Ergodic Distribution Research Articles

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.

Dynamics of a Stochastic Epidemic Model with Vaccination and Multiple Time‐Delays for COVID‐19 in the UAE

In this paper, we study the dynamics of COVID‐19 in the UAE with an extended SEIR epidemic model with vaccination, time‐delays, and random noise. The stationary ergodic distribution of positive solutions is examined, in which the solution fluctuates around the equilibrium of the deterministic case, causing the disease to persist stochastically. It is possible to attain infection‐free status (extinction) in some situations, in which diseases die out exponentially and with a probability of one. The numerical simulations and fit to real observations prove the effectiveness of the theoretical results. Combining stochastic perturbations with time‐delays enhances the dynamics of the model, and white noise intensity is an important part of the treatment of infectious diseases.

Dynamic Analysis of a Stochastic Rumor Propagation Model with Regime Switching

We study the rumor propagation model with regime switching considering both colored and white noises. Firstly, by constructing suitable Lyapunov functions, the sufficient conditions for ergodic stationary distribution and extinction are obtained. Then we obtain the threshold Rs which guarantees the extinction and the existence of the stationary distribution of the rumor. Finally, numerical simulations are performed to verify our model. The results indicated that there is a unique ergodic stationary distribution when Rs>1. The rumor becomes extinct exponentially with probability one when Rs<1.

Stationary distribution and density function analysis of stochastic susceptible‐vaccinated‐infected‐recovered (SVIR) epidemic model with vaccination of newborns

Birth vaccinations are becoming more common in society. In this paper, we describe the developed stochastic susceptible‐vaccinated‐infected‐recovered (SVIR) epidemic model with vaccination of newborns that enable us to concern the stationary distribution and further density function. By constructing a series suitable Lyapunov function, we derive the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution. More importantly, under the same conditions, we creatively find further the density function which is based on solving corresponding Fokker–Planck equation. The results of numerical simulation, which is supported by pertussis disease data, show that our conclusion accords with reality. The density function throws light on the property of an epidemic after being stationary and furnishes more information about the disease.

The impact of role reversal on the dynamics of predator-prey model with stage structure

In this paper, we investigate the dynamics of a predator-prey model with role reversal and its corresponding stochastic differential equation version. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrences of transcritical bifurcation and Hopf bifurcation are studied. For the stochastic model, we investigate the existence of a unique global positive solution and establish sufficient conditions for the existence of a unique ergodic stationary distribution. Finally, numerical simulations are carried out to verify our analytical results, which indicate that both the intensity of white noise and the predation of the juvenile predators have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses.

The impact of plankton body size on phytoplankton-zooplankton dynamics in the absence and presence of stochastic environmental fluctuation

In this paper, we study the dynamics of a stochastic phytoplankton-zooplankton (PZ) model with phytoplankton cell size and zooplankton body size. In the case of PZ model without stochastic environmental fluctuations, we provide the positivity and boundedness of the solutions, investigate the dissipativity and permanence of the model, prove the existence of Hopf bifurcation, and give the local and global stability of the boundary and positive equilibria. In the case of PZ model with stochastic environmental fluctuations, the stochastic dynamics including a unique ergodic stationary distribution, stochastic permanence, stochastic extinction and persistence in the mean are explored in detail. Based on the theoretical analysis above, via numerical simulations, we find that the increase of environmental capacity can not only destabilize the deterministic model via Hopf bifurcation and induce periodic solutions, but can also stabilize the deterministic model by rejecting the periodic solutions. Interestingly, the increase of phytoplankton cell size or zooplankton body size can stabilize the deterministic model by excluding the periodic solutions induced by environmental capacity. Additionally, it is worth emphasizing that the small phytoplankton cell size can lead to the inability of plankton to survive in both deterministic and random environments, while the small zooplankton body size can destabilize the deterministic model and induce periodic solutions. Furthermore, it should be noted that the large phytoplankton cell size can weaken the effect of random environmental disturbance, but large zooplankton body size can not. These results may provide new insights in understanding the complex dynamics of phytoplankton-zooplankton models.

THE EFFECT OF STOCHASTIC VARIABILITY ON TRANSMISSION DYNAMICS OF ECHINOCOCCOSIS

Echinococcosis, one of the most serious zoonotic diseases, has a severe impact on the human health and economic development. This paper mainly focuses on the effect of stochastic variability on transmission dynamics of echinococcosis. A stochastic model describing the transmission of echinococcus granulosus in dog-livestock-human is proposed. By using the Itô formula, strong law of large numbers for martingale and the inequality analysis, the criteria on the extinction with probability one for the disease are obtained. In addition, by constructing an appropriate stochastic Lyapunov function, the existence of a unique ergodic stationary distribution is established. Furthermore, numerical simulations have been performed to not only support our analytical results but also display that noise intensities would have significant impact on the speed of the extinction for the echinococcosis disease, and as well as the shape for the probability density curve of the stationary distribution of the solution for this model.

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.

The impact of nonlinear perturbation to the dynamics of HIV model

In this paper, we developed and studied a stochastic HIV model with nonlinear perturbation. Through a rigorous analysis, we firstly showed that the solution of the stochastic model is positive and global. Then, by employing suitable stochastic Lyapunov functions, we prove that the stochastic model admit a unique ergodic stationary distribution. In addition, sufficient conditions for the extinction of HIV infection are derived. Finally, numerical simulations are employed to confirm our theoretical results.

STATIONARY DISTRIBUTION AND EXTINCTION OF STOCHASTIC CORONAVIRUS (COVID-19) EPIDEMIC MODEL

The aim of this paper is to model corona-virus (COVID-19) taking into account random perturbations. The suggested model is composed of four different classes i.e. the susceptible population, the smart lockdown class, the infectious population, and the recovered population. We investigate the proposed problem for the derivation of at least one unique solution in the positive feasible region of nonlocal solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function and the condition for the extinction of the disease is also established. The obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been simulated numerically.

Global Dynamics of a Stochastic Viral Infection Model with Latently Infected Cells

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus.

Ergodic property, extinction, and density function of an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbations

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold related to the basic reproduction number . A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if . Next, by solving the corresponding Fokker‐Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical 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.

Stochastic analysis of a SIRI epidemic model with double saturated rates and relapse

Infectious diseases have for centuries been the leading causes of death and disability worldwide and the environmental fluctuation is a crucial part of an ecosystem in the natural world. In this paper, we proposed and discussed a stochastic SIRI epidemic model incorporating double saturated incidence rates and relapse. The dynamical properties of the model were analyzed. The existence and uniqueness of a global positive solution were proven. Sufficient conditions were derived to guarantee the extinction and persistence in mean of the epidemic model. Additionally, ergodic stationary distribution of the stochastic SIRI model was discussed. Our results indicated that the intensity of relapse and stochastic perturbations greatly affected the dynamics of epidemic systems and if the random fluctuations were large enough, the disease could be accelerated to extinction while the stronger relapse rate were detrimental to the control of the disease.

Dynamics of Tumor-Immune System with Random Noise

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases.

Stochastic optimal control for norovirus transmission dynamics by contaminated food and water

Norovirus is one of the most common causes of viral gastroenteritis in the world, causing significant morbidity, deaths, and medical costs. In this work, we look at stochastic modelling methodologies for norovirus transmission by water, human to human transmission and food. To begin, the proposed stochastic model is shown to have a single global positive solution. Second, we demonstrate adequate criteria for the existence of a unique ergodic stationary distribution by developing a Lyapunov function. Thirdly, we find sufficient criteria ℜs < 1 for disease extinction. Finally, two simulation examples are used to exemplify the analytical results. We employed optimal control theory and examined stochastic control problems to regulate the spread of the disease using some external measures. Additional graphical solutions have been produced to further verify the acquired analytical results. This research could give a solid theoretical foundation for understanding chronic communicable diseases around the world. Our approach also focuses on offering a way of generating Lyapunov functions that can be utilized to investigate the stationary distribution of epidemic models with nonlinear stochastic disturbances.

The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold R 0 s is given. When R 0 s &lt; 1 , the diseases can be extincted with probability one. When R 0 s &gt; 1 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.

On the Dynamics Behaviors of a Stochastic Echinococcosis Infection Model with Environmental Noise

This paper pays main attention to the dynamics behaviors of a stochastic echinococcosis infection model with environmental noise. The existence and uniqueness of the stochastic model is showed in this paper. We obtain the sufficient condition of the ergodic stationary distribution. What is more, the condition of extinction of the stochastic model is also given.

An HIV latent infection model with cell-to-cell transmission and stochastic perturbation

During the HIV infection process in vitro cell cultures, recent experimental and mathematical investigations have revealed that there are two fundamentally distinct viral transmission modes: cell-free infection and cell-to-cell transmission. A stochastic HIV model with latent infection, cell-free infection and cell-to-cell transmission is formulated in this paper. The existence of the unique ergodic stationary distribution is established by constructing appropriate Lyapunov function. Moreover, virus can be eradicated under certain sufficient conditions. Numerical simulations are carried out to investigate three aspects: (i) the effect of stochastic perturbation; (ii) the influence of latency; (iii) the relative contributions of cell-to-cell and cell-free infections.

Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine‐induced immunity under nonlinear regime switching

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine‐induced immunity and nonlinear telegraph perturbations. First, we derive a critical value related to the basic reproduction number of the deterministic model. The key aim of this paper is to generalize the θ‐stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ‐stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under . Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.