Ergodic Stationary Distribution Research Articles

Virus infection model under nonlinear perturbation: Ergodic stationary distribution and extinction

In order to study the effect of nonlinear perturbation on virus infection of target cells, in this paper, we propose a stochastic virus infection model with multitarget cells and exposed state. Firstly, by constructing novel stochastic Lyapunov functions, we theoretically prove that the solution of the stochastic model is positive and global. Secondly, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system and the exact expression of probability density function around a quasi-endemic equilibrium if Rs>1, and we establish a sufficient condition Re<1 for the extinction of infected cells and virus. Finally, we present examples and numerical simulations to verify our theoretical results.

Stationary distribution and extinction of a stochastic two-stage model of social insects with egg cannibalism

Considering the cannibalism behavior of many groups of animals in nature, a two-stage model of social insects with egg cannibalism and nonlinear perturbations is constructed and investigated in this paper. First, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution ψ(⋅) of the stochastic model. In a biological implication, the existence of the distribution ψ(⋅) implies the long-term persistence of social insects. Moreover, sufficient conditions for exponential extinction of social insects are obtained. Finally, our theoretical results are further discussed by several numerical simulations.

An epidemic model with Beddington–DeAngelis functional response and environmental fluctuations

An epidemic model with the saturated incidence rate and the environmental fluctuations is investigated in this paper. We study the extinction and the persistence in the mean, and stationary distribution of the solution as well. By contradiction, we firstly show that stochastic epidemic model admits a unique global positive solution for any given positive initial value. Further, when R˜0>1 is valid, we derive the persistence in the mean, and also prove the existence of an ergodic stationary distribution by constructing moderate functions. By comparison theorem of stochastic differential equations and properties of inequalities, the extinction of the solution is finally derived when R0<1 and ν<0 hold, where ν indicates the exponential rate for decline. Meanwhile, the distribution for the density of the susceptible is estimated. As a consequence, numerical simulations and illustrative examples are separately carried out to support the main results of this paper.

Analysis of a stochastic predator–prey model with weak Allee effect and Holling-(n+1) functional response

Considering the profound ecological implication of Allee effect, and the effects after incorporating it into models of population dynamics, a stochastic predator–prey model with Holling-(n+1) functional response and weak Allee effect is mainly investigated in this paper. Firstly, we verify the existence of unique global positive solution to the model, and then the boundedness of θth moment of the solution is studied. Secondly, we investigate the corresponding one-dimensional model, and obtain the explicit density function of the solution to the model. Then, based on the results of the one-dimensional system and a series of appropriate Lyapunov functions, we adopt a new technique to obtain a sufficient and almost necessary condition for the existence of the unique ergodic stationary distribution and extinction. Next, we further study the dynamical behavior of the model with Markovian switching and give some main conclusions. Finally, several numerical simulations are carried out to illustrate the theoretical results.

Analysis of a new stochastic Gompertz diffusion model for untreated human glioblastomas

In this paper, we analyze a new Ito stochastic differential equation model for untreated human glioblastomas. The model was the best fit of the average growth and variance of 94 pairs of a data set. We show the existence and uniqueness of solutions in the positive spatial domain. When the model is restricted in the finite domain [Formula: see text], we show that the boundary point 0 is unattainable while the point [Formula: see text] is reflecting attainable. We prove there is a unique ergodic stationary distribution for any non-zero noise intensity, and obtain the explicit probability density function for the stationary distribution. By using Brownian bridge, we give a representation of the probability density function of the first passage time when the diffusion process defined by a solution passes the point [Formula: see text] firstly. We carry out numerical studies to illustrate our analysis. Our mathematical and numerical analysis confirms the soundness of our randomization of the deterministic model in that the stochastic model will set down to the deterministic model when the noise intensity approaches zero. We also give physical interpretation of our stochastic model and analysis.

Effects of environmental stochasticity on the foraging dynamics of ant colonies driven by physical interactions

Collective foraging is critical for the ecological and evolutionary success of social insect colonies. To evaluate the impact of environmental stochasticity on the collective foraging dynamics of social ants, this paper develops a collective foraging model based on stochastic differential equations, in which communication among workers mainly relies on physical interactions. By analyzing the stochastic dynamics of the constructed model, we obtain relaxation conditions for the existence of an ergodic stationary distribution. Theoretical results suggest that environmental stochasticity leads to a decrease in the effective recruitment rate and average foraging period of foragers, ultimately negatively affecting the average recruit recruitment ability of single returning forager.

Dynamic analysis of a stochastic eco‐epidemiological model with disease in predators

AbstractIt is widely accepted that epidemics in biological populations and stochasticity in the environments are two important components significantly influencing the real ecosystems. To more accurately reveal developmental changes of the population dynamics and explore effective control methods for infectious disease, in this paper, we first construct a stochastic eco‐epidemiological model with disease in predators, and mathematically analyze the fundamental properties of the model. Then, we establish sufficient criterion for the existence of a unique ergodic stationary distribution of the positive solution. Also, we derive sufficient conditions, respectively, for three different extinction scenarios of populations: (i) The predators go extinct, only the prey persists; (ii) both the predators and prey go extinct; and (iii) the disease is eradicated from the system, both the predators and prey persist. Finally, we perform some numerical simulations under different interferences of environment to illustrate the main theoretical results. Especially, we observe that once the disease is eradicated from the system, then the predators and prey will coexist in an oscillatory manner. Our research indicates that the environmental noises have significant impacts on the determination of the persistence and extinction of the disease and populations.

The threshold dynamics of a stochastic two-patch brucellosis model

Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations.

Threshold dynamics and density function of a stochastic epidemic model with media coverage and mean-reverting Ornstein–Uhlenbeck process

Infectious disease transmission, mainly affected by media coverage and stochastic perturbations, has imposed great social financial burden on the community in the past few decades and even threatened public health. However, there are few studies devoted to the theoretical dynamics of epidemic models with media coverage and biologically reasonable stochastic effect yet. In this sense, this paper mainly formulates and studies a stochastic epidemic model with media coverage and two mean-reverting Ornstein–Uhlenbeck processes. We first illustrate the biological implication and mathematically reasonable assumption of introducing the Ornstein–Uhlenbeck process as stochastic effect. It is theoretically proved that the solution to the stochastic model is unique and global, as well as the existence of an ergodic stationary distribution. After that, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, it is derived that the above global solution around the endemic equilibrium follows a unique probability density function. For completeness, the sufficient criteria for extinction exponentially of the model are established. Finally, several numerical simulations are provided to verify our theoretical results. Besides, the impact of stochastic noises and media coverage on epidemic transmission is studied by comparison with the previous results of a deterministic model.

Ergodic stationary distribution and extinction of a staged progression HIV/AIDS infection model with nonlinear stochastic perturbations

Ergodic stationary distribution and extinction of a staged progression HIV/AIDS infection model with nonlinear stochastic perturbations

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps.

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and Lacute{e}vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold lambda by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.

Stationary distribution of a stage-structure predator–prey model with prey’s counter-attack and higher-order perturbations

In this paper, we investigate a stochastic predator–prey model with counter-attacking behavior which means the prey can attack juvenile predator. We establish the sufficient conditions for the existence of a unique ergodic stationary distribution by constructing a series of Lyapunov functions. From the biological point of view, the ergodic results enable us to predict whether the system is generally stable in the long term.

Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives, when emergencies occur, rumors will spread widely on the internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens’ behavior and attitude, which make the transmission rates of these information among social network groups be not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation. Firstly, increasing noise intensity can effectively reduce rumor propagation when . That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens’ opinions, thus reducing the degree of rumor propagation. Secondly, speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen (i.e., increase the value of β and decrease the value of α and γ), which can effectively curb the rumor propagation.

Dynamics study of a stochastic SIQR epidemic model with vaccination and saturated incidence

The paper considers a stochastic SIQR epidemic model with vaccination and saturated incidence. First, the existence and boundedness of the global positive solution are given. Second, the threshold for disease extinction is established. Then, through building proper Lyapunov function, the existence of an ergodic stationary distribution is obtained. At last, the main results are confirmed by numerical simulations, which manifests that the large white noise gets to repress the occurrence of disease.

Dynamic Behavior of a Stochastic Tungiasis Model for Public Health Education

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using It formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.

Stationary distribution and optimal control of a stochastic population model in a polluted environment.

This paper is concerned with a stochastic population model in a polluted environment. First, within the framework of Lyapunov method, the existence and uniqueness of a global positive solution of the model are proposed, and the sufficient conditions are established for existence of an ergodic stationary distribution of the positive solution. Second, the control strategy is introduced into the stochastic population model in a polluted environment. By using Pontryagin's maximum principle, the first-order necessary conditions are derived for the existence of optimal control. Finally, some numerical simulations are presented to illustrate the analytical results.

Dynamics of a stochastic HIV/AIDS model with treatment under regime switching

&lt;p style='text-indent:20px;'&gt;This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.&lt;/p&gt;

Dynamical analysis of a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps

&lt;abstract&gt;&lt;p&gt;In this paper, a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps is studied. Firstly, it is proved that the model has a unique global solution. Secondly, sufficient conditions for weak persistence in the mean and extinction of prey and predator populations are established. Finally, sufficient conditions for the existence and uniqueness of ergodic stationary distribution are established. Moreover, several numerical simulations are presented to illustrate the main results.&lt;/p&gt;&lt;/abstract&gt;

LONG-TIME BEHAVIOR OF STOCHASTIC STAGED PROGRESSION EPIDEMIC MODEL WITH HYBRID SWITCHING FOR THE TRANSMISSION OF HIV

<abstract> In this paper, one stochastic hybrid switching SP (staged progression) model for the transmission of HIV is proposed and investigated. The system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent. Furthermore, sufficient conditions for extinction of disease are established. At last, some examples and simulations are provided to illustrate our results. </abstract>

A coral reef benthic system with grazing intensity and immigrated macroalgae in deterministic and stochastic environments

&lt;abstract&gt;&lt;p&gt;As the most diverse, productive but vulnerable marine habitats in the world, coral reefs are easily affected by the ubiquitous environmental fluctuations, which could change the population dynamics and induce phase shifts of the ecosystem. In this paper, we consider a coral reef benthic system, where macroalgae, corals and algal turfs compete for the available space on a given region of the seabed with grazing intensity and immigrated macroalgae in both deterministic and stochastic environments. For the deterministic system, we analyze the existence and stability of equilibria, as well as the existence of bifurcations. For the stochastic system, sufficient conditions for the existence of the unique ergodic stationary distribution as well as the extinction of corals are obtained, by choosing suitable Lyapunov functions. Moreover, for the scenario that the system exhibits bistability between a macroalgal-coral coexistence equilibrium and a coral-free equilibrium in the absence of environmental fluctuation, we further investigate the irreversible noise-induced transition from macroalgal-coral coexistence to coral extirpation, and numerically estimate the critical values of noise intensity for the occurrence of such transition with the aids of the technique of stochastic sensitivity functions.&lt;/p&gt;&lt;/abstract&gt;