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Scanning the whole writing, we discuss a stochastic cooperative species system with distributed delays under the influences of Ornstein–Uhlenbeck process of mean regression. We successfully obtain the existence and uniqueness of positive solutions for stochastic system at first. Secondly, by studying the Lyapunov function, we present the existence of the stationary distribution of the system. We are relatively familiar with the understanding of the density function of random systems. This paper also gives the expression of the density function of the random system near the unique positive equilibrium. In addition, the asymptotic properties of the p-moment boundedness and solution of the stochastic population system are also studied. In particular, we use numerical simulation to verify the theoretical results in the last section. [ABSTRACT FROM AUTHOR]

In this paper, by introducing environmental white noise and telegraph noise, we proposed a stochastic predator–prey model with the Beddington–DeAngelis type functional response and investigated its dynamical behavior. Persistence and extinction are two core contents of population model research, so we analyzed these two properties. The sufficient conditions of the strong persistence in the mean and extinction were established and the threshold between them was obtained. Moreover, we took stability into account and, by means of structuring a suitable Lyapunov function with regime switching, we proved that the stochastic system has a unique stationary distribution. Finally, numerical simulations were used to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stochastic dynamics of a multi‐group stochastic SEIRI epidemic model with logistic population growth, standard incidence rate, and Markovian switching. For this purpose, we first show that the solution of the stochastic system is positive and global. Then we obtain sufficient conditions for the extinction of disease. In addition, the persistence in the mean of disease is proved. Specifically, by constructing suitable stochastic Lyapunov functions, we find a domain that is positive recurrence for the solution of stochastic system. Finally, numerical simulations are employed to verify our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a high‐dimensional human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) model with treatment and standard incidence, which is perturbed by telegraph noise. The switching is formulated by a continuous time Markov chain. Firstly, we show the existence and uniqueness of the global positive solution. Then, conditions for extinction of the disease are established. Moreover, through constructing suitable Lyapunov functions, we derive sufficient conditions for the existence of positive recurrence of the solutions. Positive recurrence implies all the individuals can coexist and persist in the long term. Finally, numerical experiments are performed for supporting the theoretical results. [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]

This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Birth vaccinations are becoming more common in society. In this paper, we describe the developed stochastic susceptible‐vaccinated‐infected‐recovered (SVIR) epidemic model with vaccination of newborns that enable us to concern the stationary distribution and further density function. By constructing a series suitable Lyapunov function, we derive the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution. More importantly, under the same conditions, we creatively find further the density function which is based on solving corresponding Fokker–Planck equation. The results of numerical simulation, which is supported by pertussis disease data, show that our conclusion accords with reality. The density function throws light on the property of an epidemic after being stationary and furnishes more information about the disease. [ABSTRACT FROM AUTHOR]

In this paper, we developed and studied a stochastic HIV model with nonlinear perturbation. Through a rigorous analysis, we firstly showed that the solution of the stochastic model is positive and global. Then, by employing suitable stochastic Lyapunov functions, we prove that the stochastic model admit a unique ergodic stationary distribution. In addition, sufficient conditions for the extinction of HIV infection are derived. Finally, numerical simulations are employed to confirm our theoretical results. [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]

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine‐induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ℛ0C related to the basic reproduction number ℛ0 of the deterministic model. The key aim of this paper is to generalize the θ‐stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ‐stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ℛ0C>1. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a stochastic two-compartment model of B -cell chronic lymphocytic leukemia, which is perturbed by white noise. Firstly, by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then, conditions for extinction of the disease are derived. Furthermore, numerical simulations are presented for supporting the theoretical results. Our results show that large noise intensity may contribute to extinction of the disease. [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 a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise. First, we prove the existence of the unique global positive solution. Second, by constructing an appropriate Lyapunov function, we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for the extinction of diseases. Finally, numerical simulations are introduced to illustrate 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 study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results. [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 paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has'minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic multimolecular biochemical reaction model with lévy jumps is investigated. Firstly, we prove the existence and uniqueness of the global positive solution. Then we derive the conditions when the reaction will end and when the reaction will proceed. Moreover, the existence of positive recurrence to the solutions is studied by constructing suitable Lyapunov functions. Results show that the end and persistence of the reaction are closely related to the intensity of lévy noise. Finally, numerical simulations are carried out to illustrate the theoretical results. • A stochastic multimolecular biochemical reaction model with lévy jumps is studied. • We establish sufficient conditions for the end and persistence of the reaction. • The existence of positive recurrence to the solutions is investigated. • Numerical simulations are carried out to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic chemostat model with single-species growth on two perfectly substitutable nutrients under telegraph noise is investigated. Firstly, we prove the solution of the system is positive and global. Then we establish sufficient conditions for the existence of ergodic stationary distribution of the model by constructing suitable Lyapunov functions with regime switching. Finally, numerical simulations are carried out to demonstrate our theoretical results. • A stochastic chemostat model with two nutrients under telegraph noise is considered. • We establish sufficient conditions for the existence of stationary distribution. • The existence of stationary distribution implies stochastic weak stability. • Numerical simulations are carried out to illustrate the results. [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 focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function. [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]

We investigate the stochastic HTLV‐I infection model with CTL immune response, which is disturbed by both white and telegraph noises. First, we obtain that the model solution is positive and global. Furthermore, we obtain the existence of ergodic stationary distribution of the solution by stochastic Lyapunov functions with regime switching, and we also establish sufficient conditions for the extinction of system with regime switching. Meanwhile, the threshold R0∗ between ergodic property and extinction of the stochastic system is acquired. Finally, our theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

A stochastic HIV viral model with both logistic target cell growth and nonlinear immune response function is formulated to investigate the effect of white noise on each population. The existence of the global solution is verified. By employing a novel combination of Lyapunov functions, we obtain the existence of the unique stationary distribution for small white noises. We also derive the extinction of the virus for large white noises. Numerical simulations are performed to highlight the effect of white noises on model dynamic behaviour under the realistic parameters. It is found that the small intensities of white noises can keep the irregular blips of HIV virus and CTL immune response, while the larger ones force the virus infection and immune response to lose efficacy. [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]

This paper considers two S-DI-A models in random environments. Firstly, using Has’minskii theory of periodic solution, we show that stochastic periodic S-DI-A model has a nontrivial positive periodic solution if. Then, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to S-DI-A model perturbed by white and telephone noises. Finally, examples are introduced to illustrate the results developed. [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]

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]