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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]

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]

• 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]