Long-time behaviour of a stochastic chemostat model with distributed delay

Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

A stochastic chemostat model with an inhibitor and noise independent of population sizes

Noise-induced bifurcations in the stochastic chemostat model with general nutrient uptake functions

In this paper, we formulate and investigate a two‐microorganism and single nutrient chemostat model with Monod‐Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic model. Finally, numerical simulations are carried out to illustrate our main results.

Dynamics of the Stochastic Chemostat Model with Monod-Haldane Response Function

Stochastic characteristics of a chemostat model with variable yield

We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.

This article presents an analysis of deterministic models of computer virus epidemic propagation (SIR models) and their classification. The main areas of research into these models are highlighted. An analysis of existing stochastic models based on the SIR model and their diversity are presented. A method for constructing a stochastic SIR model based on the classical SIR model, represented by a system of Ito stochastic differential equations with a Wiener process, is proposed. The possibility of constructing a stochastic and deterministic model of an information system protected against computer virus infection with probability 1 is demonstrated: a stochastic model, in which infection by viruses occurs continuously, and a deterministic model, in which the virus is present in the information system. A mathematical stochastic model of an information system protected from computer virus outbreaks is constructed as a system of stochastic differential equations whose first integrals are invariants preserved with probability 1. A certain functional relationship between model variables, maintaining a constant value, is considered as the system's security indicator. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Similarly, based on the proposed algorithm, a deterministic model of an information system protected from computer virus infection is constructed. A control similar to programmed control with probability 1 (PCP1) is introduced into the constructed model, which allows the invariants to be maintained. A distinctive feature of the proposed models is that they preserve invariants associated with the properties that ensure the security of the information system. The behavior of the constructed models is studied using numerical simulation in the MathCad environment. Based on the research results, conclusions were drawn on the possibility of using the proposed method in constructing stochastic models based on other models of epidemic spread, as well as for models of protecting an information system from the spread of a computer virus epidemic.

In this paper, we consider a stochastic SI epidemic model with regime switching. The Markov semigroup theory is employed to obtain the existence of a unique stable stationary distribution. We prove that if mathcal{R}^{s}<0, then the disease becomes extinct exponentially; whereas if mathcal{R}^{s}>0 and beta(i)>alpha(i), iinmathbb{S}, then the densities of the distributions of the solution can converge in L^{1} to an invariant density.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

Due to the negative impact of infectious diseases on population growth, it is important to understand the dynamic behavior of such diseases. Mathematical deterministic SIR models are widely used to study the spread of infectious diseases. In real life, there is a lot of randomness and stochasticity such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. in the considered models. In that case, we must introduce Poisson noises into the population models for describing such discontinuous systems. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous SIR model disturbed by white noise, centered and non-centered Poisson noises. In the deterministic case there is a threshold of the system for an epidemic to occur, so called the basic reproduction number. Depending on the value of the reproduction number there is the disease-free equilibrium, or there is an endemic equilibrium, which implies the disease always remains. In the case of the autonomous stochastic SIR model, we study the asymptotic behavior of the solution to the corresponding system of stochastic differential equations around these points of equilibrium.

Deterministic models have played an important role in biology over the past few centuries, and continue to play. However, there is a need to introduce a random factor due to the nature of biological processes, which led to the emergence of stochastic models. To bridge the gap between deterministic and stochastic models, and in appreciation of the huge efforts made in deterministic models, which have played a fundamental role in understanding, predicting, and controlling the transmission dynamics of infectious diseases, this study aims to clarify a method for creating stochastic model for one of most vital and recent deterministic, that is Covid-19 model with two doses, which showed the effect of receiving both doses in containing the disease. To achieve this goal, the deterministic model was studied extensively before converting to corresponding stochastic. The deterministic and stochastic systems were solved numerically and represented using MATAB tools. Keywords: Covid-19 deterministic model, stochastic model, next generation matrix method (NGMM), The diffusion matrix G.

Classification of asymptotic behavior in a stochastic SIS epidemic model with vaccination

In modeling terrain images corresponding to infrared scenes it has been found the images are characterized by a long-range dependence structure and high variability. The long-range dependence manifests itself in a `1/f' type behavior in the power spectral density and statistical self-similarity, both of which suggest the use of a stochastic fractal model. The traditional stochastic fractal model is fractional Brownian motion, which assumes the increment process arises from a Gaussian distribution. This model has been found to be rather limiting due to this restriction and therefore is incapable of modeling processes possessing high variability and emanating from long-tailed non-Gaussian distributions. Stable distributions have been shown to be good models of such behavior and have been incorporated into the stochastic fractal model, resulting in the fractional Levy stable motion model. The model is demonstrated on a terrain image and is used in an interpolation scheme to improve the resolution of the image.