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This paper studies a stochastic HIV/AIDS model with nonlinear incidence rate. In the model, the infection rate coefficient and the natural death rates are affected by white noise, and infected people are affected by an intervention strategy. We derive the conditions of extinction and permanence for the stochastic HIV/AIDS model, that is, if $ R_0^s < 1, $ HIV/AIDS will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; if $ R_0^s > 1 $, HIV/AIDS will be persistent almost surely and there exists a unique stationary distribution. The conclusions are verified by numerical simulation.

The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results. [ABSTRACT FROM AUTHOR]

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed. [ABSTRACT FROM AUTHOR]

Two types of susceptible, exposed, infectious, vaccinated/recovered, susceptible (SEIVS) epidemic models with saturation incidence and temporary immunity, driven by higher order white noise and telegraph noise, are investigated. The key aim of this work is to explore and obtain the existence of the unique ergodic stationary distribution for the above two models, which reveals whether the disease will be prevalent and persistent under some noise intensity assumptions. We also use meticulous numerical examples to validate the feasibility of the analytical findings. Finally, a brief biological discussion shows that the intensities of noises play a significant role in the stationary distributions of the two models.

This paper studies a stochastic two-species Schoener's competitive model with regime switching. We first investigate the sufficient conditions for the existence of a unique stationary distribution of the model. Then we prove that the convergence of transition probability to the stationary distribution is exponentially under some mild assumptions. Moreover, we also introduce several numerical simulations to validate the model against the biological significance.

In this paper, we propose a stochastic differential equation model which is used to explore how the environmental noise affects the spread of hepatitis B virus. Firstly, we show that there exists a unique global positive solution of the stochastic system with any positive initial value. Then, we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model. Especially, under the same conditions as the existence of a stationary distribution, it is worth noting that we obtain the specific form of probability density around the quasi-infected steady state of the stochastic system. Thirdly, we obtain sufficient criteria for extinction of the infected hepatocytes. Finally, numerical simulations are introduced to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

We formulate a stochastic Wright's mutualism model with regime switching in this paper. Under some conditions, we testify that the model admits a unique stationary distribution that is ergodic, and the transition probability of the solution of the model exponentially converges to the stationary distribution. Some recent results are extended and improved greatly. An example and several simulations are also provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior of a stochastic SQEIAR epidemic model is investigated. First of all, we establish sufficient conditions for extinction of the diseases. Then the sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to the model are obtained by constructing a suitable stochastic Lyapunov function. The existence of stationary distribution implies stochastic weak stability. Furthermore, the optimal control problem is considered to provide a theoretical basis for the prevention and control of the disease. Finally, the theoretical results are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this article, a stochastic differential equation model is proposed to investigate how the environmental noise affects the transmission dynamics of the interaction between crime, criminality and victimization with the influence of case reporting through the short message service (SMS). We prove that there is a unique global positive solution of the stochastic system with any positive initial value which is fundamental in the dynamics of population modeling. Moreover, we get 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 both criminal and victim will be persistent and coexistent in the long term. [ABSTRACT FROM AUTHOR]

• The stochastic SEQAIJR delayed model with quarantined, isolation and vaccination strategies is created to control COVID-19. • A persistence of the disease for the model is presented. • For controlling the rapid spread of the disease, optimal control strategy is constructed with three control measures. • Parameter estimation are presented with real data and model simulation. The actual ideal of the literature is to probe the effect of external undulation on prevalence of COVID-19. Owing to some environmental influences, we enhanced our model to the stochastic perturbations. In this article, we scrutinize a stochastic epidemic COVID-19 model with seven states of epidemiological classification. Incipiently, a sophisticated biological system with the effect of time lags put forward copious strategies. At the outset, aim of this work is to explore the existence and uniqueness of global positive solution. Moment exponential stability are upheld. By fabricating a suitable lyapunov function, the sufficient conditions for the persistence of the disease and the existence of an ergodic stationary distribution are established. To regulate the widespread of COVID-19 infection, the main intent is to curb the disease with the succor of three control measures. Finally, the theoretical results bring into being alive in the form of graphical visualization. By nurturing the comparison study, the parameters of the model system come up with real data of most afflicted countries that layout premier consequences. [ABSTRACT FROM AUTHOR]

• A stochastic post COVID-19 was developed. • An eradication of Omicron variant as well as post COVID-19 with heart attack. • Persistence of the disease was presented. • The numerical simulation was obtained by utilizing real data. In this study, a latest version of COVID-19 pandemic is hand overed. A Stochastic post COVID-19 delayed model is developed to explore the spread of COVID-19 as well as omicron variant with the correlation of heart attack. This article provides an eradication of the COVID-19 and omicron variant as well as the population who have heart attack after post COVID-19 of these epidemic diseases. Then the existence and uniqueness of global positive solution are studied. Ensuing, In this article, we classify COVID-19 virus and omicron variant which go to extinction and become persistent in mean. By using Lyapunov function, the existence of ergodic stationary distribution are established. Later from the persistent disease as well as extinction, heart disease are ready to develop in the human body. By Eventually, an optimal control strategies are introduced in the form of stochastic post COVID-19 delayed model to control two different types of virus. Finally, the numerical simulation are presented to determine the behavior of dynamical system by utilizing the real data of United Kingdom. [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 features of a stochastically perturbed two-species predator-prey patch-system with ratio-dependent functional response. We first prove that the system which we investigate has a unique global positive solution. Then, the sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system are presented by establishing a series of suitable Lyapunov functions. [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]

Phytoplankton is an important indicator organism to evaluate the quality of water environment, which may reflect the nutritional level of the sea area. Conversely, environmental conditions can directly affect the community structure of phytoplankton. A stochastic phytoplankton–zooplankton model considering non-degenerate and degenerate diffusions is formulated in this paper. What’s more, in both systems, we obtain sufficient conditions of extinction and ergodicity. Our results demonstrate that the weaker white noise can ensure the permanence of zooplankton and the stronger white noise can lead to the disappearance of zooplankton. Moreover, the threshold value of extinction and persistence can serve as a theoretical basis for controlling phytoplankton and zooplankton. Numerical examples are performed on the analysis results of the two cases to confirm our theoretical results. In addition, we also provide a real-life case study, the validity of the model is verified based on experimental data, and it is shown that fluctuation of external environment and the consumption of phytoplankton by zooplankton will affect the growth and number of phytoplankton. Effectively controlling the quantity of phytoplankton can delay the occurrence of water bloom or red tide. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-dimensional stochastic vegetation–water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution ϖ (·) . In a biological interpretation, the existence of the distribution ϖ (·) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium D ¯ ∗ related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker–Planck equation, we obtain the approximate expression of the distribution ϖ (·) around the equilibrium D ¯ ∗ . In addition, we obtain sufficient condition R 0 E < 1 for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics. [ABSTRACT FROM AUTHOR]

• Both the deterministic model and its corresponding stochastic version are investigated. • The prey kills juvenile predators and it also contributes to the growth of the prey. • We obtain the ergodic property by constructing a suitable stochastic Lyapunov function. In this paper, we investigate the dynamics of a predator-prey model with role reversal and its corresponding stochastic differential equation version. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrences of transcritical bifurcation and Hopf bifurcation are studied. For the stochastic model, we investigate the existence of a unique global positive solution and establish sufficient conditions for the existence of a unique ergodic stationary distribution. Finally, numerical simulations are carried out to verify our analytical results, which indicate that both the intensity of white noise and the predation of the juvenile predators have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions 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 point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate conditions for complete eradication and wiping out of the infectious disease. [ABSTRACT FROM AUTHOR]

Recently, the stochastic θ -threshold method has been proposed to analyze the impact of second-order perturbations on disease persistence. In this paper, considering the complexity of the environmental variations in the actual situation, we construct and study a stochastic staged progression HIV/AIDS infection model with third-order perturbations. First, we generalize the stochastic θ -threshold method to eliminate the effect of third-order perturbation of the stochastic model and obtain a critical value R 0 S related to the basic reproduction number R 0 of its deterministic model. By constructing several suitable θ -Lyapunov functions, it is theoretically proved that the stochastic model has a unique ergodic stationary distribution if R 0 S > 1 . In a biological interpretation, the existence of a stationary distribution implies the long-term persistence of HIV/AIDS. Moreover, for some small perturbations, by means of the exponential martingale inequality, we establish sufficient condition ϱ 0 < 0 for disease extinction. Then, for some large perturbations, another sufficient condition R 0 C < 1 for disease extinction of the stochastic model is also derived. For completeness, we provide some examples and numerical simulations to verify our analytical results. What is more, the generalized stochastic θ -threshold method can be successfully applied to the dynamical analysis of other complex high-dimensional epidemic models with nonlinear perturbations. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long term. Some numerical simulations are introduced to illustrate our theoretical result. [ABSTRACT FROM AUTHOR]

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine‐induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ℛ0C related to the basic reproduction number ℛ0 of the deterministic model. The key aim of this paper is to generalize the θ‐stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ‐stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ℛ0C>1. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper. [ABSTRACT FROM AUTHOR]

In this manuscript, a new approach of a stochastic predator-prey interaction with protection zone for the prey is developed and studied. The considered mathematical model consists of a system of two stochastic differential equations, SDEs, describing the interaction between the prey and predator populations where the prey exhibits a social behavior called also by "herd behavior." First, according to the theory of the SDEs, some properties of the solution are obtained, including: the existence and uniqueness of the global positive solution and the stochastic boundedness of the solutions. Then, the sufficient conditions for the persistence in the mean and the extinction of the species are established, where the extinction criteria are discussed in two different cases, namely, the first case is the survival of the prey population, while the predator population goes extinct; the second case is the extinction of all prey and predator populations. Next, by constructing a suitable stochastic Lyapunov function and under certain parametric restrictions, it has been proved that the system has a unique stationary distribution which is ergodic. Finally, some numerical simulations based on the Milstein's higher-order scheme are performed to illustrate the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical behaviors of a model of syphilis transmission disturbed by both white noises and telegraph noises. Multiple infections and treatment stages are considered, which include and extend the existing ones. The existence and ergodicity of the stationary distribution are obtained by constructing a suitable Lyapunov function, which determines a critical value R 0 ∗ corresponding to the control reproduction number R c of the corresponding determined system. In addition, a sufficient criteria for extinction of the diseases is derived. Finally, the numerical simulations illustrate our theoretical results, which show that, the stronger white noises can result in the extinction of the diseases and telegraph noises can strength the stability of the system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for SIR epidemic system with multi noises { d S (t) = [ a 1 − b 1 S (t) − I (t) f (S (t) , I (t)) ] d t + σ 1 S (t) d B 1 (t) − I (t) g (S (t) , I (t)) d B 3 (t) , d I (t) = [ − b 2 I (t) + I (t) f (S (t) , I (t)) ] d t + σ 2 I (t) d B 2 (t) + I (t) g (S (t) , I (t)) d B 3 (t). Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

• Classify the asymptotic behavior SIR models with noise perturbing to linear terms. • Construct a threshold value based on coefficients. • The negativity of threshold implies the extinction of the model at exponential rate. • If it is positive, the solution converges to unique stationary measure in total variation. • We can use this technique to classify models with noise perturbing to non linear terms. In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations. [ABSTRACT FROM AUTHOR]

In this paper, we focus on ergodicity and transience of the Cox–Ingersoll–Ross interest rate model driven by stable processes with random switching. Under some assumptions, we prove that the interest rate process and switching process has a unique stationary distribution. In addition, some sufficient conditions for transience of such interest rate model are given. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing, a real-valued threshold. We show that if < 0, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value > 0, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all t 2 [0;1), approximation in a long but finite interval [0; T] can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a two prey and one predator model with logistic growth in prey species have been considered in presence of stochastic perturbation. The cooperation among the preys against predation is assumed. Necessary conditions for asymptotic behavior around the equilibrium points of its deterministic system with and without stochasticity have been discussed. Finally, to illustrate the analytical results, numerical simulations have been performed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the ergodicity and transience of the Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the theory of M -matrices, we give some necessary and sufficient conditions for ergodicity of the CIR interest rate model with Markov switching. Besides, we show that the transition semigroup converges to the stationary distribution at an exponential rate in the Wasserstein distance. Finally, two examples are presented to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the solutions to the model. Then we obtain sufficient conditions for extinction of the disease in two cases, that is, the first case is that the susceptible population survival and infected and exposed populations extinction; the second case is that all the populations extinction. • A stochastic SEI epidemic model with saturation incidence and logistic growth is proposed and studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • We obtain sufficient conditions for extinction of the disease in two cases. [ABSTRACT FROM AUTHOR]

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations. [ABSTRACT FROM AUTHOR]

In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract A one predator and two independent preys system with stochastic perturbations is considered. First, we show that there is a positive solution of this system. Then, we investigate the long time behavior of this system. Sufficient criteria for extinction and persistence in time average are established. Moreover, it is also shown that this system has a stationary distribution and it is ergodicity. Finally, examples and simulations are carried on to verify these results. Highlights • A stochastic one predator and two independent preys system is considered. • It is difficult for the model ignoring the effect of predator's intraspecific competition. • We establish sufficient and nearly necessary criteria for extinction and persistence. • We also show that this system has a stationary distribution and it is ergodicity. • We give some simulations to verify our results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a regime-switching predator–prey model with anti-predator behaviour and higher-order perturbations. We obtain the ergodic property by constructing a suitable stochastic Lyapunov function with regime-switching, which provides us a biological perspective of cycling phenomena of a population system, and can better describe the stochastic persistence of a population system in practice. We find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions. Highlights • A regime-switching predator–prey model with anti-predator behaviour is studied. • We obtain the ergodic property by constructing a suitable stochastic Lyapunov function. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

Abstract In this work we investigate the existence of a unique global positive solution for a stochastic SIRS epidemic model with a general awareness-induced incidence. We also study the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Numerical simulations are presented to confirm the theoretical results. Highlights • We formulate a stochastic epidemic model with a general awareness-induced incidence. • This model admits a unique global positive solution. • Sufficient conditions for fluctuations around equilibriums are obtained. • sufficient conditions for the existence of an ergodic stationary distribution are obtained. • Numerical simulations are introduced to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Taking white noises, Markovian switching and Lévy jumps into account, an n − species Lotka–Volterra competitive model with random perturbations is proposed and studied. Sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the model are established. These sufficient conditions are sharp in some cases. Some interesting results are revealed: the white noises and Lévy jumps could make the stationary distribution disappear as well as appear; the Markovian switching could make the stationary distribution appear. Some numerical simulations are also introduced to illustrate the main results. [ABSTRACT FROM AUTHOR]

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, α (t) , is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold λ ∈ R that depends on the model parameters. It turns out that if λ < 0 the number I (t) of infected individuals converges to zero exponentially fast (extinction). However, if λ > 0 , the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we present some simulations to illustrate our theoretical results. We note that the unique equilibrium of the original ordinary differential equation model is globally asymptotically stable under appropriate conditions of the parameter value f, while the stochastic model is ergodic regardless of the value of f.

This paper is concerned with a stochastic mutualism system with toxicant substances and saturation terms. We obtain the sufficient conditions for the existence of a unique stationary distribution to the equation and it has an ergodic property. It is interesting and surprising that toxicant substances have no effect on the stationary distribution of the stochastic model. Simulations are also carried out to confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic predator-prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]