Unique Ergodic Stationary Distribution Research Articles

Extinction Dynamics and Equilibrium Patterns in Stochastic Epidemic Model for Norovirus: Role of Temporal Immunity and Generalized Incidence Rates

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic SEIQR epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when R^s>1. Extinction occurs when R0E<1. Computer simulations confirm that noise level significantly influences epidemic spread.

Dynamics of a ratio-dependent three-species food-chain system in a random environment

We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution.

A stochastic predator–prey eco-epidemiological model with hunting cooperation

Cooperative hunting can increase the predator’s hunting ability. In this work, we study a stochastic eco-epidemiological predator–prey model incorporating hunting cooperation. By constructing appropriate auxiliary functions, we establish a sufficient criterion for the existence of a unique ergodic stationary distribution. The findings imply that cooperative hunting has a significant effect on the density of infected predators when the epidemic is protracted.

Stochastic dynamic effects of media coverage and incubation on a distributed delayed epidemic system with Lévy jumps

A stochastic SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic system with Lévy jumps and distributed time delay is established, which is utilized to investigate hybrid dynamic effects of media coverage and incubation on infectious disease transmission. Stochastically ultimate boundedness of the positive solution is discussed. Existence of a unique global positive solution is studied. By constructing appropriate stochastic Lyapunov functions, existence of a unique ergodic stationary distribution is investigated. Sufficient conditions are derived to discuss exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, sufficient conditions for persistence in mean and extinction of infectious disease are investigated. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model

In this paper, a stochastic HIV/AIDS epidemic model is presented to study the synthetic effect of ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments among MSM ( men who have sex with men). Firstly, we give the global stability of disease-free equilibrium and the endemic equilibrium in terms of basic reproduction number R0 for deterministic model. And then the existence of global positive solutions and the existence of unique ergodic stationary distribution under R0S>1 for stochastic model are given. Further, the long-time stochastic dynamic of the model is investigated, including the criteria on the extinction and persistence in mean for the stochastic model. Finally, we give some numerical simulations to illustrate our theoretical results, and the sensitive analysis shows that ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments can effectively control the spread of AIDS among MSM population.

The impact of temperature variation on the algae–zooplankton dynamics with size-selective disturbance

The phenomenon of algal blooms caused by global warming frequently occurs in global water bodies, however, the mechanism by which temperature affects algal growth is still unclear. In this paper, we investigate the dynamics of a deterministic algae–zooplankton model with water temperature, as well as its corresponding stochastic version. Mathematical theoretical work mainly analyzes the stability, persistence and Hopf bifurcation of the deterministic model, and performs a survival analysis on the stochastic model and shows the existence of a unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulations. Numerical analysis indicates that the variation of temperature can generate complex impacts on the fundamental mechanisms underlying the growth of algae, whether in deterministic or stochastic environment. One of the most interesting results indicates that the decrease of temperature can destabilize the algae–zooplankton interactions, but increase their permanence; the increase of temperature can stabilize the algae–zooplankton interactions, but reduce their permanence.

Dynamics and density function for a stochastic anthrax epidemic model

<abstract><p>In response to the pressing need to understand anthrax biology, this paper focused on the dynamical behavior of the anthrax model under environmental influence. We defined the threshold parameter $ R^s $, when $ R^s > 1 $; the disease was almost certainly present and the model exists a unique ergodic stationary distribution. Subsequently, statistical features were employed to analyze the dynamic behavior of the disease. The exact representation of the probability density function in the vicinity of the quasi-equilibrium point was determined by the Fokker-Planck equation. Finally, some numerical simulations validated our theoretical results.</p></abstract>

Stochastic analysis of survival and sensitivity in a competition model influenced by toxins under a fluctuating environment

<abstract><p>This paper proposed a stochastic toxin-dependent competition model to investigate the impact of environmental noise on species interaction dynamics. First, a survival analysis was conducted to establish the sufficient conditions for population extinction and persistence. Second, we proved the existence of a unique ergodic stationary distribution. Finally, the spatial arrangement of random states near the deterministic attractor was investigated using the stochastic sensitivity functions technique. This analytical approach facilitates constructing confidence ellipses and estimating critical noise intensity corresponding to the onset of transition. Both theoretical and numerical findings demonstrated that significant levels of noise experienced by one species lead to its extinction while promoting persistence in its competitor; conversely, negligible levels of noise did not alter the original competition outcomes in the deterministic model. However, when both species encounter moderate levels of noise, various modifications can occur in competition outcomes. These findings have significant implications for preserving ecosystem diversity.</p></abstract>

Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching

<p>This paper is concerned with a stochastic Mycobacterium tuberculosis model, which is perturbed by both white noise and colored noise. First, we prove that the stochastic model has a unique global positive solution. Second, we derive an important condition $ R_0^* $ depending on environmental noise for this stochastic model. We construct an appropriate Lyapunov function, and show that the model possesses a unique ergodic stationary distribution when $ R_0^* < 0 $, in other words, it indicates the long-term persistence of the disease. Finally, we investigate the related conditions of extinction.</p>

New Results on the Asymptotic Behaviour of a Stochastic SEI Model of Lymphatic Filariasis

The goal of this article is to examine a new stochastic epidemic model for lymphatic filariasis, susceptible‐exposed‐infected (SEI). Like other diseases, the spread of lymphatic filariasis is subject to a degree of randomness due to fluctuations in the natural environment. This provides us with the opportunity to formulate a mathematical model of lymphatic filariasis that comprehensively considers external stochastic influences. By using carefully designed stochastic Lyapunov functions, we establish sufficient conditions for the eradication of lymphatic filariasis infections. Moreover, under specific conditions regarding the intensity of white noise, we also demonstrate the existence of a unique ergodic stationary distribution and the persistence of the mean for the model. Numerical simulations are conducted to substantiate the validity of our findings.

DETERMINISTIC AND STOCHASTIC ANALYSIS OF A SIZE-DEPENDENT PHYTOPLANKTON–ZOOPLANKTON MODEL WITH ADDITIVE ALLEE EFFECT

In this paper, a deterministic size-dependent phytoplankton–zooplankton (PZ) model with additive Allee effect as well as its stochastic differential equation version is formulated to explore the growth dynamic of phytoplankton. A stability and Hopf bifurcation analysis is performed towards deterministic PZ model, and stochastic dynamic analysis is carried out on the stochastic PZ model, including the stochastic extinction, persistence in mean and the unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulation analysis. Numerical simulations reveal that the synergistic effect of plankton body size and Allee effect can have a complex impact on the growth of phytoplankton in the deterministic and stochastic environments. One of the most interesting findings suggests that the reduced phytoplankton cell size can trigger bistable phenomenon, while the increased phytoplankton cell size or the weakened Allee effect has the ability to destroy such bistable phenomenon.

Deterministic-stochastic analysis of fractional differential equations malnutrition model with random perturbations and crossover effects

To boost the handful of nutrient-dense individuals in the societal structure, adequate health care documentation and comprehension are permitted. This will strengthen and optimize the well-being of the community, particularly the girls and women of the community that are welcoming the new generation. In this article, we extensively explored a deterministic-stochastic malnutrition model involving nonlinear perturbation via piecewise fractional operators techniques. This novel concept leads us to analyze and predict the process from the beginning to the end of the well-being growth, as it offers the possibility to observe many behaviors from cross over to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative. The threshold parameter is developed and the role of malnutrition in society is examined. Through a rigorous analysis, we first demonstrated that the stochastic model’s solution is positive and global. Then, using appropriate stochastic Lyapunov candidates, we examined whether the stochastic system acknowledges a unique ergodic stationary distribution. The objective of this investigation is to design a nutritional deficiency in pregnant women using a piecewise fractional differential equation scheme. We examined multiple options and outlined numerical methods of coping with problems. To exemplify the effectiveness of the suggested concept, graphical conclusions, including chaotic and random perturbation patterns, are supplied. Consequently, fractional calculus’ innovative aspects provide more powerful and flexible layouts, enabling us to more effectively adapt to the system dynamics tendencies of real-world representations. This has opened new doors to readers in different disciplines and enabled them to capture different behaviors at different time intervals.

Dynamics of a stochastic prey–predator system with prey refuge, predation fear and its carry-over effects

This paper proposes and studies the dynamics of a Holling-type II predator–prey interaction system that incorporates the following three components: (1) a prey refuge; (2) predation fear and its carry-over effects; and (3) environmental noise in both prey and predator populations. The impacts of those three components are studied through both rigorous analysis and numerical simulations. Analytical results show that the introduction of prey refuge, predation fear, and its carry-over effects can generate Hopf bifurcation. It is found that increasing prey refuge and predation fear effect in a reasonable region can stabilize the system, while excessive refuge strength would lead to the extinction of predators. The theoretical results of the corresponding system with environmental noise include (1) sufficient conditions for the existence of a unique ergodic stationary distribution of the SDE system by constructing appropriate stochastic Lyapunov functions; (2) the explicit probability density function of the distribution by solving the Fokker–Planck equation; and (3) the extinction conditions of prey and/or predator species at an exponential rate in the long run. Our work shows that the proposed model, incorporating prey refuge, predation fear, carry-over effect, and environmental noise, exhibits rich and complex dynamic behaviors. Moreover, our results indicate that small environmental noise can save the prey and predator from extinction, while large environmental noise can drive the species to extinction. These interesting findings provide more perspectives on the protection and control of species in complex communities.

Dynamical behavior of a stochastic regime-switching epidemic model with logistic growth and saturated incidence rate

This work aims to study the effect of environmental noises like white noise and telegraph noise modeled by Markov switching on the dynamical properties of a modified logistic-growth-type epidemic model. Applying the Khasminskii method via a suitable construction of the Lyapunov function, we 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 epidemic model which is the stability of stochastic system in the weak sense. It is noteworthy that regime switching can induce state transitions of diseases between extinction and persistence. Further, the pulse control scheme is developed to drive the disease to extinction when χ<0. Several numerical simulations are provided to validate our analytical findings.

Dynamical behavior and density function of a stochastic model of HPV infection and cervical cancer with a case study for Xinjiang, China

In this article, combining the transmission features of HPV infection and secondary cervical cancer in Xinjiang, China, a stochastic dynamical model for the HPV infection and secondary cervical cancer with environmental white noise is proposed. Firstly, the stochastic extinction of disease is investigated. A sufficient criterion for the asymptotic behavior of any positive solution of stochastic model revolving around the disease-free equilibrium of corresponding deterministic model is established. Secondly, a threshold criterion for the existence of unique ergodic stationary distribution is obtained by means of the auxiliary function. Furthermore, a new technique of partitioned matrix for the calculation of probability density function is proposed, the expression of a log-normal density function around the quasi-endemic equilibrium of stochastic model is calculated. Lastly, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the cervical cancer data in Xinjiang province, China. The basic reproduction number is estimated as 1.3496 (95% CnI: (1.3458, 1.3716)). Then, to determine the key parameters of the model, the sensitivity analysis is explored. Some possible interventions and control measures are provided to reduce the HPV infection spread and cervical cancer in Xinjiang of China.

Analysis and simulation of a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures: A case study of the outbreak in Urumqi, China in August 2022.

In this paper, a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures is proposed. Firstly, the existence and uniqueness of the global positive solution is obtained. Secondly, threshold criteria for the stochastic extinction and persistence in the mean with probability one are established. Moreover, a sufficient condition for the existence of unique ergodic stationary distribution for any positive solution is also established. Finally, numerical simulations are carried out in combination with real COVID-19 data from Urumqi, China and the theoretical results are verified.

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.

DYNAMICAL BEHAVIOR OF A STOCHASTIC FOOD CHAIN CHEMOSTAT MODEL WITH VARIABLE YIELDS

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.

Dynamical behaviors of a stochastic HIV/AIDS epidemic model with treatment

In this paper, by using Itô's formula, we study the dynamic behaviors of a stochastic human immune deficiency virus (HIV) model, which describes Acquired Immune Deficiency Syndrome (AIDS) with the effect of treatment. First, we discuss the global positivity of the solution to such a model with positive initial value. Then, we consider the limiting behavior of the solution. Moreover, we obtain the sufficient condition to guarantee the extinction of AIDS. Meanwhile, by constructing an appropriate Lyapunov function, we prove that the solution has a unique ergodic stationary distribution under a certain assumptions. Finally, we illustrate the validity of our results by numerical simulations.

Stationary distribution, density function and extinction of stochastic vegetation-water systems

In this paper, two stochastic vegetation-water dynamic systems are proposed and studied. First, for the deterministic system, the possible equilibria and the related local stability are analyzed. Then for the stochastic system perturbed by Gaussian white noise, we establish sufficient conditions for the existence and uniqueness of ergodic stationary distribution, which reflects the long-term persistence of vegetation. By defining a stochastic quasi-positive equilibrium E¯∗ and solving the Kolmogorov–Fokker–Planck equation, an approximate expression of probability density function of the stationary distribution around E¯∗ is derived. Besides, we obtain some sufficient criteria for vegetation extinction. In terms of the stochastic system perturbed by both white and colored noises, the related extinction law and stationary distribution are investigated. Finally, several numerical examples are performed to substantiate our theoretical results and analyze the mean first passage time (MFPT).