Stochastic characteristics of a chemostat model with variable yield

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

<abstract><p>The spread of infectious diseases are inevitably affected by natural and social factors, and their evolution presents oscillations and other uncertainties. Therefore, it is of practical significance to consider stochastic noise interference in the studies of infectious disease models. In this paper, a stochastic SIR model with nonlinear incidence and recovery rate is studied. First, a unique global positive solution for any initial value of the system is proved. Second, we provide the sufficient conditions for disease extinction or persistence, and the influence of threshold $ \tilde{R_{0}} $ of the stochastic SIR model on disease state transition is analyzed. Additionally, we prove that the system has a stationary distribution under some given parameter conditions by building an appropriate stochastic Lyapunov function as well as using the equivalent condition of the Hasminskii theorem. Finally, the correctness of these theoretical results are validated by numerical simulations.</p></abstract>

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.

In this paper, we establish and analyze a stochastic human immunodeficiency virus model with both virus-to-cell and cell-to-cell transmissions and Ornstein–Uhlenbeck process, in which we suppose that the virus-to-cell infection rate and the cell-to-cell infection rate satisfy the Ornstein–Uhlenbeck process. First, we validate that there exists a unique global solution to the stochastic model with any initial value. Then, we adopt a stochastic Lyapunov function technique to develop sufficient criteria for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of all CD4+ T cells and free viruses. In particular, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-chronic infection equilibrium of the stochastic system. Finally, numerical simulations are conducted to validate these analytical results. Our results suggest that the methods used in this paper can be applied to study other viral infection models in which the infected CD4+ T cells are divided into latently infected and actively infected subgroups.

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> which is a sharp threshold for the dynamics: If R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> <; 1, the disease will die out; if R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> > 1, the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.

In this paper, single-species population diffusion models with chemotaxis in polluted environment are proposed and studied. For the deterministic single-species population diffusion model, the sufficient conditions for the extinction and strong persistence of the single-species population are established. For the stochastic single-species population diffusion model. First, we show that system has unique global positive solution. And then, the sufficient conditions for extinction and strongly persistent in the mean of the single-species are obtained. Numerical simulations are used to confirm the efficiency of the main results.

In this paper, we consider the chemostat model with inhibitory exponential substrate, variable yield and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. The Hopf bifurcation of solutions to the system is studied. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Based on some known techniques on limit sets of differential dynamical systems, we show that, for any time delay, the chemostat model is permanent if and only if only one positive equilibrium exits.

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

In this paper, we derive sufficient conditions that the origin of a stochastic system is stable in some meanings by a particular shape of a stochastic Lyapunov function without smoothness at some points. The treated stability properties are transient stability and transient asymptotic stability for the origins being non-equilibria, and stability in probability and asymptotic stability in probability for the origins being equilibria. Furthermore, we also discuss the relationship between our stochastic Lyapunov functions and viscosity supersolutions, which are the notion often used for analyzing non-smooth Lyapunov functions for deterministic systems. Then, we propose a relaxed notion of viscosity weak supersolution for our stochastic Lyapunov function.

Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching

In order to study the impact of random environments during the spread of the disease, we propose a novel stochastic SIQS model on scale-free networks, which introduces stochastic perturbations to the infected rates. We first obtain the existence of global positive solutions. Moreover, by constructing appropriate stochastic Lyapunov functions, we prove sufficient conditions for extinction and persistence of the disease. Finally, we verify the analysis results through numerical simulations. In addition, the results of previous studies are also improved in our research.

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

In our life there are various infectious diseases which lead to negative effects on the economy. Therefore, it is highly important to develop mathematical models to describe the spread of the pandemic. In this study, we first develop a deterministic SEI1I2R epidemic model incorporating double epidemic hypothesis. We first verify that there exists a unique global positive solution of the system with any positive initial value, then we investigate the local and global asymptotical stability of the equilibria. Following that, we incorporate the inherent stochasticity of the virus spread by reinterpreting the previous SEIR-type epidemic model of infectious diseases as stochastic systems modelled via the log-normal standard Ornstein–Uhlenbeck process. After that we study the stochastic dynamics of the model in detail. More precisely, we first analyze the existence and uniqueness of the global solution of the system. Then by using the Markov semigroup theory, we establish sufficient criteria for the existence and uniqueness of an invariant probability measure of the system, which implies the persistence of the disease in a community. Especially, under the same conditions as the local asymptotical stability of the endemic equilibrium, we get the specific form of the local probability density function around the quasi-endemic equilibrium of the stochastic system. Meanwhile, we also verify that the invariant global probability density function can be approximated by the local probability density function as the noise intensity tends to zero. In addition, we give some sufficient conditions for elimination of the disease. Finally, numerical simulations are carried out to demonstrate the validity of the derived results and the availability of the proposed models by using MATLAB. This work can improve our insight into epidemic dynamics and help us implement interventions to regulate the disease dynamics.

In this paper, we consider a stochastic food chain chemostat model with variable yields. First, we prove the stochastic model has a unique global positive solution. Second, by employing suitable Lyapunov functions, Itô[Formula: see text] formula and some other important inequalities, the existence of a unique ergodic stationary distribution of a stochastic food chain chemostat model is researched, which can help us better understand the statistical characteristics of stochastic food chain chemostat models. Second, we investigate the extinction of the microorganism and the bacteria. Moreover, the case of extinction for bacteria but persistence for microbial species is considered. Finally, some numerical simulations are carried out to illustrate our theoretical results and the influence of the variable yields on the microorganism and the bacteria.