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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]

Infectious disease transmission, mainly affected by media coverage and stochastic perturbations, has imposed great social financial burden on the community in the past few decades and even threatened public health. However, there are few studies devoted to the theoretical dynamics of epidemic models with media coverage and biologically reasonable stochastic effect yet. In this sense, this paper mainly formulates and studies a stochastic epidemic model with media coverage and two mean-reverting Ornstein–Uhlenbeck processes. We first illustrate the biological implication and mathematically reasonable assumption of introducing the Ornstein–Uhlenbeck process as stochastic effect. It is theoretically proved that the solution to the stochastic model is unique and global, as well as the existence of an ergodic stationary distribution. After that, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, it is derived that the above global solution around the endemic equilibrium follows a unique probability density function. For completeness, the sufficient criteria for extinction exponentially of the model are established. Finally, several numerical simulations are provided to verify our theoretical results. Besides, the impact of stochastic noises and media coverage on epidemic transmission is studied by comparison with the previous results of a deterministic model. [ABSTRACT FROM AUTHOR]

In this paper, we examine a stochastic vegetation-water model, where the Black–Karasinski process is introduced to characterize the random fluctuations in vegetation evolution. It turns out that Black–Karasinski process is a both mathematically and biologically reasonable assumption by comparison with existing stochastic modeling approaches. First, it is theoretically proved that the solution of the stochastic model is unique and global. Then two critical values ℛ 0 E and ℛ 0 S are obtained to classify the dynamical behavior of vegetation. It is shown that: (i) If ℛ 0 S > 1 , the stochastic model has a stationary distribution ℓ (⋅) , which reflects the long-term coexitence of vegetation and the water environment. (ii) The vegetation will go extinct exponentially if ℛ 0 E < 1. (iii) ℛ 0 E = ℛ 0 S = ℛ 0 if there are no random noises in vegetation dynamics, where ℛ 0 is the basic reproduction number of deterministic model. Furthermore, by solving the associated Fokker–Planck equation, the approximate expression for probability density function of the distribution ℓ (⋅) around a quasi-positive equilibrium is studied. Finally, several numerical examples are provided to support our theoretical findings. • We develop a stochastic vegetation-water model, where the surface water and the soil water are considered. • This paper is the first mathematical attempt to introduce the Black–Karasinski process to characterize the random fluctuations in vegetation dynamics. • We prove that the stochastic system has a stationary distribution if R0S > 1. • We obtain that the vegetation will go extinct exponentially if R0E < 1. • An approximate expression of a local density function of the stationary distribution is derived. • Our techniques and theoretical methods used can be successfully applied to many complex ecological models perturbed by Black–Karasinski process. [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]

In this paper, we analyze the threshold R v S of a stochastic SIS epidemic model with partially protective vaccination of efficacy e ∈ [ 0 , 1 ] . Firstly, we show that there exists a unique global positive solution of the stochastic system. Then R v S > 1 is verified to be sufficient for persistence in the mean of the system. Furthermore, three conditions for the disease to die out are given, which improve the previously-known results on extinction of the disease. We also obtain that large noise will exponentially suppress the disease from persisting regardless of the value of the basic reproduction number R v S . [ABSTRACT FROM AUTHOR]

In this paper, a stochastic HBV infection model with virus-to-cell infection, cell-to-cell transmission and CTL immune response is proposed. The model has a general form of infection rate, in which the contact rate is governed by the log-normal Ornstein–Uhlenbeck process. First, it is proved that the stochastic model has a unique positive global solution. The dynamic properties of the solutions around the equilibrium points are also analysed. It further follows that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 , while the endemic equilibrium is globally asymptotically stable if R 0 > 1. Then, we establish sufficient conditions for the stationary distribution and extinction of the model by constructing suitable Lyapunov functions, respectively. After that, we calculate the exact analytical expression for the probability density function of stationary distribution near the quasi-endemic equilibrium. Finally, the effect of the Ornstein–Uhlenbeck process on the dynamical behaviour of the model is verified by numerical simulations. One of the most interesting findings is that larger regression speeds and smaller volatility intensities can significantly reduce major outbreaks of HBV infection within the host, which may have important implications for future HBV therapeutic regimens. • A HBV infection model disturbed by Ornstein–Uhlenbeck process is presented. • Dynamic properties of the solutions around the equilibrium points are analysed. • Stationary distribution and extinction are studied. • The density function around the quasi-endemic equilibrium is discussed. • Comprehensive numerical simulations are represented. [ABSTRACT FROM AUTHOR]

• stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is established. • The existence of a unique global positive solution for the stochastic model are obtained. • We establish the sufficient conditions for the unique ergodic stationary distribution. • We derive the exact expression of probability density function to the stochastic model. Focusing on the results of Rajasekar (2020) and the continuous dynamics of stochastic differential equation (SDE) developed by Mao (1997), a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is investigated in this paper. First, we propose and prove that the unique solution of stochastic model is globally positive. By constructing some suitable Lyapunov functions, the sufficient condition R 0 h > 1 is obtained for the unique stationary distribution which has ergodicity property. Next, by solving the corresponding Fokker-Planck equation, we derive the approximate probability density function around the quasi-endemic equilibrium of the stochastic system. The above stationary distribution and density function can reveal all statistical properties of the disease persistence. In addition, by comparison with other existing articles, our developed theoretical results and some numerical simulations are introduced at the end of this paper. [ABSTRACT FROM AUTHOR]

• A stochastic staged progression AIDS model with staged treatment and second-order perturbation is studied. • The existence and uniqueness of the ergodic stationary distribution is obtained under the condition of. • If, we obtain that the AIDS epidemic will go to extinction in long-term. Focusing on deterministic AIDS model proposed by Hyman (2000) and the detailed data from the World Health Organization (WHO), there are three stages of AIDS process which are described as Acute infection period, Asymptomatic phase and AIDS stage. Our paper is therefore concerned with a stochastic staged progression AIDS model with staged treatment. In view of the complexity of random disturbances, we reasonably take second-order perturbation into consideration for realistic sense. By means of our creative transformation technique and stochastic Lyapunov method, a critical value R 0 H > 1 is firstly obtained for the existence and uniqueness of ergodic stationary distribution to the stochastic system. Not only does it respectively reveal the corresponding dynamical effects of the linear and second-order perturbations to the model, but the unified form of second-order and linear fluctuations is derived. Next, some sufficient conditions about extinction of stochastic system are established in view of the basic reproduction number R 0. Finally, some examples and numerical simulations are introduced to illustrate our analytical results. In addition, some advantages of our new method and theory are highlighted by comparison with other existing results at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic hepatitis B virus (HBV) infection model and the dynamic behaviors of the model are investigated. When the fraction of vertical transmission μ ω ν C is not considered to be new infections, the existence and ergodicity of the stationary distribution of the model are obtained by constructing a suitable Lyapunov function, which determines a critical value ρ 0 s corresponding to the basic reproduction number of ODE system. This implies the persistence of the diseases when ρ 0 s > 1. Meanwhile, the sufficient conditions for the extinction of the diseases are derived when ρ 0 T < 0. What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. Finally, the numerical simulations are illustrated to verify the theoretical results and match the HBV epidemic data in China. • A stochastic HBV infection model is proposed and studied. • Existence and ergodicity of stationary distribution are proved. • Critical value ρ 0 s corresponding to basic reproduction number ρ 0 is obtained. • Density function of stochastic model near quasi-equilibrium is given. • Simulations match the HBV epidemic data in China. [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]

Considering the transmission rate perturbed by log-normal Ornstein–Uhlenbeck process, we develop a stochastic HBV model with vertical transmission term. For higher-dimensional deterministic system, the local asymptotic stability of the endemic equilibrium is given by proving the global stability of the corresponding linearized system. For stochastic system, the existence of stationary distribution is obtained by constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process and the critical value corresponding to the basic reproduction number for determined system is derived, which means the persistence of the disease. And sufficient conditions for disease extinction are given. Furthermore, by solving five-dimensional Fokker–Planck equation, the exact expression of the probability density function near the quasi-equilibrium is provided to reveal the statistical properties. In the end, numerical simulations illustrate our theoretical results and exhibit the trends of the critical values for persistence and extinction of diseases along with the change of noise intensity and reversion speed. • A stochastic HBV model with OU process and vertical transmission term is proposed. • Local stability of endemic equilibrium for deterministic model is demonstrated. • Stationary distribution and extinction of diseases for stochastic model are obtained. • Exact expression of probability density function near the quasi-equilibrium is given. • Simulations exhibit the trends of critical values for persistence and extinction of diseases. [ABSTRACT FROM AUTHOR]

• The stochastic SIR epidemic model with general nonlinear incidence and general stochastic perturbations is established. • If R0S > 1, the existence of the unique ergodic stationary distribution is obtained. • If R0H > 1, we prove that there is a unique log-normal density function of the stochastic system. • We derive the disease extinction of the general stochastic SIR model when R0C < 1. Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value R 0 S related to the basic reproduction number R 0. Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if R 0 S > 1. Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value R 0 H > 1. Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R 0 C < 1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper. [ABSTRACT FROM AUTHOR]