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In this paper, the dynamical behaviors of a stochastic eco-epidemic predator-prey model with Lévy jump and Allee effect for prey population are investigated. First, the existence and uniqueness of the global positive solution are built. Then, the long-term behaviors of the prey and predator populations are obtained. Furthermore, we demonstrate the stochastic ultimate boundedness of all species and the ergodic stationary distribution without Lévy jump. Finally, numerical examples are provided to support the theoretical analysis results. [ABSTRACT FROM AUTHOR]

As the prevalence of viral infection in body, human T‐cell leukemia virus type 1 (HTLV‐1) is receiving increasing attention. Research on the corresponding virus models is of great significance to tackle the challenges of understanding HTLV‐1 development and treatment. This paper focuses on the dynamic analysis for a stochastic model with nonlinear cytotoxic T lymphocyte (CTL) response, which is driven by Ornstein–Uhlenbeck (OU) process to model the progression of HTLV‐1 in vivo. Rich dynamic behaviors such as the extinction of infected CD4+ T cells (ITCs), stationary distribution (SD), probability density, and finite‐time stability (FTS) of the model are established to reveal the interaction of cell populations. The optimal therapeutic strategy based on the cost‐benefit viewpoint is further obtained. Finally, illustrative numerical simulations are represented to corroborate the effectiveness of treatment and the ambient perturbation's impact that strengthening the noise strength can lead to rapid virus clearance. [ABSTRACT FROM AUTHOR]

Vaccination is an important tool in disease control to suppress disease, and vaccine-influenced diseases no longer conform to the general pattern of transmission. In this paper, by assuming that the infection rate is affected by the Ornstein–Uhlenbeck process, we obtained a stochastic SIRV model. First, we prove the existence and uniqueness of the global positive solution. Sufficient conditions for the extinction and persistence of the disease are then obtained. Next, by creating an appropriate Lyapunov function, the existence of the stationary distribution for the model is proved. Further, the explicit expression for the probability density function of the model around the quasi-equilibrium point is obtained. Finally, the analytical outcomes are examined by numerical simulations. [ABSTRACT FROM AUTHOR]

A mathematical model has been proposed to consider two distinct forms of psychological pressure that arise due to the COVID-19 situation and their impact on individuals’ lives, including their mental well-being and happiness. For a more realistic situation, the effect of stochasticity should be taken into account. Hence, our paper mainly investigates the effect of stochastically environmental variability on the transmission dynamics of psychological stress. We obtain two thresholds ℛs1 and ℛs2. If ℛs1 > 1, the psychological stress will persist and there will be a unique stationary distribution; whereas if ℛs2 < 1, the extinction of psychological stress is obtained. Moreover, we display the mean first passage time from the initial value to the state of disappearance in order to examine the impact of environmental disturbances; meanwhile, we conduct numerical simulations to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper delves into the analysis of a stochastic epidemic model known as the susceptible–vaccinated–exposed–infectious–recovered (SVEIR) model, where transmission dynamics are governed by a nonlinear function. In the theoretical analysis section, by suitable stochastic Lyapunov functions, we establish that when the threshold value, denoted as R 0 s , falls below 1, the epidemic is destined for extinction. Conversely, if the reproduction number R 0 of the deterministic model surpasses 1, the model manifests an ergodic endemic stationary distribution. In the numerical simulations and data interpretation section, leveraging a graphical analysis with COVID-19 data, we illustrate that random fluctuations possess the capacity to quell disease outbreaks, underscoring the role of vaccines in curtailing the spread of diseases. This study not only contributes to the understanding of epidemic dynamics but also highlights the pivotal role of stochasticity and vaccination strategies in epidemic control and management. The inherent balance and patterns observed in epidemic spread and control strategies, reflect a symmetrical interplay between stochasticity, vaccination, and disease dynamics. [ABSTRACT FROM AUTHOR]

Considering stochastic perturbations and spatial diffusion in both virus‐to‐cell and cell‐to‐cell transmissions, a stochastic reaction–diffusion HIV model is developed. Firstly, the existence and uniqueness of a global positive solution and stationary distribution are demonstrated. Secondly, considering the effect of drug therapy on the disease, the control strategy is introduced into the stochastic HIV model. By employing Pontryagin's stochastic maximum principle, the sufficient and necessary conditions are obtained for near‐optimal control. Finally, numerical simulations are reported to support and supplement our theoretical results. [ABSTRACT FROM AUTHOR]

A stochastic three species predator–prey model with intermediate predator and anti-predator behavior is established and studied in this paper. By constructing suitable Lyapunov functions and combining knowledge of stochastic differential equations, the behavior of global positive solutions to the model are investigated. Firstly, we prove that there is unique global positive solution to the model, and establish the stochastic boundedness of the positive solution as well. Secondly, the sufficient condition for the existence of a unique ergodic stationary distribution is provided. Thirdly, the persistence and extinction of the populations are discussed. Finally, some numerical simulations demonstrate that the results obtained in this paper are true, moreover, the influence of white noise on the populations is revealed. [ABSTRACT FROM AUTHOR]

Multi-strain infectious diseases, which are usually prevented from spreading widely by vaccination, have two main transmission mechanisms: competitive exclusion and co-existence. In this paper, a stochastic multi-strain coinfection model with amplification and vaccination is developed. For the deterministic model, the basic reproduction number R 0 and fixed points are provided. For the stochastic model, we first prove the existence and uniqueness of the positive solution under any initial value. Then, a portion of those infected with the common strain will always become patients with the amplified strain, which increases the risk of death from the disease. Therefore, we verified that patients with common strains would become extinct if R 1 s < 1 . Furthermore, by constructing the Lyapunov function, we find that model (3) has a unique ergodic stationary distribution if R 0 S > 1 . Particularly, we get a concrete form of the probability density of the distribution κ (·) near equilibrium E ∗ , where E ∗ is the quasi-local equilibrium of the stochastic model. Finally, the results are verified by numerical simulation. The results show that vaccination can control disease outbreaks or even eliminate them. [ABSTRACT FROM AUTHOR]

This paper presents a stochastic vector-borne epidemic model with direct transmission and media coverage. It proves the existence and uniqueness of positive solutions through the construction of a suitable Lyapunov function. Immediately after that, we study the transmission mechanism of vector-borne diseases and give threshold conditions for disease extinction and persistence; in addition we show that the model has a stationary distribution that is determined by a threshold value, i.e., the existence of a stationary distribution is unique under specific conditions. Finally, a stochastic model that describes the dynamics of vector-borne diseases has been numerically simulated to illustrate our mathematical findings. [ABSTRACT FROM AUTHOR]

The stochastic fluctuation of information induces the dynamic behavior and disease control strategy to change in the HBV epidemic model, but how stochastic information dissemination affects them is vague. Here, we consider an HBV epidemic model with stochastic information intervention. Using the Lyapunov function, we demonstrate that the system has a unique global positive solution, in addition, we also derive the threshold dynamics to obtain sufficient conditions that can ensure the extinction and persistence, as well as stationarity of the system. Moreover, we qualify the impact of stochasticity of the information on the optimal control strategy of the HBV, indicating that while information intervention could effectively reduce the disease peak, the fluctuation will attenuate this effect, i.e., the stable information is better than the noisy one. Finally, various stochastic and optimal control simulations are performed to verify the theoretical results and verify that optimal control can accelerate the extinction of the disease. [ABSTRACT FROM AUTHOR]

This study conducts a quantitative research survey on the development of queueing networks over years. Development is a process of gradual change that takes place over many years, during which a theory slowly progress and attain a good state. Queueing theory has been through many developments which made its existence inevitable in every field. Queueing networks can be considered as a collection of nodes, where each node stands for a service facility. It has been proved to be a powerful and versatile tool for modelling facilities in manufacturing units and telecommunication networks. This paper presents the development in Queueing networks and its types over years. This paper's main objective is to give all the analysts and researchers the knowledge about the evolution that happened in Queueing networks over years. [ABSTRACT FROM AUTHOR]

Abstract Wireless sensor networks (WSNs) encounter a significant challenge in ensuring network security due to their operational constraints. This challenge stems from the potential infiltration of malware into WSNs, where a single infected node can rapidly propagate worms to neighboring nodes. To address this issue, this research introduces a stochastic $$\textsf{S}\textsf{E}\textsf{I}\textsf{R}\textsf{S}$$ S E I R S model to characterize worm spread in WSNs. Initially, we established that our model possesses a globally positive solution. Subsequently, we determine a threshold value for our stochastic system and derive a set of sufficient conditions that dictate the persistence or extinction of worm spread in WSNs based on the mean behavior. Our study reveals that environmental randomness can impede the spread of malware in WSNs. Moreover, by utilizing various parameter sets, we obtain approximate solutions that showcase these precise findings and validate the effectiveness of the proposed $$\textsf{S}\textsf{E}\textsf{I}\textsf{R}\textsf{S}$$ S E I R S model, which surpasses existing models in mitigating worm transmission in WSNs.

The stochastic fluctuation of information induces the dynamic behavior and disease control strategy to change in the HBV epidemic model, but how stochastic information dissemination affects them is vague. Here, we consider an HBV epidemic model with stochastic information intervention. Using the Lyapunov function, we demonstrate that the system has a unique global positive solution, in addition, we also derive the threshold dynamics to obtain sufficient conditions that can ensure the extinction and persistence, as well as stationarity of the system. Moreover, we qualify the impact of stochasticity of the information on the optimal control strategy of the HBV, indicating that while information intervention could effectively reduce the disease peak, the fluctuation will attenuate this effect, i.e., the stable information is better than the noisy one. Finally, various stochastic and optimal control simulations are performed to verify the theoretical results and verify that optimal control can accelerate the extinction of the disease.

In this article, a SIRS epidemic model with a general incidence rate is proposed and investigated. We briefly verify the global existence of a unique positive solution for the proposed system. Moreover, and unlike other works, we were able to find the stochastic threshold R s of the proposed model which was used for the discussion of the persistence in mean and extinction of the disease. Moreover, we utilize stochastic Lyapunov functions to show under sufficient conditions the existence and uniqueness of stationary distributions of the solution. Lastly, numerical simulation is executed to conform our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic virus infection model with nonlytic immune response, where the transmission rate is realistically modeled as being subject to continuous fluctuations, represented by the Ornstein–Uhlenbeck process. Firstly, we establish the existence and uniqueness of the global solution for the stochastic model and its invariant set, ensuring the robustness and applicability of model. Next, by constructing appropriate Lyapunov functions, we derive sufficient conditions for virus extinction and the existence of a stationary distribution for the stochastic model. These conditions elucidate the key dynamic behaviors, such as extinction and persistence, within the stochastic framework. [ABSTRACT FROM AUTHOR]

In this article, an S V I R age structure model is formulated that describes the dynamics of the Hepatitis B virus (HBV), keeping in view the vaccinated population and the standard incidence rate. After the formulation of the deterministic system, the model is extended to a stochastic system, taking into account the noises in the transmission coefficient. Utilizing the well-known formula of Itô's and the Lyapunov function theory, first, demonstrate that the perturbed system has a unique global positive solution subject to positive initial conditions. Analyze the model for the extinction of HBV, and the required condition is derived. The threshold parameter for the model is calculated, and by using this quantity, it is proved that for small values of the white noises and R s < 1 , the disease will eventually go extinct at a negative exponential rate. It is also proved that for R s 0 > 1 , the weak permanence of the disease is observed. The criterion R s 0 > 1 confirms the stationary distribution of mean infected persons, suggesting that the illness will remain throughout time. To reinforce the primary analytical findings of this paper, various illustrative examples are conducted separately through numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic Leslie-Gower model with Beddington-DeAngelis functional response driven by the Ornstein-Uhlenbeck process is studied. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: the existence and uniqueness of the global solution of the model are demonstrated, the ultimate boundedness of the model is analyzed, the existence of the stationary distribution of the model is proven, and the conditions for system extinction are discussed. Finally, numerical simulations are utilized to verify our conclusions.

This paper studies a stochastic HIV/AIDS model with nonlinear incidence rate. In the model, the infection rate coefficient and the natural death rates are affected by white noise, and infected people are affected by an intervention strategy. We derive the conditions of extinction and permanence for the stochastic HIV/AIDS model, that is, if $ R_0^s < 1, $ HIV/AIDS will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; if $ R_0^s > 1 $, HIV/AIDS will be persistent almost surely and there exists a unique stationary distribution. The conclusions are verified by numerical simulation.

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

A stochastic SEIR epidemic model with standard incidence and vertical transmission was developed in this work. The primary goal of this study was to determine whether stochastic environmental disturbances affect dynamic features of the epidemic model. The existence, uniqueness, and boundedness of global positive solutions are stated. A threshold was determined for the extinction of the infectious disease. After that, the existence and uniqueness of an ergodic stationary distribution were verified by determining the correct Lyapunov function. Ultimately, theoretical outcomes of numerical simulations are shown. [ABSTRACT FROM AUTHOR]

In this paper, assuming the certain variable satisfies the Ornstein–Uhlenbeck process, we formulate a stochastic within-host dengue model with immune response to obtain further understanding of the transmission dynamics of dengue fever. Then we analyze the dynamical properties of the stochastic system in detail, including the existence and uniqueness of the global solution, the existence of a stationary distribution, and the extinction of infected monocytes and free viruses. In particular, it is worth revealing that we get the specific form of covariance matrix in its probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Vaccination and social media play pivotal roles in affecting disease transmission. Research has shown that disease transmission can be subject to random events. Consequently, we develop a stochastic infectious disease dynamical model that incorporates saturated media coverage and vaccination age. The Itô's formula and the Lyapunov function method are applied to study the extinction behavior of the disease and the existence of a unique ergodic stationary distribution. The findings suggest that the media effect is delayed and cannot eliminate the disease completely. To directly control disease transmission, a combination of high-intensity noise disturbance and low vaccine wane rate is required. Furthermore, the shorter the disease incubation period, the more difficult it is to control. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the dynamic behavior of a novel stochastic epidemic model (SEM) that incorporates distributed delay and time switching. We prove the existence and uniqueness of a global positive solution and establish the presence of ergodic stationary distribution (ESD) through the construction of appropriate Lyapunov functions. The threshold R 0 S derived from our analysis plays a vital role in this procedure. Additionally, we conduct computer simulations to validate our theoretical discoveries, demonstrating that the incorporation of white noise can induce random fluctuations in the system variables under time switching. [ABSTRACT FROM AUTHOR]

According to the transmission characteristics of COVID-19, this paper proposes a stochastic SAIRS epidemic model with two mean reversion Ornstein–Uhlenbeck processes and saturated incidence rates. We first prove the existence and uniqueness of the global solution in the stochastic model. Using several suitable Lyapunov methods, we then derive the extinction and persistence of COVID-19 under certain conditions. Further, stationary distribution and ergodic properties are obtained. Moreover, we obtain the probability density function of the stochastic model around the equilibrium. Numerical simulations illustrate our theoretical results and the effect of essential parameters. Finally, we apply the model to investigate the latest outbreak of the COVID-19 epidemic in Guangzhou city, China. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the inevitable impact of environmental perturbations on disease transmission, we primarily investigate a stochastic model for measles infection with nonlinear incidence. The transmission rate in this model follows a logarithmic normal distribution influenced by an Ornstein–Uhlenbeck (OU) process. To analyze the dynamic properties of the stochastic model, our first step is to establish the existence and uniqueness of a global solution for the stochastic equations. Next, by constructing appropriate Lyapunov functions and utilizing the ergodicity of the OU process, we establish sufficient conditions for the existence of a stationary distribution, indicating the prevalence of the disease. Furthermore, we provide sufficient conditions for disease elimination. These conditions are derived by considering the interplay between the model parameters and the stochastic dynamics. Finally, we validate the theoretical conclusions through numerical simulations, which allow us to assess the practical implications of the established conditions and observe the dynamics of the stochastic model in action. By combining theoretical analysis and numerical simulations, we gain a comprehensive understanding of the stochastic model's behavior, contributing to the broader understanding of measles transmission dynamics and the development of effective control strategies. [ABSTRACT FROM AUTHOR]

Savannas are widely distributed on the earth, and the growth of plants in savannas often encounters the environmental perturbations. In this research, based on a deterministic tree–grass savanna model proposed by Tamen et al. (Biomath 3: 1407191, 2014), we use white noise and a continuous-time Markov chain to model the slight environmental perturbations and environmental regime switching, respectively, and formulate a stochastic tree–grass savanna model under regime switching. We first give conditions under which the model is permanent (i.e., the tree species and the grass species coexist) in the sense that the model admits a unique ergodic non-boundary stationary distribution and the transition probability of the solution of the model converges to this stationary distribution exponentially under total variation norm. Then, we provide conditions under which the grass species or the tree species die out. Finally, we discuss the impacts of random environmental perturbations on the coexistence and disintegration of the model with the help of exemplary data and numerical simulations. [ABSTRACT FROM AUTHOR]

Releasing Wolbachia-infected mosquitoes to invade the wild mosquito population is a method of mosquito control. In this paper, a stochastic mosquito population model with Wolbachia invasion perturbed by environmental fluctuation is studied. Firstly, the well-posedness, positivity, and Markov-Feller property of the solution for this model are proved. Then a group of sharp threshold-type conditions is provided to characterize the long-term behavior of the model, which pinpoints the almost necessary and sufficient conditions for the persistence and extinction of Wolbachia-infected and uninfected mosquito populations. Our results indicate that even for a low initial Wolbachia infection frequency, a successful Wolbachia invasion into the wild mosquito population can be driven by stochastic environmental fluctuations. Finally, some numerical experiments are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results. [ABSTRACT FROM AUTHOR]

Queueing systems where certain inventory items are required to provide service to a customer have become popular in the literature from early 1990th. Such systems are similar to those models analysed in the literature models with paired customers, assembly-like queues, passenger–taxi models, etc. During the last few years they are considered in the context of modelling operation of the nodes of a wireless sensor network with energy harvesting. Distinguishing feature of the model considered in this paper, besides the suggestion that arrival flow of customers is described by the Markovian arrival process, is the assumption about a general distribution of the service time while only exponential or phase-type distribution was previously assumed in the existing literature. We apply the well-known technique of M/G/1 type Markov chains and semi-regenerative processes to obtain the ergodicity criterion in a transparent form, stationary distribution of the system under study and the Laplace–Stieltjes transform of the sojourn time distribution. This creates an opportunity to formulate and solve various optimization problems. A number of numerical examples illustrate the computational tractability of the theoretical results and illustrate the behavior of the system performance measures depending on its parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic phytoplankton–zooplankton system considering phytoplankton defensive and zooplankton offensive effects. The aim of this paper is to study the effects of environmental fluctuations on plankton population dynamics. We prove the existence, uniqueness and stochastically ultimately boundedness of global positive solutions, and the extinction and persistence in the mean of plankton populations. When the system is persistent in the mean, there exists a unique stationary distribution. To further investigate the dynamics of the stochastic plankton system, we perform some numerical simulations and find that the white noise can directly affect the survival of plankton populations. The phytoplankton defense can strengthen the capability of phytoplankton protection that will benefit the plankton survival and weaken the impact of environmental fluctuations, but it has a negative effect on zooplankton population. Our findings reveal that zooplankton offense is beneficial to the survival of phytoplankton but may threaten the persistence of zooplankton population. An appropriate increase of phytoplankton defense or decrease of zooplankton offense can potentially change the survival state of the plankton system. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process is formulated and analysed, which is used to obtain a better understanding of the population dynamics. At first, we validate that the stochastic system has a unique global solution with any initial value. Then we analyse the stochastic dynamics of the model in detail, including pth moment boundedness, asymptotic pathwise estimation in turn. After that, we obtain sufficient conditions for the existence of a stationary distribution of the system by adopting stochastic Lyapunov function methods. In addition, under some mild conditions, we derive the specific form of covariance matrix in the probability density near the quasi-positive equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities v 1 and v 2 . We assume that the sequences of random times concerning the motions along each velocity follow two independent geometric counting processes of intensity λ , and that the resetting times are Poissonian with rate ξ > 0 . Under these assumptions we obtain the probability laws of the modified telegraph process describing the position and the velocity of the running particle. Our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. We obtain also the asymptotic distribution of the particle position when (i) λ tends to infinity, and (ii) the time goes to infinity. In the latter case the asymptotic distribution arises properly as an effect of the resetting mechanism. A quite different behavior is observed in the two cases when v 2 < 0 < v 1 and 0 < v 2 < v 1 . Furthermore, we focus on the determination of the moment-generating function and on the main moments of the process describing the particle position under reset. Finally, we analyse the mean-square distance between the process subject to resets and the same process in absence of resets. Quite surprisingly, the lowest mean-square distance can be found for ξ = 0 , for a positive ξ , or for ξ → + ∞ depending on the choice of the other parameters. [ABSTRACT FROM AUTHOR]

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and $ \widetilde{\mathcal{R}}_0 < 1 $, smokers will eventually become extinct. If $ \widetilde{\mathcal{R}}_0 > 1 $, smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as $ R_0^s > 1 $. Finally, conclusions are explained by numerical simulations.

Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.

This work examines a stochastic viral infection model with a general distributed delay. We transform the model with weak kernel case into an equivalent system through the linear chain technique. First, we establish that a global positive solution to the stochastic system exists and is unique. We establish the existence of a stationary distribution of a positive solution under the stochastic condition $ R^s > 0 $, also referred to as a stationary solution, by building appropriate Lyapunov functions. Finally, numerical simulation is proved to verify our analytical result and reveals the impact of stochastic perturbations on disease transmission.

Zika is an infectious disease with multiple transmission routes, which is related to severe congenital disabilities, especially microcephaly, and has attracted worldwide concern. This paper aims to study the dynamic behavior and optimal control of the disease. First, we establish a stochastic reaction-diffusion model (SRDM) for Zika virus, including human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, and prove the existence, uniqueness, and boundedness of the global positive solution of the model. Then, we discuss the sufficient conditions for disease extinction and the existence of a stationary distribution of positive solutions. After that, three controls, i.e. personal protection, treatment of infected persons, and insecticides for spraying mosquitoes, are incorporated into the model and an optimal control problem of Zika is formulated to minimize the number of infected people, mosquitoes, and control cost. Finally, some numerical simulations are provided to explain and supplement the theoretical results obtained.

Abstract The classical compartment model is often used to study the spread of an epidemic with one virus. However, there are few types of research on epidemic models with multiple viruses. The article aims to propose two new deterministic and stochastic SIIIRS models with multiple viruses and saturation incidences. We obtain asymptotic properties of disease-free and several endemic equilibria for the deterministic model. In the stochastic case, we prove the existence and uniqueness of positive global solutions. The extinction and persistence of diseases are obtained under different threshold conditions. We analyze the existence of stationary distribution through a suitable Lyapunov function. The results indicate that the extinction or persistence of the two viruses is closely related to the intensity of white noise interference. Specifically, considerable white noise is beneficial for the extinction of diseases, while slight one can lead to long-term epidemics of diseases. Finally, numerical simulations illustrate our theoretical results and the effect of essential parameters. [ABSTRACT FROM AUTHOR]

The large-time behavior of a nonlinearly coupled pair of measure-valued transport equations with discontinuous boundary conditions, parameterized by a positive real-valued parameter λ, is considered. These equations describe the hydrodynamic or fluid limit of many-server queues with reneging (with traffic intensity λ), which model phenomena in diverse disciplines, including biology and operations research. For a broad class of reneging distributions with finite mean, and service distributions with finite mean and hazard rate function that is either nonincreasing or bounded away from zero and infinity, it is shown that if the fluid equations have a unique invariant state, then the Dirac measure at this invariant state is the unique invariant distribution of the fluid equations. In particular, this implies that the stationary distributions of scaled N-server systems converge to the unique invariant state of the corresponding fluid equations. Moreover, when the mean arrival rate is not equal to the mean service rate, that is, when \lambda \not= 1, it is shown that the solution to the fluid equation starting from any initial condition converges to this unique invariant state in the large-time limit. The proof techniques are different under the two sets of assumptions on the service distribution, as well as under the two regimes λ < 1 and λ > 1. When the hazard rate function is nonincreasing, a reformulation of the dynamics in terms of a certain renewal equation is used, in conjunction with recursive asymptotic estimates. When the hazard rate function is bounded away from zero and infinity, the proof uses an extended relative entropy functional as a Lyapunov function. Analogous large-time convergence results are also established for a system of coupled measure-valued equations modeling a multiclass queue. [ABSTRACT FROM AUTHOR]

This paper considered a multi-stage sheep brucellosis model with incomplete immunity. First, we established a deterministic model, calculated the basic reproduction number ℛ 0 , 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. [ABSTRACT FROM AUTHOR]

In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

We consider the sequence {ξn,t}t≥1 of controlled critical branching processes with immigration, where n = 1, 2, ... is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n converge to the distribution of a random variable whose square has a gamma distribution. [ABSTRACT FROM AUTHOR]