Mean-reverting Ornstein-Uhlenbeck Process Research Articles

Smoothing in linear multicompartment biological processes subject to stochastic input.

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

Understanding the dynamics of hepatitis B transmission: A stochastic model with vaccination and Ornstein-Uhlenbeck process

In this paper, we explore and analyze a stochastic epidemiological model with vaccination and two mean-reverting Ornstein-Uhlenbeck processes to describe the dynamics of HBV transmission. Our study begins by deriving a sufficient condition for the extinction of hepatitis B. Additionally, we determine the presence of a stationary distribution using the Markov process stability theory. Next, we utilize algebraic equation theory and the associated Fokker-Planck equation to derive an explicit expression for the unique probability density function. To validate our theoretical findings, we conduct several numerical simulations using hepatitis B data provided by the World Health Organization (WHO). The results of these simulations support and complement our analytical approach.

A stochastic epidemic model with individual heterogeneity and mean-reverting Ornstein-Uhlenbeck process

A stochastic epidemic model with individual heterogeneity and mean-reverting Ornstein-Uhlenbeck process

First-exit-time problems for two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses

We study the first-exit-time problem for the two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses which run according to specific rules depending on the processes. We obtain the Laplace Transform of the first-exit-time probability density function and the corresponding moments. For both processes, some computational results on the first-exit-time densities are provided by means of the numerical inversion of the relevant Laplace Transforms. Moreover, we also investigate the asymptotic behaviour of the first-exit-time moments when the ellipse grows. In particular, an asymptotic exponential trend holds for the first-exit-time density of the mean-reverting Ornstein–Uhlenbeck process.

Introducing a novel mean-reverting Ornstein–Uhlenbeck process based stochastic epidemic model

The major objective of this paper is to examine a novel mean-reverting Ornstein–Uhlenbeck process-based stochastic SIRD model for transmission the epidemic disease that is a great crisis in numerous societies. For this purpose, the deterministic model is further converted into the stochastic form by allowing the infection rate satisfies the mean-reverting Ornstein–Uhlenbeck process to account the uncertainties involved in epidemic spread. At first using Lyapunov functions, the solution’s uniqueness and positivity will be demonstrated. Subsequently, the stochastic epidemic threshold ℜ0S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S}$$\\end{document} that controls the disease’s extinction and persistence in the mean is identified analytically. It has been established that when ℜ0S<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S} < 1$$\\end{document} the disease will extinguish, whereas if ℜ0S>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S} > 1$$\\end{document} the disease is persistent. At last, several numerical simulations are presented to demonstrate the findings of the hypothetical investigation results. These simulations served to vividly illustrate and validate the implications derived from the hypothetical analysis.

Dynamics caused by the mean-reverting Ornstein–Uhlenbeck process in a stochastic predator–prey model with stage structure

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.

A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps.

By using the Ornstein-Uhlenbeck (OU) process to simulate random disturbances in the environment, and considering the influence of jump noise, a stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala mutualism model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala mutualism model is proved by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala mutualism model is discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala mutualism model is obtained. Finally, the extinction of the stochastic Gilpin-Ayala mutualism model is proved. The theoretical results were verified by numerical simulations.

Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process

&lt;abstract&gt;&lt;p&gt;This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein–Uhlenbeck process. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results.&lt;/p&gt;&lt;/abstract&gt;

Dynamical complexity driven by water temperature in a size-dependent phytoplankton–zooplankton model with environmental variability

In this paper, we propose a stochastic phytoplankton–zooplankton (PZ) system modeling the growth dynamic of phytoplankton, where the death rate of zooplankton and the growth rate of phytoplankton are considered to be dependent on water temperature, and random environmental perturbation is characterized by mean-reverting Ornstein–Uhlenbeck process. Mathematical theoretical work mainly studies the stability, persistence, and Hopf bifurcation of deterministic PZ model, as well as analyzes the stochastic extinction, persistence in mean, and ergodic stationary distribution of stochastic PZ model, which provides a theoretical basis for numerical simulations. Numerical simulation work mainly reveals that water temperature plays a crucial role in influencing the dynamic of size-dependent phytoplankton–zooplankton interactions, whether in deterministic or random environments. An important conclusion of this study is that the increase of water temperature is able to induce bubbling effect, however, the small phytoplankton cell size can weaken this bubbling effect, and the large phytoplankton cell size can strengthen this bubbling effect, which supports the experimental conclusion that body size plays a key role in mitigating the negative impact of warming on predator–prey interactions. Further, it is found that the plankton with large body size has the advantages of stable and sustainable survival in the cold environment, while the plankton with small body size has advantages of stable and sustainable survival in the warm environment. Therefore, it can be inferred that plankton will evolve into large body size in the cold environment, but they will evolve into small body size in the warm environment.

Stationary distribution of a reaction-diffusion hepatitis B virus infection model driven by the Ornstein-Uhlenbeck process.

A reaction-diffusion hepatitis B virus (HBV) infection model based on the mean-reverting Ornstein-Uhlenbeck process is studied in this paper. We demonstrate the existence and uniqueness of the positive solution by constructing the Lyapunov function. The adequate conditions for the solution's stationary distribution were described. Last but not least, the numerical simulation demonstrated that reversion rates and noise intensity influenced the disease and that there was a stationary distribution. It was concluded that the solution tends more toward the stationary distribution, the greater the reversion rates and the smaller the noise.

Dynamics analysis of a stochastic HIV model with non-cytolytic cure and Ornstein–Uhlenbeck process

In this paper, we introduce an HIV infection model with virus-to-cell infection, cell-to-cell infection and non-cytolytic cure. Two mean-reverting Ornstein–Uhlenbeck processes are also taken into account in the model. Firstly, it is proved that the stochastic model has a unique positive global solution. The model is found to have at least one stationary distribution by constructing suitable Lyapunov functions if the critical condition R0s>1. Then, the probability density function near the quasi-positive equilibrium is obtained by solving the corresponding Fokker–Planck equation. The spectral radius method is used to derive the virus extinction under a sufficient condition R0e<1. Finally, some numerical simulations are carried out.

A generalized stochastic SIRS epidemic model incorporating mean-reverting Ornstein–Uhlenbeck process

The aim of this work is to study a new stochastic SIRS epidemic model that includes the mean-reverting Ornstein–Uhlenbeck process and a general incidence rate. First, we prove the global existence and positivity of the solution by using Lyapunov functions. Second, we analytically make out the stochastic epidemic threshold T̃0S which pilots the extinction and persistence in mean of the disease. We have proven that the disease extinguishes when T̃0S<1. Otherwise, if T̃0S>1, then disease is persistent in mean. For the critical case T̃0S=1, we have shown that the disease dies out by using an approach involving some appropriate stopping times. Finally, we present a series of numerical simulations to confirm the feasibility and correctness of the theoretical analysis results.

&lt;p&gt;Estimation of Stochastic Trend on Ukrainian Stock Market&lt;/p&gt;

The paper presents a new approach for price dynamic assessment on the Ukrainian stock market based on a qualitative modification of the Black-Scholes-Merton model for estimating unknown parameters with a stochastic trend. The mean-reverting Ornstein-Uhlenbeck process was chosen as the time-dependent trend parameter since it is both mathematically convenient and has a natural economic interpretation. In order to estimate the parameters of the corresponding stochastic diffusion, the trajectory-fitting technique based on the Kalman-Bucy filter was used. The proposed approach was then tested on UX-index assets.

Dynamical analysis of a modified Leslie–Gower Holling-type II predator-prey stochastic model in polluted environments with interspecific competition and impulsive toxicant input

In this paper, we use a mean-reverting Ornstein-Uhlenbeck process to simulate the stochastic perturbations in the environment, and then a modified Leslie–Gower Holling-type II predator-prey stochastic model in a polluted environment with interspecific competition and pulse toxicant input is proposed. Through constructing V-function and applying formula, the sharp sufficient conditions including strongly persistent in the mean, persistent in the mean and extinction are established. In addition, the theoretical results are verified by numerical simulation.

Dynamical behavior of a stochastic SIQR epidemic model with Ornstein–Uhlenbeck process and standard incidence rate after dimensionality reduction

Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results.

A stochastic predator–prey system with modified LG-Holling type II functional response

In this paper, a stochastic two-predator one-prey system with modified Leslie–Gower and Holling-type II functional response is proposed, which is randomly disturbed by the well-known mean-reverting Ornstein–Uhlenbeck process. By Itoˆ’s integral formula, stochastic comparison theorem, the strong law of large number theorem for martingales, and modeling and analysis methods in stochastic differential equations, the existence and uniqueness of the global positive solution for the system are discussed. Then, the additional conditions for the persistence in the mean and extinction of the system are obtained, respectively. Besides, the effects of the speed of reversion and the intensity of volatility in the Ornstein–Uhlenbeck process on the dynamics of the system are investigated. Furthermore, the ergodic stationary distribution of the system under a low-level intensity of stochastic noise is also derived, which indicates that x, y1 and y2 will be persist and fluctuate around the positive values. Finally, a series of numerical examples are provided to verify the correctness of the theoretical analysis.

Stationary distribution and extinction of a stochastic SVEIS epidemic model incorporating Ornstein–Uhlenbeck process

In this paper, we assumed that the parameter in the SVEIS epidemic model satisfies the mean-reverting Ornstein–Uhlenbeck process, and propose a new stochastic SVEIS model. Through constructing suitable Lyapunov function, we prove that this stochastic model has a stationary distribution when the critical value R0s>1. Then the sufficient condition R0e<1 for the exponential extinction is also established. Results show that the values go to the deterministic basic reproduction number R0 as the intensity of volatility tends to 0.

A Stochastic Non-autonomous Chemostat Model with Mean-reverting Ornstein–Uhlenbeck Process on the Washout Rate

A Stochastic Non-autonomous Chemostat Model with Mean-reverting Ornstein–Uhlenbeck Process on the Washout Rate

Analysis of a stochastic population model with mean-reverting Ornstein–Uhlenbeck process and Allee effects

Considering the survival regulation mechanisms of many groups of animals and the complexity of random variations in ecosystem, in this paper, we mainly formulate and study a stochastic non-autonomous population model with Allee effects and two mean-reverting Ornstein–Uhlenbeck processes. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we give the existence and moment estimate of a global solution of the stochastic model. Then the sufficient criteria for exponential extinction and the existence of a stationary distribution of the stochastic model are established. Moreover, there are some challenges to give the explicit expression of probability density function of the stationary distribution. By solving the relevant Fokker–Planck equation, we derive the approximate expression of the density function of the stochastic model. Finally, some numerical simulations are provided to verify our analytical results and study the impact of stochastic noises on population dynamics.

Optimal filtering equations in state space model of the two factors mean reverting Ornstein-Uhlenbech process

This paper investigates a nonlinear state space model for energy markets ruled by a two factor mean reverting Ornstein Uhlenbeck process with a stochastic nonlinear autoregressive drift term. A recursive approach using Taylor series based approximations for filtering, prediction, and smoothing problem of the hidden mean reverting factor from the noisy observations is proposed. The closed-form solutions are obtained for estimating the hidden mean reverting factor. Also, the existence and uniqueness of the solution for this nonlinear system are proved. Finally, the performance of the developed methods is checked in a simulation study.