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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]

• 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]