Search

In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number R 0 was greater than unity and the disease free equilibrium was globally asymptotically stable when R 0 was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as R 0 s and R 0 E , were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model. In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number was greater than unity and the disease free equilibrium was globally asymptotically stable when was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as and , were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model. [ABSTRACT FROM AUTHOR]

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the p$$ p $$ order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

Although the long‐term dynamics of the prey‐predator model has analyzed in the passing decades, the threshold of extinction and persistence has not been well unified, especially the higher order perturbation affects the phytoplankton‐zooplankton dynamics. Here, we formulate a stochastic phytoplankton‐zooplankton model with nonlinear perturbation. A uniform expression of threshold value of extinction and ergodicity for this model is derived. In addition, the sufficient conditions of strong and weak persistence in the mean of phytoplankton and zooplankton are given. Our results indicate that white noise influences the dynamics of both phytoplankton and zooplankton species, which further affects the stability of phytoplankton‐zooplankton models. Some numerical examples are provided to support our theoretical results, and their explanations from the perspective of ecology are presented in this paper. [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]

Since November 2019, each country in the world has been affected by COVID-19, which has claimed more than four million lives. As an infectious disease, COVID-19 has a stronger transmission power and faster propagation speed. In fact, environmental noise is an inevitable important factor in the real world. This paper mainly gives a new random infectious disease system under infection rate environmental noise. We give the existence and uniqueness of the solution of the system and discuss the ergodic stationary distribution and the extinction conditions of the system. The probability density function of the stochastic system is studied. Some digital simulations are used to demonstrate the probability density function and the extinction of the system. [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]

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold R0P related to the basic reproduction number R0. A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if R0P>1. Next, by solving the corresponding Fokker‐Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition R0Q<1 for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results. [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]

Echinococcosis, one of the most serious zoonotic diseases, has a severe impact on the human health and economic development. This paper mainly focuses on the effect of stochastic variability on transmission dynamics of echinococcosis. A stochastic model describing the transmission of echinococcus granulosus in dog-livestock-human is proposed. By using the Itô formula, strong law of large numbers for martingale and the inequality analysis, the criteria on the extinction with probability one for the disease are obtained. In addition, by constructing an appropriate stochastic Lyapunov function, the existence of a unique ergodic stationary distribution is established. Furthermore, numerical simulations have been performed to not only support our analytical results but also display that noise intensities would have significant impact on the speed of the extinction for the echinococcosis disease, and as well as the shape for the probability density curve of the stationary distribution of the solution for this model. [ABSTRACT FROM AUTHOR]

Due to the complex set of abiotic and biotic interaction variables in the ecosystems, the dynamics of animal populations are generally considered to be driven in astate‐dependent manner. This paper presents an analysis of a stochastic Nicholson's blowflies model with distributed delay, which consists of the existence and uniqueness of globally positive solution, extinction, existence of positive T‐periodic solution and positive recurrence. Our results suggest that a smaller white noise is necessary to cause the existence of positive T‐periodic solution and positive recurrence of the stochastic model, while a larger white noise will accelerate the extinction of the population. In addition, numerical simulation is used to support our main results. [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]

To capture the realistic dynamics of vegetation pattern, we study a stochastic vegetation-water model, where the vegetation consumption of general nonlinear response type is considered. This paper is the first mathematical attempt to introduce the Black–Karasinski process as the random effect in vegetation evolution. It is shown that Black–Karasinski process is a both biologically and mathematically reasonable assumption by comparison with existing stochastic modeling methods. We obtain a critical value ℛ 0 S to classify the long-term dynamical properties of the stochastic vegetation model, including the existence of stationary distribution (i.e., a reflection of vegetation persistence) and the exponential extinction of vegetation. [ABSTRACT FROM AUTHOR]

This paper discusses the dynamics of a Gilpin–Ayala competition model of two interacting species perturbed by white noise. We obtain the existence of a unique global positive solution of the system and the solution is bounded in p th moment. Then, we establish sufficient and necessary conditions for persistence and the existence of an ergodic stationary distribution of the model. We also establish sufficient conditions for extinction of the model. Moreover, numerical simulations are carried out for further support of present research. [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]