Ergodic Distribution Research Articles

Dynamical behaviors of a stochastic HTLV-I infection model with general infection form and Ornstein–Uhlenbeck process

In this study, considering the Ornstein–Uhlenbeck process to perturb the infection rate, we develop a HTLV-I infection model with general infection form. By constructing several suitable Lyapunov functions and a compact set, and then using the strong law of numbers and Fatou’s lemma, we obtain sufficient conditions for the existence and uniqueness of the ergodic stationary distribution η(⋅) for the stochastic model. This implies long-term persistence of HTLV-I infection in a biological sense. Moreover, by using Itô’s integral stochastic model is transformed into the corresponding linearized system. Then solving the Fokker–Planck equation, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. In addition, sufficient conditions for the extinction of HTLV-I infection are established. Finally, considering different incidence rate functions, we employ numerical simulations to support our results.

Dynamics of a stochastic delayed chemostat model with nutrient storage and Lévy jumps

In this paper, a stochastic delayed chemostat model with nutrient storage and Lévy jumps is proposed. Firstly, the existence and uniqueness of the positive global solution of the model are discussed. Then, the threshold λ̄ and optimal control conditions for the persistence in the mean and extinction of the microorganism x are obtained. Besides, the ergodic stationary distribution of the SDDE model under a low-level intensity of stochastic noise is deduced. Finally, some numerical examples are given to support the theoretical analysis results. The simulation results show that stochastic noise and time delay play a vital role in controlling the persistence and extinction of microorganisms, respectively. On the one hand, high-intensity noise can inhibit the growth of microorganisms. On the other hand, if τ>τ∗, the corresponding deterministic model will become unstable and produce a Hopf bifurcation. Moreover, the solutions of the SDDE model will oscillate around the non-constant T−periodic solution of the corresponding deterministic model.

Ergodic stationary distribution of two stochastic tuberculosis models with imperfect vaccination and early and late latency

This contribution probes into ergodic stationary distribution for two stochastic SVELIT (susceptible-vaccinated-early latent-late latent-infective-treated) tuberculosis (TB) models to observe the impact of white noises and color noises on TB control in random environments. We first investigate the existence and uniqueness of ergodic stationary distribution (EUESD) for the autonomous SVELIT model subject to white noises via the proper Lyapunov functions, and sufficient conditions on the extinction of disease are acquired. Next, sufficient conditions for the EUESD and the extinction of disease for the SVELIT model with Markov switching are also established. Eventually, some numerical examples validate the theoretical findings. What’s more, it has been observed that higher amplitude noises may lead to the eradication of TB, which is conducive to TB control.

A stochastic SICA model for HIV/AIDS transmission

In this paper, we present a stochastic HIV/AIDS model with standard incidence rate to investigate the impact of random fluctuations on the HIV viral dynamics. Firstly, the sufficient conditions for the stochastic extinction of the disease are obtained. Secondly, after defining the threshold parameter Rs, we prove that the model has a unique ergodic stationary distribution when Rs>1, which reveals that the disease will persist almost surely. Subsequently, we give the exact expression of the probability density function around a quasi-equilibrium point by applying four-dimensional Fokker–Planck equation. Finally, through the numerical simulations of several examples, we not only verify the correctness of the theoretical results, but also explore how the disease goes for the case of not giving clear results theoretically. Epidemiologically, we find that increasing the noise intensities may be conducive to control the spread of the HIV.

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.

Stochastic Analysis of a Hantavirus Infection Model

In this paper, a stochastic Hantavirus infection model is constructed. The existence, uniqueness, and boundedness of the positive solution of the stochastic Hantavirus infection model are derived. The conditions for the extinction of the Hantavirus infection from the stochastic system are obtained. Furthermore, the criteria for the presence of a unique ergodic stationary distribution for the Hantavirus infection model are established using a suitable Lyapunov function. Finally, the importance of environmental noise in the Hantavirus infection model is illustrated using the Milstein method.

Asymptotic behavior and extinction of a stochastic predator–prey model with Holling type II functional response and disease in the prey

In this paper, we formulate and investigate a stochastic one‐prey and two‐predator model with Holling II functional response and disease in the prey, in which the predators only feed on infected prey. The existence and uniqueness of global positive solution is proved by using conventional methods. The corresponding deterministic model has a disease‐free equilibrium point if the basic reproduction number , and it has three boundary equilibrium points and one positive equilibrium point if . For the stochastic model, we investigate the asymptotic behavior around all of the five equilibrium points and prove that there is a unique ergodic stationary distribution under certain conditions. Moreover, we obtain the condition on which the population of the infected prey and the two predators will die out in the time mean sense. Finally, numerical simulations are conducted to illustrate our analysis results.

Dynamical behavior and density function analysis of a stochastic HIV/AIDS model with general incidence rate

In this paper, we mainly present the dynamic properties of a stochastic SICA model with general incidence rate. Through rigorous analysis and reasoning of the stochastic model, we obtain that the solution of the model is global, positive, and unique. By constructing the threshold value , which includes the influence of white noise, we obtain a sufficient condition for the ergodicity of this model. Furthermore, we show that the model has a unique ergodic stationary distribution while by adopting a novel method. The extinction of the system is also established. Besides, the expression of probability density function of the stochastic model with bilinear incidence rate around the unique stable positive equilibrium of the deterministic model is derived. Finally, numerical simulations are presented to illustrate the theoretical results.

Dynamic behavior of stochastic prostate cancer system with comprehensive therapy under regime switching

The random volatility, including continuous perturbation and regime switching, inevitably occurs in tumor microenvironment. In this paper, by introducing Gaussian white noise and regime switching, the combination of the two environmental disturbances is considered in the prostate cancer (PCa) model with comprehensive treatment. The Itô’s formula, Lyapunov function method and ergodicity of Markov chain are employed to research the extinction-persistence behavior of PCa cells and the existence of a unique ergodic stationary distribution. Our results show high-intensity Gaussian white noise type of continuous perturbation can suppress PCa development and that when the regime switching is on, patients can obtain a good quality of life and improve the survival rate by improving immunity and then maintaining in such high-intensity noise state for a long time.

Optimal control theory of a novel stochastic human norovirus model and vaccine development

Norovirus is a contagious disease that causes diarrhea and vomiting and has a significant impact on humanity in terms of medical costs, deaths and mortalities. In this research work, the stochastic modeling techniques are utilized and a novel human norovirus model is investigated keeping in view the contamination of food and water with vaccination effects. It is proved that the model has a unique global solution and by using Lyapunov function theory, sufficient conditions are derived under which the proposed model has a unique ergodic stationary distribution for [Formula: see text]. It is observed that the disease will extinct out of the population whenever [Formula: see text]. To demonstrate the analytical results, two independent examples are depicted graphically by using standard numerical algorithm. To reduce the spread of the virus, we employed certain control measures and an optimal control problem is presented. To further validate the obtained analytical results, additional graphical solutions were generated. This study might provide a robust theoretical framework for a global understanding of chronic communicable illnesses. Our method also aims to provide a method for producing Lyapunov functions, which can be used to explore the stationary distribution of disease models having nonlinear stochastic perturbations.

On the Capacity-Achieving Input of Channels With Phase Quantization

Several information-theoretic studies on channels with output quantization have identified the capacity-achieving input distributions for different fading channels with 1-bit in-phase and quadrature (I/Q) output quantization. However, an exact characterization of the capacity-achieving input distribution for channels with multi-bit phase quantization has not been provided. In this paper, we consider four different channel models with multi-bit phase quantization at the output and identify the optimal input distribution for each channel model. We first consider a complex Gaussian channel with $b$-bit phase-quantized output and prove that the capacity-achieving distribution is a rotated $2^b$-phase shift keying (PSK). The analysis is then extended to multiple fading scenarios. We show that the optimality of rotated $2^b$-PSK continues to hold under noncoherent fast fading Rician channels with $b$-bit phase quantization when line-of-sight (LoS) is present. When channel state information (CSI) is available at the receiver, we identify $\frac{2\pi}{2^b}$-symmetry and constant amplitude as the necessary and sufficient conditions for the ergodic capacity-achieving input distribution; which a $2^b$-PSK satisfies. Finally, an optimum power control scheme is presented which achieves ergodic capacity when CSI is also available at the transmitter.

Global asymptotic stability, extinction and ergodic stationary distribution in a stochastic model for dual variants of SARS-CoV-2

Several mathematical models have been developed to investigate the dynamics SARS-CoV-2 and its different variants. Most of the multi-strain SARS-CoV-2 models do not capture an important and more realistic feature of such models known as randomness. As the dynamical behavior of most epidemics, especially SARS-CoV-2, is unarguably influenced by several random factors, it is appropriate to consider a stochastic vaccination co-infection model for two strains of SARS-CoV-2. In this work, a new stochastic model for two variants of SARS-CoV-2 is presented. The conditions of existence and the uniqueness of a unique global solution of the stochastic model are derived. Constructing an appropriate Lyapunov function, the conditions for the stochastic system to fluctuate around endemic equilibrium of the deterministic system are derived. Stationary distribution and ergodicity for the new co-infection model are also studied. Numerical simulations are carried out to validate theoretical results. It is observed that when the white noise intensities are larger than certain thresholds and the associated stochastic reproduction numbers are less than unity, both strains die out and go into extinction with unit probability. More-over, it is observed that, for weak white noise intensities, the solution of the stochastic system fluctuates around the endemic equilibrium (EE) of the deterministic model. Frequency distributions are also studied to show random fluctuations due to stochastic white noise intensities. The results presented herein also reveal the impact of vaccination in reducing the co-circulation of SARS-CoV-2 variants within a given population.

Plankton growth dynamic driven by plankton body size in deterministic and stochastic environments

In this paper, we establish a new phytoplankton–zooplankton model by considering the effects of plankton body size and stochastic environmental fluctuations. Mathematical theory work mainly gives the existence of boundary and positive equilibria and shows their local as well as global stability in the deterministic model. Additionally, we explore the dynamics of V‐geometric ergodicity, stochastic ultimate boundedness, stochastic permanence, persistence in the mean, stochastic extinction, and the existence of a unique ergodic stationary distribution in the corresponding stochastic version. Numerical simulation work mainly reveals that plankton body size can generate great influences on the interactions between phytoplankton and zooplankton, which in turn proves the effectiveness of mathematical theory analysis. It is worth emphasizing that for the small value of phytoplankton cell size, the increase of zooplankton body size can not change the phytoplankton density or zooplankton density; for the middle value of phytoplankton cell size, the increase of zooplankton body size can decrease zooplankton density or phytoplankton density; for the large value of phytoplankton cell size, the increase of zooplankton body size can increase zooplankton density but decrease phytoplankton density. Besides, it should be noted that the increase of zooplankton body size cannot affect the effect of random environmental disturbance, while the increase of phytoplankton cell size can weaken its effect. There results may enrich the dynamics of phytoplankton‐zooplankton models.

Dynamics Analysis of a Predator–Prey Model with Hunting Cooperative and Nonlinear Stochastic Disturbance

This paper proposes a stochastic predator–prey model with hunting cooperation and nonlinear stochastic disturbance, and focuses on the effects of nonlinear white noise and hunting cooperation on the populations. First, we present the thresholds R1 and R2 for extinction and persistence in mean of the predator. When R1 is less than 0, the predator population is extinct; when R2 is greater than 0, the predator population is persistent in mean. Moreover, by establishing suitable Lyapunov functions, we investigate the threshold R0 for the existence of a unique ergodic stationary distribution. At last, we carry out the numerical simulations. The results show that white noise is harmful to the populations and hunting cooperation is beneficial to the predator population.

Regional income convergence in Colombia: population, space, and long-run dynamics

We examine the trajectory of regional income dynamics in Colombia. Using data on all 33 Colombian departments from 2000 to 2016, we employ extensions of (spatial) Markov chains, space-time mobility measures, along with a fully weighted version of the distribution analysis approach. By considering these extensions, our analysis enables us to answer questions such as whether the role of spatial context influences the distributional dynamics of Colombian departments, or the magnitude of the moderating effect of department’s population. The inclusion of additional measures such as the asymptotic half-life of convergence provides additional results, informing on how long it would take to reach the hypothetical long-run distribution of per capita income. Results, which are reported for both pre- and post-2008 trends, complement previous literature on regional economic convergence in a relevant South American context, showing stronger convergence patterns when controlling for the population living in each department. The patterns do not particularly intensify when controlling for spatial spillovers, since the role of spatial context was already playing a relevant role from the beginning of the period analyzed. Therefore, although the ergodic distributions show a conditional-convergence pattern, addressing the problems of spatial exclusion fully, persistent polarization among geographical departments and populations, along with the relevant core-periphery gaps, still requires the design and implementation of specific policies.

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.

Probing a Stochastic Epidemic Hepatitis C Virus Model with a Chronically Infected Treated Population.

The hepatitis C virus is hitherto a tremendous threat to human beings, but many researchers have analyzed mathematical models for hepatitis C virus transmission dynamics only in the deterministic case. Stochasticity plays an immense role in pathology and epidemiology. Hence, the main theme of this article is to investigate a stochastic epidemic hepatitis C virus model with five states of epidemiological classification: susceptible, acutely infected, chronically infected, recovered or removed and chronically infected, and treated. The stochastic hepatitis C virus model in epidemiology is established based on the environmental influence on individuals, is manifested by stochastic perturbations, and is proportional to each state. We assert that the stochastic HCV model has a unique global positive solution and attains sufficient conditions for the extinction of the hepatotropic RNA virus. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the solutions to the stochastic HCV model. Moreover, this article confirms that using numerical simulations, the six parameters of the stochastic HCV model can have a high impact over the disease transmission dynamics, specifically the disease transmission rate, the rate of chronically infected population, the rate of progression to chronic infection, the treatment failure rate of chronically infected population, the recovery rate from chronic infection and the treatment rate of the chronically infected population. Eventually, numerical simulations validate the effectiveness of our theoretical conclusions.

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.

Analysis of an SQEIAR stochastic epidemic model with media coverage and asymptomatic infection

In this paper, the dynamical behavior of a stochastic SQEIAR epidemic model is investigated. First of all, we establish sufficient conditions for extinction of the diseases. Then the sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to the model are obtained by constructing a suitable stochastic Lyapunov function. The existence of stationary distribution implies stochastic weak stability. Furthermore, the optimal control problem is considered to provide a theoretical basis for the prevention and control of the disease. Finally, the theoretical results are verified by numerical simulations.