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This study explores an epidemic model elucidating the dynamics of COVID-19 transmission amidst vaccination and saturated treatment interventions. The investigation encompasses both deterministic and stochastic frameworks, considering constant and fluctuating environments, utilizing COVID-19 data from India for empirical validation. Through rigorous mathematical and numerical analyses, we ascertain pivotal insights. Our deterministic model unveils a critical phenomenon: the occurrence of backward bifurcation at ℛ0 = 1, underscoring that merely reducing the basic reproduction number below unity does not ensure disease eradication. Sensitivity analyses underscore the acceleration of epidemic spread with higher transmission rates, yet mitigation measures such as vaccination and comprehensive treatment can effectively reduce the basic reproduction number below unity. Within the stochastic framework, we establish the existence of a unique global positive solution. We delineate conditions for disease extinction or persistence and identify criteria for the emergence of stationary distribution, reflecting the sustained presence of infection within the community. Our findings elucidate that while smaller noise intensities sustain disease prevalence, heightened noise levels lead to complete eradication of the infection. [ABSTRACT FROM AUTHOR]

In this article, we develop a Hepatitis B virus model with six compartments affected by environmental fluctuations since the Hepatitis B virus produces serious liver infections in the human body, putting many people at high risk. The existence of a global positive solution is shown to prove the positivity of solutions. We demonstrate that the system experiences the extinction property for a specific parametric restriction. Besides that, we obtain the stochastic stability region for the proposed model through the stationary distribution. To determine the appearance and disappearance of infection in the population, we find and analyze the reproduction ratio R0S$$ {R}_0^S $$. In addition, we have verified the condition of the reproduction ratio through the graphical simulations. [ABSTRACT FROM AUTHOR]

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein–Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus. [ABSTRACT FROM AUTHOR]

This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0. [ABSTRACT FROM AUTHOR]

Influenza remains one of the most widespread epidemics, characterized by serious pathogenicity and high lethality, posing a significant threat to public health. This paper focuses on an influenza A infection model that includes vaccination and asymptomatic patients. The deterministic model examines the existence and local asymptotic stability of equilibria. In light of the influence of environmental disruption on the spread of disease, we develop a stochastic model in which the transmission rate follows a log-normal Ornstein–Uhlenbeck process. To demonstrate the dynamic behavior of the stochastic model, we verify the existence and uniqueness of the global positive solution. The establishment of suitable Lyapunov functions allows for the determination of sufficient conditions for the stationary distribution and extinction of the disease. Furthermore, the expression of the local density function around the quasi-endemic equilibrium is represented. Eventually, numerical simulations are conducted to support theoretical results and explore the effect of environmental noise. Our findings indicate that high noise intensity can expedite the extinction of the disease, while low noise intensity can facilitate the disease reaching a stationary distribution. This information may be valuable in developing strategies for disease prevention and control. [ABSTRACT FROM AUTHOR]

The density of a population in a natural state often fluctuates greatly, but it is not an unlimited change. Throughout the full text, we consider a stochastic predator–prey system with fear, Holling-II response function, distributed delay and mean-reverting Ornstein–Uhlenbeck process, which can better reflect the actual situation and provide theoretical basis for species research in the actual environment. And we drew some significant conclusions. The first step is to obtain the stability of equilibrium point. Then, the existence and uniqueness of positive solution for stochastic system is got and we also reach the stationary distribution of the stochastic system. In addition, we present an exact local expression for the density function of the random system near the unique positive equilibrium. Then we obtain the pth moment boundedness and the asymptotic behavior of the solution. At last, we give the extinction condition of the population system by the asymptotic behavior of the solution. In particular, we confirm the theoretical results by numerical simulation in the corresponding section. [ABSTRACT FROM AUTHOR]

This study develops a stochastic SIRS compartmental model for exploring the transmission dynamics of infectious diseases, integrating the Beddington–DeAngelis incidence rate and vaccination. In the deterministic case, the reproduction number ℛ0 is derived, and the global dynamics is analyzed using the Lyapunov function with respect to ℛ0. The outcomes underscore that ℛ0 completely governs the overall dynamics of the system. In the stochastic case, the primary challenge arises from the two-dimensional boundary system, preventing the Fokker–Planck equation from obtaining the density function of the invariant measure. To address the weak convergence property regarding the invariant measure for both the stochastic system and its corresponding two-dimensional boundary system, the concept of limit measures is introduced. The theoretical results indicate that the persistence and extinction of the infectious disease are entirely determined by the Lyapunov exponent λ, representing the long-term growth rate. Numerical simulations further support these findings. [ABSTRACT FROM AUTHOR]

A vector–host model of dengue with multiple stages and independent fluctuations is investigated in this paper. Firstly, the existence and uniqueness of the positive solution are shown by contradiction. When the death rates of aquatic mosquitoes, adult mosquitoes, and human beings respectively control the intensities of white noises, and if R 0 s > 1 , then the persistence in the mean for both infective mosquitoes and infective human beings is derived. When R 0 s > 1 is valid, the existence of stationary distribution is derived through constructing several appropriate Lyapunov functions. If the intensities of white noises are controlled and φ < 0 is valid, then the extinction for both infective mosquitoes and infective human beings is obtained by applying the comparison theorem and ergodic theorem. Further, the main findings are verified through numerical simulations by using the positive preserving truncated Euler–Maruyama method (PPTEM). Moreover, several numerical simulations on the infection scale of dengue in Fuzhou City were conducted using surveillance data. The main results indicate that the decrease in the transfer proportion from aquatic mosquitoes to adult mosquitoes reduces the infection scale of infective human beings with dengue virus, and the death rates of aquatic mosquitoes and adult mosquitoes affect the value of the critical threshold R 0 s . Further, the controls of the death rates of mosquitoes are the effective routes by the decision-makers of the Chinese mainland against the spread of dengue. [ABSTRACT FROM AUTHOR]

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. [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]

Motivated by the notion of isotropic α -stable Lévy processes confined, by reflections, to a bounded open Lipschitz set D ⊂ R d , we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time. [ABSTRACT FROM AUTHOR]

AbstractWe consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process {(M1(t),M2(t)):t≥0}, which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process {(M1(t),M2(t)):t≥0} to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of {(M1(t),M2(t)):t≥0}. Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples. [ABSTRACT FROM AUTHOR]

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]

This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. 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 given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito’s formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion. [ABSTRACT FROM AUTHOR]

With the vigorous development of information technology, online rumors are spreading recklessly. In this article, by comprehensive consideration of the stochastic disturbance, generalized nonlinear incidence, education age and punishment age, the dynamical behaviors of a novel stochastic SEIFR (susceptible-educated-infected-forced silence-removed) rumor spreading model with class-age structure under the generalized nonlinear incidence is analyzed and discussed. Firstly, given that the ubiquity of stochastic disturbance in the process of rumor spreading, a stochastic rumor spreading model is established. Secondly, the existence of the unique global positive solution for the stochastic model is manifested. Thirdly, the sufficient condition of rumor extinction is attained by virtue of Itô's formula and strong law of large numbers. Additionally, the existence of a unique stationary distribution implying the rumor persistence is investigated based on the method of Khasminskii. Finally, some simulations and a practical application are carried to validate the results. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number $ R_0 $ was greater than unity and the disease free equilibrium was globally asymptotically stable when $ R_0 $ was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as $ R_0^s $ and $ R_0^E $, were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model.

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.

AIDS is a chronic infectious disease that has been having a major impact on human health and social stability. In this paper, based on the deterministic SIATR model, a stochastic SIATR model is developed to account for the spread of AIDS by introducing the Ornstein–Uhlenbeck process. First, the existence and uniqueness of the global positive solution of the model are investigated. Second, a threshold R0E is set: when R0E < 1, the disease will become extinct; when R0E > 1, the disease will persist. Then, it is shown that there exists a stationary distribution for the model when R0E > 1. On this basis, we derive the exact expression for the probability density function of the model in the neighborhood of the quasi-equilibrium state. Finally, the results of the previous proof are verified by several numerical simulations. [ABSTRACT FROM AUTHOR]

The random changes in the environment play a crucial role in the sustainability of ecosystems. Usually, the construction of stochastic models does not take into account the non-linear growth of intrinsic growth rate. In addition, prey only considers the collective response of the population when encountering predators and ignores the role of individual prey. To address this issue, we contemplate the dynamics of a stochastic prey–predator model with Smith growth rate and cooperative defense. The population density of prey is measured by mass, and the growth limitations are based on the proportion of unused available resources. Additionally, the grazing pattern of the predator incorporates cooperative characteristics into the functional response. We carry out existence and uniqueness analysis for the global positive solution. Then, we construct sufficient conditions for the existence of an ergodic stationary distribution of positive solutions for investigating whether prey and predator populations continue to survive. Numerical examples indicate that the Smith growth rate, cooperative defense and environmental disturbance play crucial roles in the coexistence of interacting populations. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate a novel stochastic SIHR epidemic model, which includes a general incidence rate and mean-reversion Ornstein–Uhlenbeck process. Firstly, the existence of global positivity of the solution is testified by Lyapunov function. Secondly, this disease will be eradicated if the reproduction number R 0 s < 1 . Otherwise, if the reproduction number R 0 ∗ > 1 , then the system has a stationary distribution, which means that the pandemic will persist. In addition, an explicit expression of the probability density function for a linear system near quasi-endemic equilibrium is obtained under certain conditions. Finally, a series of numerical simulations is carried out to validate the theoretical conclusions. [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]

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple H 2 ⊂ H 1 ⊂ H 0 to obtain a bound on the difference between the original solution and the perturbed solution. [ABSTRACT FROM AUTHOR]

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results. [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 ℛ s 1 and ℛ s 2 . If ℛ s 1 > 1 , the psychological stress will persist and there will be a unique stationary distribution; whereas if ℛ s 2 < 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]

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 paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number R 0 was greater than unity and the disease free equilibrium was globally asymptotically stable when R 0 was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as R 0 s and R 0 E , were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model. In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number was greater than unity and the disease free equilibrium was globally asymptotically stable when was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as and , were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model. [ABSTRACT FROM AUTHOR]

Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed. [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, 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]

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]

In this paper, a stochastic SEIQ model with nonlinear incidence rate is proposed. The existence and uniqueness of global positive solution of the stochastic model are proved. Then, by constructing some Lyapunov functions, we derive a sufficient condition for the ergodic stationary distribution when R 0 s is greater than one. By solving a four-dimensional Fokker–Planck equation, we get the exact expression of log-normal probability density function of stationary distribution for the stochastic model. Sufficient condition for the extinction of the exposed and infected population is also provided. Finally, numerical simulations are presented to verify the above theoretical results. [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]

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]

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]

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]

In this paper, the stochastic fluctuation of buffer stock level at time t is investigated. Therefore, random walk processes X(t) and Y(t) with two specific barriers have been defined to describe the stochastic fluctuation of the product level. Here X(t) = Y(t) -- a and the parameter a specifies half capacity of the buffer stock warehouse. Next, the one-dimensional distribution of the process X(t) has calculated. Moreover, the ergodicity of the process X(t) has been proven and the exact formula for the characteristic function has been found. Then, the weak convergence theorem has been proven for the standardized process W(t) = X(t)/a, as a → ∞. Additionally, exact and asymptotic expressions for the ergodic moments of the processes X(t) and Y (t) are obtained. [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]

The stochastic SEIR model was employed to investigate the dynamics of influenza transmission. By incorporating transmission rates and prevalence ratios, this model provides the most comprehensive explanation of the virus’s unpredictable dissemination. To simulate the stochastic components of influenza transmission, we implemented conventional Brownian motions and stochastic differential equations. The investigation examines the uniqueness and presence of the solutions to demonstrate the conditions needed for eliminating the infection under random disturbances. The transmission rate coefficient (β) strongly impacts disease transmission speed. as demonstrated by the simulation results.Thus, the proper usage of safe transmission control methods is another decisive factor that determines the outcome of epidemics. Actual data of the Kingdom of Saudi Arabia was used. The results highlighted practicality of stochastic models and their usefulness to address and formulate and even execute the public health related policies. Regarding this, this study sets a high bar for other studies on modeling viral diseases on the grounds that stochastic and dynamic factors are also very important. These subsequent improvements in the model shall enable us to pinpoint the best strategies for the prevention and eradication of influenza and any other subsequent epidemic diseases, with reference to epidemic, epidemiology and public health.

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 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.