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To capture the underlying dynamics of the COVID-19 pandemic, we develop a stochastic SEIABR compartmental model, where the concentration of coronaviruses in the environment is considered. This paper is the first attempt to introduce the Black-Karasinski process as the random fluctuations in the modeling of epidemic transmission, and it is shown that Black-Karasinski process is a both mathematically and biologically reasonable assumption compared with existing stochastic modeling methods. We first obtain two critical values R 0 S and R 0 E related to the basic reproduction number R 0 of deterministic system. It is theoretically proved that (i) if R 0 S > 1 , the stochastic model has a stationary distribution ℓ (·) , which implies the long-term persistence of COVID-19; (ii) the disease will go extinct exponentially when R 0 E < 1 ; (iii) R 0 S = R 0 E = R 0 if there is no environmental noise in COVID-19 transmission. Then, we study the local stability of the endemic equilibrium P * of deterministic system under R 0 > 1. By developing an important lemma for solving the relevant Fokker-Planck equation, an approximate expression of probability density function of the distribution ℓ (·) around P * is further derived. Finally, several numerical examples are performed to substantiate our theoretical results. It should be mentioned that the techniques and methods of analysis in this paper can be applied to other complex high-dimensional stochastic epidemic systems. [ABSTRACT FROM AUTHOR]

In order to study the effect of nonlinear perturbation on virus infection of target cells, in this paper, we propose a stochastic virus infection model with multitarget cells and exposed state. Firstly, by constructing novel stochastic Lyapunov functions, we theoretically prove that the solution of the stochastic model is positive and global. Secondly, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system and the exact expression of probability density function around a quasi-endemic equilibrium if R s > 1 , and we establish a sufficient condition R e < 1 for the extinction of infected cells and virus. Finally, we present examples and numerical simulations to verify our theoretical results. [ABSTRACT FROM AUTHOR]

• We propose a stochastic delayed SIS epidemic model with vaccination and double diseases which make the research more difficult. • We establish sufficient conditions for extinction and persistence in the mean of the two diseases. • We obtain the threshold between persistence in the mean and extinction of the stochastic system. • Time delay and environmental white noise have important effects on the persistence and extinction of the two diseases. • We give the survival analysis of the two diseases under appropriate conditions. In this paper, we investigate the threshold dynamics of a stochastic delayed SIS epidemic model with vaccination and double diseases which make the research more difficult. We establish sufficient conditions for extinction and persistence in the mean of the two diseases. We also obtain the threshold between persistence in the mean and extinction of the stochastic system. It is shown that: (i) time delay and environmental white noise have important effects on the persistence and extinction of the two diseases; (ii) the two diseases can coexist under certain conditions. Finally, some numerical simulations are provided to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

• A stochastic epidemic model for cholera is proposed and investigated. • We establish sufficient conditions for extinction of the disease. • We establish sufficient criteria for the existence of a unique ergodic stationary distribution. • The existence of an ergodic stationary distribution implies that all the individuals can be coexistent. In this paper, a stochastic epidemic model for cholera is proposed and investigated. Firstly, we establish sufficient conditions for extinction of the disease. Then we establish sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The existence of an ergodic stationary distribution implies that all the individuals can be coexistent in the long run. Finally, some examples together with numerical simulations are introduced to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

• A stochastic SIR epidemic model with distributed delay and degenerate diffusion is studied. • The Markov semigroup theory is used to obtain the existence of a unique stable stationary distribution. • We prove the densities of the distributions of the solutions can converge in L 1 to an invariant density. • We establish sufficient conditions for extinction of the disease. In this paper, we study a stochastic SIR epidemic model with distributed delay and degenerate diffusion. Firstly, we transform the stochastic model into an equivalent system which contains three equations. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We verify the densities of the distributions of the solutions can converge in L 1 to an invariant density. Then we establish sufficient conditions for extinction of the disease. Some examples and numerical simulations are introduced to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

• A stochastic HIV model with general nonlinear incidence rate is formulated. • The positive and global solution of the model is rigorously verified. • Existence of the unique ergodic stationary distribution of the model is studied. • Two examples are given to illustrate the feasibility of the theoretical results. • The effects of the noise intensity on model behavior are investigated. We present a stochastic HIV infection model with logistic target cell growth, general nonlinear incidence rate, CTL immune response and parameter perturbations. Through a rigorous analysis of the model, we obtain that the solution of the model is positive and global. A critical condition R 0 s of the model is derived, which depends not only on the general incidence function but also on the noise intensities. Under certain assumptions, by constructing suitable Lyapunov functions, we find that the system has a unique ergodic stationary distribution when R 0 s > 1. We further explore the effect of the noise intensity on model behavior. Our conclusion improves and generalizes the results of the existing HIV stochastic models. [ABSTRACT FROM AUTHOR]

Highlights • A stochastic multigroup SIQR epidemic model with standard incidence rates 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. • A stationary distribution implies stochastic weak stability of the system. Abstract In this paper, we consider a stochastic multigroup SIQR epidemic model with standard incidence rates. By using the stochastic Lyapunov function method, 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. A stationary distribution means that all the individuals can be coexistent and persistent in the long term. Finally, some examples and numerical simulations are introduced to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • A stochastic dengue epidemic model is proposed and studied. • We obtain sufficient conditions for the existence of an ergodic stationary distribution to the model. • We establish sufficient conditions for extinction of the diseases. • The existence of stationary distribution implies stochastic weak stability. Abstract In this paper, we investigate the dynamical behavior of a stochastic dengue epidemic model. First of all, by constructing a suitable stochastic Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the diseases. The existence of stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose and study a stochastic predator–prey model with herd behavior. Firstly, 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 establish sufficient conditions for extinction of the predator population in two cases, that is, the first case is the prey population survival and the predator population extinction; the second case is all the prey and predator populations extinction. Finally, some examples together with numerical simulations are introduced to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study two stochastic multigroup S-DI-A epidemic models for the transmission of HIV. For the stochastic S-DI-A epidemic model with periodic coefficients, we first obtain sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we show that the model admits a positive T -periodic solution by using Khasminskii theory of periodic solution. Moreover, we establish sufficient conditions for exponential extinction of the infectious disease. For the stochastic S-DI-A epidemic model disturbed by both white and telegraph noises, we first establish sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we obtain sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions by constructing a suitable stochastic Lyapunov function with regime switching and we also obtain sufficient conditions for exponential extinction of the system with regime switching. [ABSTRACT FROM AUTHOR]

• A vector-host transmission model with general incidence rates in deterministic and stochastic environments is developed. • For the deterministic model, the equilibrium points and their local asymptotic stability are obtained. • Sufficient conditions for the existence of ergodic stationary distribution and extinction of stochastic system are obtained. • The exact expression of the density function around a quasi-equilibrium is derived. • Numerical results are carried out to illustrate the theoretical results. In this study, we develop a vector-host transmission model with general incidence rates for the dynamics of pine wilt disease in deterministic and stochastic environments. The existence and local asymptotic stability of equilibria are investigated in the deterministic case. We show the required conditions for the ergodic stationary distribution and extinction of the model in the stochastic case by constructing appropriate Lyapunov functions. Furthermore, by solving the corresponding Fokker-Planck equation, we obtain exact expressions of probability density function around the quasi-equilibrium of the stochastic model. Finally, we employ comprehensive numerical simulations to support our results and compare deterministic and stochastic situations. [ABSTRACT FROM AUTHOR]

• A stochastic compartmental cholera with multiple transmission pathways is studied. • A unique stationary distribution and ergodicity are proved. • New method in constructing Lyapunov function on high-dimensional system is provided. • Critical value R 0 * corresponding to a basic reproduction number R 0 is obtained. • Sufficient conditions for extinction of diseases are derived. A stochastic compartmental model for cholera is investigated, which incorporates the effect of two transmission pathways on the diseases via contaminated water. Multiple stages of infection and multiple states of pathogen and environmental white noises are considered, which include or extend the cholera models in some existing articles. The existence of a unique stationary distribution and ergodicity of the model are obtained by using a suitable Lyapunov function, which determines a critical value R 0 * corresponding to a basic reproduction number R 0 of the ordinary differential equation. The results mean if R 0 * > 1 , the system still retains the stability and all individuals are persistent in a long term. In addition, the sufficient conditions for extinction of diseases are derived. What's more, we provide a new method in constructing a Lyapunov function, which can be successfully applied to other complex high-dimensional systems. [ABSTRACT FROM AUTHOR]