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In this paper, by using Itô's formula, we study the dynamic behaviors of a stochastic human immune deficiency virus (HIV) model, which describes Acquired Immune Deficiency Syndrome (AIDS) with the effect of treatment. First, we discuss the global positivity of the solution to such a model with positive initial value. Then, we consider the limiting behavior of the solution. Moreover, we obtain the sufficient condition to guarantee the extinction of AIDS. Meanwhile, by constructing an appropriate Lyapunov function, we prove that the solution has a unique ergodic stationary distribution under a certain assumptions. Finally, we illustrate the validity of our results by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a periodic stochastic Lotka‐Volterra model with distributed delay and dispersal is investigated. First, by Itô's formula, we show that the solution with any positive initial value is global and positive. Then, we obtain the sufficient conditions guaranteeing the extinction of the system. Furthermore, we discuss the existence of the nontrivial periodic solution by Khasminskii theory. Finally, numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic food chain chemostat model with variable yields. First, we prove the stochastic model has a unique global positive solution. Second, by employing suitable Lyapunov functions, Itô ′ s formula and some other important inequalities, the existence of a unique ergodic stationary distribution of a stochastic food chain chemostat model is researched, which can help us better understand the statistical characteristics of stochastic food chain chemostat models. Second, we investigate the extinction of the microorganism and the bacteria. Moreover, the case of extinction for bacteria but persistence for microbial species is considered. Finally, some numerical simulations are carried out to illustrate our theoretical results and the influence of the variable yields on the microorganism and the bacteria. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear stochastic eco-epidemic model is presented to study the dynamics of harvested prey–predator model with group defence. One of the key roles of forming a group is to defend themselves from the attack of the predators. Since prey population spreads infection through direct contact; we consider two groups of prey population. Using some certain assumptions, a stochastic model is formulated by considering noisy intrinsic growth rate of susceptible pest population, death rate of infected pest and predator populations, respectively. Next, the existence and uniqueness of positive solution are performed using comparison theorem of stochastic differential equations (SDEs). The boundedness and uniform continuity of the system are analyzed. Then, the bionomic equilibrium and optimal harvesting policy are obtained by using stochastic maximum principle. We verify the conditions of extinction and persistence in mean with the numerical results. To verify our consideration with the reality, the effects of harvesting effort and catchability coefficient are observed. In addition, infected and uninfected regions are reported to specify the infection free system. [ABSTRACT FROM AUTHOR]

A class of stochastic SIRS infectious disease models with both logistic birth and Markov switching were investigated. The uniqueness of the existence of a globally positive solution to the stochastic infectious disease model was first analyzed through construction of suitable V functions and then by means of Itô’s formula. Afterwards, the results of the existence of an ergodic smooth distribution for the solution of the model and the sufficient conditions for the extinction of the disease were discussed. Finally, numerical examples were given to illustrate the conclusions. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with vaccination effect is investigated. Our purpose is to study the dynamical behaviors of the model. First, the existence and uniqueness of the positive solution of the model is proved. Then, the dynamic properties of the model, such as extinction and persistence, are analyzed by regarding certain conditions on the parameters. Some sufficient conditions for extinction and persistence are given by using Itô's formula and martingale methods. The results show that low-intensity white noise may cause disease to spread, while high-intensity white noise may drive infective, quarantined individuals to extinction. Finally, numerical simulations are illustrated to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and propose a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. For each species, sharp sufficient conditions for persistence in the mean and extinction are respectively obtained. The results demonstrate that the persistence and extinction of the species have close relationships with the environmental stochasticity. In addition, the theoretical results are numerically illustrated by some simulations. [ABSTRACT FROM AUTHOR]

The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mutualism ecological model with stochastic perturbations. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded, uniformly continuous and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistent are also established. At last, some figures are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of stationary distribution and extinction for a stochastic generalized logistic system. Sufficient and necessary conditions for the existence of a stationary distribution and extinction are obtained. (a) The system has a unique stationary distribution if and only if the noise intensity is less than twice the intrinsic growth rate. The probability density function has been solved by the stationary Fokker-Planck equation. (b) The system will become extinct when and only when the noise intensity is no less than twice the intrinsic growth rate, and the exponential extinction rate is estimated precisely by two parameters of the systems. A new perspective is provided to explain the recurrence phenomenon in practice. Nontrivial examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic SIS model with saturated incidence rate. We investigate the existence and uniqueness of the positive solution to the system, and we show the condition for the infectious individuals to be extinct. Moreover, we prove that the system has the ergodic property and derive the expression for its invariant density. The simulation results are illustrated finally. [ABSTRACT FROM AUTHOR]

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R 0 is obtained. Furthermore, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable and if R 0 > 1 , then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R 0 > 1 , we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R 0 ≤ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R 0 > 1 , then there is a unique stationary distribution to stochastic system. [ABSTRACT FROM AUTHOR]

Abstract: A stochastic ratio-dependent predator–prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itô’s formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

In this article we stochastically perturb the classical non-autonomous delay Lotka-Volterra model [image omitted] into the stochastic delay population system (SDPS) [image omitted] Different from most of the existing papers [A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl. 292 (2004), 364-380, A. Bahar and X. Mao, Stochastic delay population dynamics, J. Pure Appl. Math. 11 (2004), 377-400, X. Mao, Delay population dynamics and environmental noise, Stochastics Dyn. 5(2) (2005), pp. 149-162], the system parameters in this article are time-dependent. We will give a sufficient condition under which the SDPS will have a unique global positive solution. We will then establish some new asymptotic properties for the moments of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction. [ABSTRACT FROM AUTHOR]

Abstract: This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69–84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population. [Copyright &y& Elsevier]

In this article, we stochastically perturb the classical non-autonomous Lotka-Volterra model [image omitted] into the stochastic differential equation [image omitted] Different from most of existing articles, for example, [3, 20] the system parameters in this article are time-dependent. We will give a sufficient condition under which the stochastic differential equation will have a unique global positive solution. We will then establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction. [ABSTRACT FROM AUTHOR]

In this paper we review our recent research on stochastic delay population dynamics. The main aim is to reveal that a different type of environmental noise has a different effect on delay population systems. [ABSTRACT FROM AUTHOR]

Acknowledging many effects on humans, which are ignored in deterministic models for COVID-19, in this paper, we consider stochastic mathematical model for COVID-19. Firstly, the formulation of a stochastic susceptible–infected–recovered model is presented. Secondly, we devote with full strength our concentrated attention to sufficient conditions for extinction and persistence. Thirdly, we examine the threshold of the proposed stochastic COVID-19 model, when noise is small or large. Finally, we show the numerical simulations graphically using MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with quarantine and standard incidence is investigated. We first consider the existence and uniqueness of the global positive solution to the stochastic system. Further, we obtain conclusions that the large noise can make the disease die out exponentially, and the stationary distribution will exist under certain conditions. • A stochastic SIQR epidemic model with quarantine and standard incidence is studied. • We consider the existence and uniqueness of the global positive solution to the system. • Large noise can make the disease die out exponentially. • The stationary distribution exists under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, stochastic effect on the spread of the infectious disease with saturated incidence rate and the special transfer from infectious is discussed. The threshold dynamics is explored for the case of relatively small noise. Our results show that large noise will cause the elimination of the disease, which will help suppress the spread of the disease. [ABSTRACT FROM AUTHOR]