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Cholera is a global epidemic infectious disease that seriously endangers human life. It is disturbed by random factors in the process of transmission. Therefore, in this paper, a class of stochastic SIRB cholera model with Ornstein–Uhlenbeck process is established. On the basis of verifying that the model exists a unique global solution to any initial value, a sufficient criterion for the existence of a stationary distribution of the positive solution of the random model is established by constructing an appropriate random Lyapunov function. Furthermore, under the same condition that there is a stationary distribution, the specific expression of the probability density function of the random model around the positive equilibrium point is calculated. Finally, the theoretical results are verified by numerical model. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a stochastic Lotka–Volterra competitive model with the Ornstein–Uhlenbeck process and infinite delays. First, we verify the existence and uniqueness of the global solution of the system with any initial value. Then, we investigate the pth moment boundedness, asymptotic pathwise estimation, and asymptotic behavior of the solutions of the stochastic system in turn. In addition, we develop sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system by establishing a series of suitable Lyapunov functions. Finally, by solving the corresponding six-dimensional Fokker–Planck equation, we obtain the accurate expression of the local density function of the linear system corresponding to the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we make up adequate conditions for complete eradication and wiping out of the diseases. In a biological viewpoint, the existence of a stationary distribution shows that the diseases will be prevalent and persistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that the diseases will be prevalent and persistent in the long term. In addition, we make up adequate conditions for complete eradication and wiping out of the diseases. Some numerical simulations are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

To analyze the impacts of dispersion and environmental noise on population dynamics, in the current paper, we propose a stochastic Lotka–Volterra competition model in a patchy environment, where the fluctuations in the varying environment are characterized by an Ornstein–Uhlenbeck process. Besides that, we establish sufficient criteria for the asymptotic stability of the positive equilibrium of the system by adopting the Lyapunov function method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate conditions for complete eradication and wiping out of the infectious disease. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long term. Some numerical simulations are introduced to illustrate our theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a higher-order stochastically perturbed multigroup staged-progression model for the transmission of HIV with saturated incidence rate. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. In addition, we make up adequate conditions for complete eradication and wiping out the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, examples and numerical simulations are introduced to validate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze two stochastic predator–prey models with distributed delay and stage structure for prey. For the nonautonomous periodic case of the model, by using Khasminskii's theory of periodic solution, we show that the system has at least one positive T -periodic solution. For the model which is disturbed by both white and telegraph noises, we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The positive recurrence implies that both prey and predator populations will be persistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The theoretical results show that large white noise is beneficial for eliminating TB endemic while small white noise is not. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by using the stochastic Lyapunov function with regime switching. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic HIV-1 infection model with distributed delay and logistic growth. Firstly, we transfer the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Then, we establish sufficient conditions for the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Moreover, we obtain sufficient criteria for extinction of the infected cells; that is, the uninfected cells are survival and the infected cells are extinct. Our results show that the smaller white noise can ensure the existence of a stationary distribution when the basic reproduction number R 0 S of the stochastic system is bigger than one, while the larger white noise can lead to the extinction of the infected cells when the basic reproduction number R 0 of the deterministic system is smaller than one. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a multigroup SIRS epidemic model with random perturbations and varying total population size. By utilizing the stochastic Lyapunov function method, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Since our model is multidimensional, it is extremely difficult to construct an appropriate stochastic Lyapunov function to prove the existence of the stationary distribution, which implies stochastic weak stability. Then we establish sufficient conditions for extinction of the diseases. These conditions are related to the basic reproduction number in its corresponding deterministic system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Firstly, we obtain sufficient conditions for extinction of the diseases. Then we establish sufficient conditions for persistence in the mean of the diseases. Moreover, in the case of persistence, we derive sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. [ABSTRACT FROM AUTHOR]

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a regime-switching predator–prey model with anti-predator behaviour and higher-order perturbations. We obtain the ergodic property by constructing a suitable stochastic Lyapunov function with regime-switching, which provides us a biological perspective of cycling phenomena of a population system, and can better describe the stochastic persistence of a population system in practice. We find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions. Highlights • A regime-switching predator–prey model with anti-predator behaviour is studied. • We obtain the ergodic property by constructing a suitable stochastic Lyapunov function. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic predator-prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and analyze a stochastic predator–prey model with additional food and nonlinear perturbation. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solution to the model. Then we obtain sufficient conditions for extinction of the predator species in two cases, one is the prey species surviving and the predator species extinction; the other is both the prey and predator species are extinct. Finally, some examples together with numerical simulations are provided to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic cell-to-cell HIV-1 model with distributed delay. Firstly, we show that there is a global positive solution of this model before exploring its long-time behavior. Then sufficient conditions for extinction of the disease are established. Moreover, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the model by constructing a suitable stochastic Lyapunov function. The stationary distribution implies that the disease is persistent in the mean. Finally, we provide some numerical examples to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical behavior of a stochastic SIQR epidemic model with standard incidence which is disturbed by both white and telegraph noises. Firstly, we obtain sufficient conditions for persistence in the mean of the disease. Then we establish sufficient conditions for extinction of the disease. In addition, in the case of persistence, we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, some numerical simulations are introduced to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a stochastic Susceptible-Infective-Removed-Infective (SIRI) epidemic model with relapse. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Moreover, sufficient conditions for extinction of the disease are also obtained. [ABSTRACT FROM PUBLISHER]

In this paper, a stochastic delayed HIV-1 infection model with nonlinear incidence is proposed and investigated. First of all, we prove that there is a unique global positive solution as desired in any population dynamics. Then by constructing some suitable Lyapunov functions, we show that if the basic reproduction number R 0 ≤ 1 , then the solution of the stochastic system oscillates around the infection-free equilibrium E 0 , while if R 0 > 1 , then the solution of the stochastic system fluctuates around the infective equilibrium E ∗ . Sufficient conditions of these results are established. Finally, we give some examples and a series of numerical simulations to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SEIR epidemic model with standard incidence. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution to the model. Our result does not depend on the positive endemic equilibrium E ∗ of the deterministic system, which improves the previous result to a greater extent. Moreover, we also establish sufficient conditions for extinction of the disease. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic HIV-1 infection model with logistic growth. Firstly, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HIV-1 infection model. Then we obtain sufficient conditions for extinction of the infection. The stationary distribution shows that the infection can become persistent in vivo. [ABSTRACT FROM AUTHOR]

In this paper, we develop a mathematical model for a tuberculosis model with constant recruitment and varying total population size by incorporating stochastic perturbations. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of an ergodic stationary distribution as well as extinction of the disease to the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a mathematical model of cell-to-cell HIV-1 infection to C D 4 + T cells perturbed by stochastic perturbations. First of all, we investigate that there exists a unique global positive solution of the system for any positive initial value. Then by using Lyapunov analysis methods, we study the asymptotic property of this solution. Moreover, we discuss whether there is a stationary distribution for this system and if it owns the ergodic property. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic delayed SEIR epidemic model with nonlinear incidence. Firstly we verify that there exists a unique global positive solution of the system. Then by constructing some suitable Lyapunov functions, we study the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium respectively. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the dynamics of a stochastic nonautonomous SIR epidemic model, in which population growth is subject to logistic growth in absence of disease. For the periodic system, we present sufficient conditions for persistence of the epidemic and in the case of persistence, by constructing some suitable Lyapunov functions, we show that there is at least one nontrivial positive periodic solution. One of the most important findings is that random perturbations may be beneficial to formate the periodic solution to the stochastic nonautonomous SIR epidemic model. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

An n-species stochastic Gilpin-Ayala cooperative model was investigated in this study. The Lyapunov function and the M-matrix method were applied to study the stability of the solutions. Sufficient conditions for the existence of a global positive solution of the Gilpin-Ayala cooperative model were established. Certain asymptotically stable results of a global positive solution of the cooperative model and its domain of attraction were estimated. That main objective of this study is to provide corrections for errors in some theorems given in the work of . The errors of Theorems 2, 3, 5, and 6 from the published work appeared in the parameters θ and p. [ABSTRACT FROM AUTHOR]

In this paper, a general stochastic non-autonomous logistic model with delays and Lévy jumps is proposed and studied. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the model are established. The threshold between weak persistence and extinction is also obtained. Finally, numerical simulations are introduced to illustrate the theoretical analysis results. Our results illustrate that both persistence and extinction have close relationships with time-dependent delay and Lévy noise. [ABSTRACT FROM AUTHOR]

In this paper, a three-species food web stochastic model with generalist predator is proposed and studied. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution implies that all species can coexist in a community. • A three-species food web stochastic model with generalist predator is studied. • We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. • The existence of a stationary distribution implies that all species can coexist in a community. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a stochastic predator–prey system where apart from direct predation the prey population is affected by the fear induced from predators. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological interpretation, the existence of a stationary distribution indicates that both the prey and predator species will be persistent and coexistent in the long term. • A stochastic predator–prey system with fear effect is proposed and studied. • Criteria for the existence of a unique ergodic stationary distribution are obtained. • A stationary distribution shows that both the prey and predator species will persist. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. We find that the hypothetical conditions on the nonlinear function are relative weak and valid for many forms of incidence rate. • A stochastic SIRS model with logistic growth and general nonlinear incidence rate is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • The conditions on the nonlinear function are relative weak and valid for many forms of incidence rate. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic cholera model with imperfect vaccination is proposed and investigated. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. Our result may provide some new insights for elimination of cholera. Some numerical simulation is provided to demonstrate our main result. • A stochastic cholera model with imperfect vaccination is proposed and investigated. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • Our result may provide some new insights for elimination of cholera. • Some numerical simulation is provided to demonstrate our main result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIR epidemic model with Logistic birth. By using the stochastic Lyapunov function method, we show that the stochastic basic reproduction number R 0 S can be used to determine the threshold dynamics of the stochastic system. If R 0 S > 1 , we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. While if R 0 S < 1 , under some extra conditions, we obtain sufficient conditions for extinction of the disease. Finally, some examples and numerical simulations are provided to illustrate the theoretical results. • A stochastic SIR epidemic model with logistic birth is studied. • We establish sufficient conditions for the existence of a stationary distribution. • We obtain sufficient conditions for extinction of the disease. • Some examples and numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a multigroup SIS epidemic model with standard incidence rates and Markovian switching. First of all, we establish sufficient conditions for extinction of the diseases. Then we obtain sufficient conditions for persistence in the mean of the diseases. In addition, in the case of persistence, we obtain sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. • A multigroup SIS epidemic model with standard incidence rates and Markovian switching is studied. • We establish sufficient conditions for extinction of the diseases. • We obtain sufficient conditions for persistence in the mean of the diseases. • We obtain sufficient conditions for the existence of positive recurrence of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic predator–prey model with distributed delay and Markovian switching. Firstly, we obtain sufficient criteria for weak persistence of the predator population. Then in the case of weak persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. Finally, we establish sufficient conditions for extinction of the predator population. • A stochastic predator–prey model with distributed delay and Markovian switching is studied. • We obtain sufficient criteria for weak persistence of the predator population. • We establish sufficient conditions for the existence of positive recurrence of the solutions. • We establish sufficient conditions for extinction of the predator population. [ABSTRACT FROM AUTHOR]

In this paper, we study a multigroup SIR epidemic model in a stochastic environment. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the diseases. The results reveal that the smaller white noise can ensure the persistence of the diseases while the larger white noise can lead to the extinction of the diseases. Finally, some examples and numerical simulations are provided to illustrate the theoretical results. • A multigroup SIR epidemic model in a stochastic environment is studied. • We establish sufficient conditions for the existence of a stationary distribution to the model. • We establish sufficient conditions for extinction of the diseases. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]