The threshold of a stochastic SIRS epidemic model with saturated incidence

In this paper, we investigate the dynamics of a stochastic SIRS epidemic model with saturated incidence. When the noise is small, we obtain a threshold of the stochastic system which determines the extinction and persistence of the epidemic. Besides, we find that large noise will suppress the epidemic from prevailing.     

Threshold dynamics in a stochastic SIRS epidemic model with nonlinear incidence rate

We discuss the dynamic of a stochastic Susceptible-Infectious-Recovered-Susceptible (SIRS) epidemic model with nonlinear incidence rate.The crucial threshold $\tilde{R}_0$ is identified and this will determine the extinction and persistence of the epidemic when the noise is small. We also discuss the asymptotic behavior of the stochastic model around the endemic equilibrium of the corresponding deterministic system. When the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. Finally, these results are illustrated by computer simulations.     

Stability analysis of a stochastic delayed SIR epidemic model with general incidence rate

In this paper, a stochastic delayed epidemic model with a generalized incidence rate is proposed and discussed. The positivity of solutions is established. A linearized form of the model is given and the stability conditions of the endemic equilibrium are obtained by using the technique of Lyapunov functionals.     

一类具有扩散和Beddington-DeAngelis反应函数的病毒模型的全局稳定性

研究了一类具有扩散和Beddington-DeAngelis反应函数的病毒模型.通过构造Lyapunov函数,证明了模型的感染平衡点是全局渐近稳定的.     

The effect of a generalized nonlinear incidence rate on the stochastic SIS epidemic model

In this paper, we aim to analyze the classical SIS epidemic model with a generalized force of infection (including nonmonotonic cases), where the transmission rate is perturbed by white noise. Using Feller's test for explosions, we prove that the disease dies out with probability one without any restriction on the model parameters.     

Global properties of a discrete viral infection model with general incidence rate

In this paper, we propose a discrete viral infection model with a general incidence rate. The discrete model is derived from a continuous case by using a 'mixed' Euler method, which is a mixture of both forward and backward Euler methods. We prove that the mixed Euler method preserves the qualitative properties of the corresponding continuous system, such as positivity, boundedness, and global behaviors of solutions. Furthermore, the model and mathematical results presented in another previous study are extended and generalized. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate

In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.     

The threshold between permanence and extinction for a stochastic Logistic model with regime switching

A stochastic logistic model under regime switching is proposed and investigated. Sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The threshold between weak persistence and extinction is obtained. Then we show that this threshold also is the threshold between stochastic permanence and extinction under a simple additional condition. The results show that firstly, the stationary probability distribution of the Markov chain plays a key role in determining the permanence and extinction of the population. Secondly, different types of environmental noises have different effects on the permanence and extinction of the population. Thirdly, the more the stochastic noises, the easier the population goes to extinction.     

Asymptotic properties of a HIV-1 infection model with time delay

Based on some important biological meanings, a class of more general HIV-1 infection models with time delay is proposed in the paper. In the HIV-1 infection model, time delay is used to describe the time between infection of uninfected target cells and the emission of viral particles on a cellular level as proposed by Herz et al. [A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247-7251]. Then, the effect of time delay on stability of the equilibria of the HIV-1 infection model has been studied and sufficient criteria for local asymptotic stability of the infected equilibrium and global asymptotic stability of the viral free equilibrium are given.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

一类具有Beddington-DeAngelis型功能反应函数的HIV病毒动力学系统模型的稳定性

基于Nowak等于1996年提出的一类经典的HIV病毒动力学模型,考虑了一类具有Beddington-DeAngelis功能反映函数的HIV病毒动力学模型,并研究了无病毒平衡点的全局稳定性与感染平衡点的局部稳定性等.     

The global dynamics of a model about HIV-1 infection in vivo

A four dimension ODE model is built to study the infection of human immunodeficiency virus (HIV) in vivo. We include in this model four components: the healthy T cells, the latent-infected T cells, the active-infected T cells and the HIV virus. Two types of HIV transmissions in vivo are also included in the model: the virus-to-cell transmission, and the cell-to-cell HIV transmission. There are two possible equilibriums: the healthy equilibrium, and the infected steady state. The basic reproduction number R ₀ is introduced. When R ₀ < 1, the healthy equilibrium is globally stable and when R ₀ > 1, the infected equilibrium exists and is globally stable. Through simulations, we find that, the cell-to-cell HIV transmission is very important for the final outcome of the HIV attacking. Some important clinical observations about the HIV infection situation in lymph node are also verified.      

An SIRS model with a nonlinear incidence rate

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.     

An SIRS model with a nonlinear incidence rate

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.     

The threshold of a non‐autonomous SIRS epidemic model with stochastic perturbations

In this paper, we investigate a stochastic non‐autonomous SIRS (susceptible‐infected‐recovered‐susceptible) model. The extinction and the prevalence of the disease are discussed, and so, the threshold is given. Especially, we show there is a positive nontrivial periodic solution. At last, some examples and simulations are provided to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response

In this paper, we discuss a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results.     

Study of an S-I epidemic model with nonlinear incidence rate: Discrete and stochastic version

Discrete and stochastic version of a susceptible-infective model system with nonlinear incidence rate is investigated. We observe that the discrete system converges to a unique equilibrium point for certain effective transmission rate of the disease and beyond which stability of the system is disturbed. Stochastic analysis suggests that the model system is globally asymptotically stable in probability for certain strengths of white noise. Numerical simulations are also performed to validate the results.     

Optimal threshold density in a stochastic resource management model with pulse intervention

Human activities and agricultural practices are having huge impacts on the development of fishery and land resources through different ways. To model such systems that involve harvesting, an impulsive model of natural resources with a stochastic noise perturbation element is formulated to study the relationship between (a) the maximal expectation of biomass after harvesting or fishing events and (b) the minimal expectation of pest biomass and the number of times pesticide is applied. Using a detailed analytical treatment, time estimation, and numerical demonstrations, we establish that the proposed mechanism is capable of maximizing fish populations at the end of a fishing season and minimizing pest numbers after a crop harvesting season once the intensity of the noise is relatively small. Investigations of the effects of different parameters reveal that theoretical predictions from the new stochastic model accord with those from the deterministic case. Recommendations for Resource Managers

• Various measures can be implemented to manage natural resources, such as adjusting fishing quantity and intensity to maximize fish population.
• In the natural environment, population growth is inevitably affected by the environment noise. So it is important to understand the noise effect to maintain sustainability of resources.
• Investigated methods are useful to converse resources and can be widely applied to control pests.

     

Epidemic transmission on SEIR stochastic models with nonlinear incidence rate

Our interest is to quantify the spread of an infective process with latency period and generic incidence rate that takes place in a finite and homogeneous population. Within a stochastic framework, two random variables are defined to describe the variations of the number of secondary cases produced by an index case inside of a closed population. Computational algorithms are presented in order to characterize both random variables. Finally, theoretical and algorithmic results are illustrated by several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of a nonautonomous model for migratory birds with saturation incidence rate

In this paper, an eco-epidemiological model is newly proposed to consider the role of migratory birds by incorporating the temporal pattern of the avian migration into the model. In the new model, population of birds varies because they are migratory in nature. Under quite weak assumptions, sufficient conditions for the permanence and extinction of the disease is obtained. Moreover, by constructing a Liapunov function, the global attractivity of the model is discussed.