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In this paper, we consider the stochastic nutrient–phytoplankton–zooplankton model with nutrient cycle. In order to take stochastic fluctuations into account, we add the stochastic increments to the variations of biomass of nutrition, phytoplankton and zooplankton during time interval Δ t , thus we obtain the corresponding stochastic model. Subsequently, we explore the existence, uniqueness and stochastically ultimate boundness of global positive solution. By constructing suitable Lyapunov function, we also obtain V-geometric ergodicity of this model. In addition, the sufficient conditions of exponential extinction and persistence in the mean of plankton are established. At last, we present some numerical simulations to validate theoretical results and analyze the impacts of some important parameters. [ABSTRACT FROM AUTHOR]

This study explores the asymptotic behavior of a stochastic epidemic model that accounts for the impact of media coverage. The initial focus lies on determining the conditions leading to the exponential extinction of the disease. Additionally, we investigate the weak convergence of the probability distribution of the stochastic process N(t), representing the total population, to a one-dimensional stochastic process with density calculated through the Fokker–Planck equation. Subsequently, we demonstrate the persistent nature of the disease and utilize Has’minskii theory to establish the presence of a unique ergodic stationary distribution for our stochastic epidemic model. Finally, numerical simulations are conducted to validate the theoretical findings.

This paper aims to improve the analysis of a stochastic SIVR epidemic model with an imperfect vaccination process, taking into consideration the fact that a fraction of vaccinated individuals becomes susceptible to infection. The uniqueness of the positive solution is shown. Further, we obtain the threshold of the stochastic SIVR model which determines whether the epidemic will persist or die out. In the extinction case, we prove that the solution converges almost surely toward the disease-free equilibrium of the deterministic SIVR model. Some numerical illustrations are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic hybrid differential infectivity epidemic model with standard incidence perturbed by mean‐reverting Ornstein–Uhlenbeck process. Applying Lyapunov method, we first show existence and uniqueness of the global solution. Then sufficient conditions for persistence in the mean and exponential extinction of the infectious disease are obtained. Furthermore, by solving the corresponding Fokker–Planck equation, we derive that the global solution around the endemic equilibrium follows a unique probability density function. Finally, numerical simulations are employed to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic delay Lotka-Volterra model whose coefficients are dependent on expectations. By applying the M-matrix technique, the existence and uniqueness of the global positive solution and its exponential extinction are studied. An example is provided to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Direct volume rendering has become a popular method for visualizing volumetric datasets. Even though computers are continually getting faster, it remains a challenge to incorporate sophisticated illumination models into direct volume rendering while maintaining interactive frame rates. In this paper, we present a novel approach for advanced illumination in direct volume rendering based on GPU ray-casting. Our approach features directional soft shadows taking scattering into account, ambient occlusion and color bleeding effects while achieving very competitive frame rates. In particular, multiple dynamic lights and interactive transfer function changes are fully supported. Commonly, direct volume rendering is based on a very simplified discrete version of the original volume rendering integral, including the development of the original exponential extinction into a-blending. In contrast to a-blending forming a product when sampling along a ray, the original exponential extinction coefficient is an integral and its discretization a Riemann sum. The fact that it is a sum can cleverly be exploited to implement volume lighting effects, i.e. soft directional shadows, ambient occlusion and color bleeding. We will show how this can be achieved and how it can be implemented on the GPU. [ABSTRACT FROM AUTHOR]