Existence Of Ergodic Stationary Distribution Research Articles

Long-term effect of SARS-CoV-2 variant : Challenging issues and controlling strategies

In this study, a latest version of COVID-19 pandemic is hand overed. A Stochastic post COVID-19 delayed model is developed to explore the spread of COVID-19 as well as omicron variant with the correlation of heart attack. This article provides an eradication of the COVID-19 and omicron variant as well as the population who have heart attack after post COVID-19 of these epidemic diseases. Then the existence and uniqueness of global positive solution are studied. Ensuing, In this article, we classify COVID-19 virus and omicron variant which go to extinction and become persistent in mean. By using Lyapunov function, the existence of ergodic stationary distribution are established. Later from the persistent disease as well as extinction, heart disease are ready to develop in the human body. By Eventually, an optimal control strategies are introduced in the form of stochastic post COVID-19 delayed model to control two different types of virus. Finally, the numerical simulation are presented to determine the behavior of dynamical system by utilizing the real data of United Kingdom.

On a stochastic generalized delayed SIR model with vaccination and treatment

In this paper we consider a stochastic susceptible-infectious-recovered (SIR) epidemiological model which is obtained on the basis of the deterministic SIR epidemiological model with general incidence rate, distributed delay, general treatment and vaccination. For our model we first prove existence and uniqueness of the global positive solution and then we consider conditions under which disease survives in population by proving the existence of ergodic stationary distribution. Also, the stochastic model adopts the disease-free equilibrium from it is deterministic analogue, and we investigate conditions under which it is stable in probability. Finally, the numerical simulations with real life date are carried out to illustrate the theoretical results.

A confidence ellipse analysis for stochastic dynamics model of Alzheimer's disease

The Alzheimer’s disease (AD) is a neurodegenerative disease, which is caused by the aggregation of beta-amyloid peptide ( $$A\beta$$ ) in the patient’s brain and the disorder of $${\text{Ca}}^{2 + }$$ homeostasis in neurons. Caluwé and Dupont (Theor Biol 331:12–18, 2013) proposed a deterministic AD model to explore the effect of $${\text{Ca}}^{2 + }$$ on AD. They demonstrated the positive feedback loop between $$A\beta$$ and $${\text{Ca}}^{2 + }:$$ and the occurrence of bistability. Based on their results, we further discuss the bistable behaviors. We present two periodically feasible drug strategies to alleviate the AD and screen out more effective one. In this paper, we also formulate a stochastic AD model, analyze the existence and uniqueness of global positive solutions and establish sufficient conditions for the existence of ergodic stationary distribution. Furthermore, the confidence ellipses describing the configurational arrangement of stochastic coexistence equilibria are constructed by stochastic sensitivity function technique, and tipping threshold is estimated as well. Noise-induced stochastic switching between two coexistence equilibria is observed in bistability region. Our results provide a new idea to control noise to alleviate AD through physical therapy.

Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Dynamics of a stochastic multi-stage sheep brucellosis model with incomplete immunity

This paper considered a multi-stage sheep brucellosis model with incomplete immunity. First, we established a deterministic model, calculated the basic reproduction number [Formula: see text], set out the conditions for the global stability of the disease-free equilibrium and endemic equilibrium. Second, considering the influence of environmental white noise on brucellosis infection, we further established the stochastic version of the model. By constructing a suitable Lyapunov function, we proved the existence and uniqueness of the global positive solution. Further, we got the sufficient conditions for disease extinction and the existence of ergodic stationary distribution. Finally, we carried out some numerical simulations to verify the theoretical results.

Vaccination effect on a stochastic epidemic model with healing and relapse

In this work, we consider a stochastic epidemic model with vaccination, healing and relapse. We prove the existence and the uniqueness of the positive solution. We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system. Moreover, we also establish sufficient conditions for the existence of ergodic stationary distribution to the model, which reveals that the infectious disease will persist. The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.

A nonlinear HCV model in deterministic and randomly fluctuating environments

Two novel deterministic and stochastic nonlinear hepatitis C virus (HCV) models with nonlinear incidence rates as , and , containing acute and chronic infections, are developed, and an endeavor to understand their dynamics has been accomplished. For the former model, both the existence of unique equilibria and the sharp sufficient conditions of globally stable equilibria under different cases are established. Whereafter, the efforts taken to detect the asymptotic behaviors around equilibria, the existence of ergodic stationary distribution, and stochastic extinction of the latter model have been supplemented with meticulous theoretical proofs. At last, several numerical examples and discussions are illustrated to sustain and visualize our theoretical results. Compared and analyzed the numerical and theoretical results, we can find that the noise can influence the dynamics of disease and turn the disease from persistence to extinction. Moreover, several control measures are suggested based on our study.

Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling

A stochastic nutrient–phytoplankton–zooplankton model with instantaneous nutrient recycling is proposed and analyzed in this paper. When the nutrient uptake function and the grazing function are linear and the ingested phytoplankton is completely absorbed by the zooplankton, we establish two stochastic thresholds R0S and R1S, which completely determine the persistence and extinction of the plankton. That is, if R0S<1, both the phytoplankton and the zooplankton eventually are eliminated; if R0S>1 and R1S<1, the phytoplankton is persistent in mean but the zooplankton is extinct; while for R1S>1, the entire system is persistent in mean. Furthermore, sufficient criteria for the existence of ergodic stationary distribution of the model are obtained and the persistent levels of the plankton are estimated. Numeric simulations are carried out to illustrate the theoretical results and to conclude our study. Our results suggest that environmental noise may cause the local bloom of phytoplankton, which surprisingly can be used to explain the formation of algal blooms to some extent. Moreover, we find that the nonlinear nutrient uptake function and grazing function may take credit for the periodic succession of blooms regardless of whether they are in the absence or presence of the environmental noises.

Stationary distribution and extinction of a stochastic cattle brucellosis model

In this paper, we propose a deterministic cattle brucellosis model and analyze the global stability of disease-free equilibrium and the persistence of the disease. Secondly, we consider environmental perturbations and present a stochastic version of the model, study the existence and uniqueness of positive solution, investigate the sufficient conditions for the existence of ergodic stationary distribution and extinction of the model. Finally, we use numerical simulation to observe the influence of the disturbance term on the original model.

The impact of environmental fluctuations on a plankton model with toxin-producing phytoplankton and patchy agglomeration

In aquatic ecosystem, plankton populations are easily affected by environmental fluctuations due to the unpredictability of many physical factors such as nutrient availability, acidity, water temperature and so on. To reveal how environmental fluctuations influence plankton populations, in this paper, we develop and analyze a stochastic toxin-producing phytoplankton (TPP)-zooplankton model with patchy agglomeration. We first perform the survival analysis of the model, including the persistence and extinction as well as the existence of ergodic stationary distribution. Then, for the scenario when the model exhibits bistability between a positive equilibrium or a limit cycle and TPP-only equilibrium in the absence of noise, we further study the phenomenon of noise-induced transition from plankton coexistence to zooplankton extirpation. Also, we numerically estimate the critical values of noise intensities for the occurrence of transitions with the aids of the technique of stochastic sensitivity functions. Our research indicates that zooplankton population becomes more vulnerable and easily goes to extinction in the presence of environmental noise, and as a result, phytoplankton could benefit from the decrease of grazing pressure. This partially explains the often seen phenomenon in aquatic ecosystems that many toxin-releasing microalgal species with patchy agglomeration are more likely to cause blooms.

Stationary Distribution and Extinction of a Stochastic Brucellosis Model with Standard Incidence

In this paper, we proposed a stochastic SVEI brucellosis model with stage structure by introducing the effect of environmental white noise on transmission dynamics of brucellosis. By Has'minskii theory and constructing suitable Lyapunov functions, we established sufficient conditions on the existence of ergodic stationary distribution for the considered model. Moreover, we also established sufficient condition for extinction of the disease. Finally, two examples with numerical simulations are given to illustrate the main results of this paper.

Dynamics of interacting plankton induced by plankton body size in deterministic and stochastic environments

In this paper, we study the dynamics of a nutrient–phytoplankton–zooplankton model incorporating the factor of plankton body size in deterministic and stochastic environments. The value of this paper lies in two aspects: Mathematically, for the model without stochastic environmental fluctuations, we give the existence of boundary and positive equilibria, show the local and global stability of these equilibria, and prove the existence of Hopf bifurcation. For the model with stochastic environmental fluctuations, we provide the results related to the stochastic extinction and persistence in the mean, as well as the existence of ergodic stationary distribution. Ecologically, via numerical simulations, we find that plankton body size plays a key role in regulating the dynamics of interacting plankton in the deterministic and stochastic environments. It is worth emphasizing that the increase of phytoplankton cell size can stabilize the model by excluding the periodic solution induced by nutrient removal rate, while the increase of zooplankton body size can remain such periodic phenomena. Additionally, it should be noted that the very small phytoplankton cell size or zooplankton body size has a negative effect on the plankton growth in the stochastic fluctuation environments. These results may help to understand the complex dynamics of plankton models.

Modeling and analysis of SIR epidemic dynamics in immunization and cross-infection environments: Insights from a stochastic model

We propose a stochastic SIR model with two different diseases cross-infection and immunization. The model incorporates the effects of stochasticity, cross-infection rate and immunization. By using stochastic analysis and Khasminski ergodicity theory, the existence and boundedness of the global positive solution about the epidemic model are firstly proved. Subsequently, we theoretically carry out the sufficient conditions of stochastic extinction and persistence of the diseases. Thirdly, the existence of ergodic stationary distribution is proved. The results reveal that white noise can affect the dynamics of the system significantly. Finally, the numerical simulation is made and consistent with the theoretical results.

Stochastic Hybrid Hepatitis B Epidemic Model with Markovian Switching

In this paper, we consider a stochastic epidemic model under regime‐switching. We show that our model is well‐posed by proving the existence of solution, positivity, and uniqueness. Furthermore, we investigate sufficient conditions for the existence of ergodic stationary distribution to the model, which shows that the hepatitis B epidemic will persist. In addition, we establish sufficient conditions for the extinction of the hepatitis B epidemic. We perform numerical simulations to confirm and illustrate the theoretical results.

A stochastic mussel-algae model under regime switching.

We investigate a novel model of coupled stochastic differential equations modeling the interaction of mussel and algae in a random environment, in which combined effect of white noises and telegraph noises formulated under regime switching are incorporated. We derive sufficient condition of extinction for mussel species. Then with the help of stochastic Lyapunov functions, a well-grounded understanding of the existence of ergodic stationary distribution is obtained. Meticulous numerical examples are also employed to visualize our theoretical results in detail. Our analytical results indicate that dynamic behaviors of the stochastic mussel-algae model are intimately associated with two kinds of random perturbations.

Extinction and stationary distribution of a stochastic SIQR epidemic model with demographics and non-monotone incidence rate on scale-free networks.

By taking full consideration of contact heterogeneity of individuals, quarantine measures, demographics, information transmission and random environments, we present a stochastic SIQR epidemic model with demographics and non-monotone incidence rate on scale-free networks, which introduces stochastic perturbations to death rate. The formula of the basic reproduction number of the deterministic model is obtained by utilizing the existence of the endemic equilibrium. Next, we define a stopping time, then the existence of a unique global positive solution for the stochastic model is proved by constructing appropriate Lyapunov function to demonstrate the stopping time is infinite. In addition, we also manifest sufficient conditions for diseases extinction and the existence of ergodic stationary distribution by constructing appropriate stochastic Lyapunov functions. At last, numerical simulations illustrate the analytical results.

Complex dynamics of a stochastic uni-directional consumer-resource mutualism system

Like predation and competition, mutualism is recognized as a consumer-resource interaction, which includes bi-directional and uni-directional mutualisms. In this paper, we firstly propose a stochastic uni-directional consumer-resource system of two species in which the consumer has both positive and negative effects on the resource, while the resource has only a positive effect on the consumer. We then mathematically analyze the system, to demonstrate the existence, uniqueness, asymptotic pathwise behavior and stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the global attractivity and the existence of ergodic stationary distribution of the system. We also establish sufficient conditions for the extinction and persistence in mean of the resource, the consumer or the entire system. Numerical simulations are carried out to demonstrate the analytical results.

Periodicity and stationary distribution of two novel stochastic epidemic models with infectivity in the latent period and household quarantine.

Two types of stochastic epidemic models are formulated, in which both infectivity in the latent period and household quarantine on the susceptible are incorporated. With the help of Lyapunov functions and Has’minskii’s theory, we derive that, for the nonautonomous periodic version with white noises, it owns a positive periodic solution. For the other version with white and telephone noises, we construct stochastic Lyapunov function with regime switching to present easily verifiable sufficient criteria for the existence of ergodic stationary distribution. Also, we introduce a series of numerical simulations to support our analytical findings. At last, a brief discussion of our theoretical results shows that the stochastic perturbations and household quarantine measures can significantly affect both periodicity and stationary distribution.

Dynamics of stochastic HTLV‐I infection model with nonlinear CTL immune response

In this paper, the dynamics of stochastic human T‐cell leukemia virus type I (HTLV‐I) infection model with cytotoxic T lymphocyte (CTL) immune response is investigated. First, we show that the stochastic model exists as a unique positive global solution originating from the positive initial value. Second, we demonstrate that the stochastic model is stochastically permanent and stochastically ultimately bounded for any positive initial value. Third, we establish sufficient conditions for the existence of ergodic stationary distribution of the stochastic model. Fourth, the threshold between extinction and persistence of the virus is obtained. Finally, numerical simulations are carried out to illustrate the theoretical results.

Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.