zuo, Wenjie, and Jiang, Daqing
- Subjects
- *
ORNSTEIN-Uhlenbeck process , *PROBABILITY density function , *BASIC reproduction number , *STOCHASTIC systems , *NUMBER systems , *STOCHASTIC analysis - Abstract
Considering the transmission rate perturbed by log-normal Ornstein–Uhlenbeck process, we develop a stochastic HBV model with vertical transmission term. For higher-dimensional deterministic system, the local asymptotic stability of the endemic equilibrium is given by proving the global stability of the corresponding linearized system. For stochastic system, the existence of stationary distribution is obtained by constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process and the critical value corresponding to the basic reproduction number for determined system is derived, which means the persistence of the disease. And sufficient conditions for disease extinction are given. Furthermore, by solving five-dimensional Fokker–Planck equation, the exact expression of the probability density function near the quasi-equilibrium is provided to reveal the statistical properties. In the end, numerical simulations illustrate our theoretical results and exhibit the trends of the critical values for persistence and extinction of diseases along with the change of noise intensity and reversion speed. • A stochastic HBV model with OU process and vertical transmission term is proposed. • Local stability of endemic equilibrium for deterministic model is demonstrated. • Stationary distribution and extinction of diseases for stochastic model are obtained. • Exact expression of probability density function near the quasi-equilibrium is given. • Simulations exhibit the trends of critical values for persistence and extinction of diseases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
199. A stochastic analysis of a SIQR epidemic model with short and long-term prophylaxis.
- Author
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Sekkak, Idriss, Nasri, Bouchra R., Rémillard, Bruno N., Kong, Jude Dzevela, and El Fatini, Mohamed
- Subjects
- *
EPIDEMICS , *PREVENTIVE medicine , *COMMUNICABLE diseases , *STOCHASTIC analysis , *STOCHASTIC models , *LOTKA-Volterra equations , *COMPUTER simulation - Abstract
This paper aims to incorporate a high order diffusion term into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and we find a condition ensuring the extinction of an infectious disease. The existence of a stationary distribution for the stochastic epidemic model is investigated as well. Numerical simulations are conducted to support our theoretical results and an example of application with COVID-19 data from Canada is used to estimate the transmission rate and reproduction number associated with the stochastic model, while constructing a model fitting the data. • A stochastic SIQR epidemic model with higher-order perturbation is investigated. • The existence and uniqueness of the global positive solution are studied. • A stochastic threshold is established for extinction. • COVID-19 data from Canada are used to estimate the transmission rate. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
200. Stochastic modeling of SIS epidemics with logarithmic Ornstein–Uhlenbeck process and generalized nonlinear incidence.
- Author
-
Shi, Zhenfeng and Jiang, Daqing
- Subjects
- *
ORNSTEIN-Uhlenbeck process , *STOCHASTIC models , *STOCHASTIC systems , *EPIDEMICS , *ORDINARY differential equations , *DISEASE outbreaks - Abstract
In this paper, we investigate a stochastic SIS epidemic model with logarithmic Ornstein–Uhlenbeck process and generalized nonlinear incidence. Our study focuses on the construction of stochastic Lyapunov functions to establish the threshold condition for the extinction and the existence of the stationary distribution of the stochastic system. We also derive the exact expression of the density function around the quasi-endemic equilibrium, which provides valuable insight into the transmission and progression of the disease within a population. Our findings demonstrate the importance of considering the impact of stochasticity on the spread of epidemics, particularly in the presence of complex incidence mechanisms and stochastic environmental factors. Additionally, the stochastic threshold reveals that ordinary differential equation models and white noise models underestimate the severity of disease outbreaks, while our proposed the stochastic epidemic model with logarithmic Ornstein–Uhlenbeck process accurately captures real-world scenarios. • A stochastic SIS epidemic model with logarithmic Ornstein–Uhlenbeck process and generalized nonlinear incidence is developed. • We obtain the stochastic threshold which determines whether the disease in the stochastic model will spread or go extinct. • Furthermore, we derive the exact expression of the density function around the quasi-endemic equilibrium. • We conduct extensive numerical simulations to illustrate the impact of parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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