Weiming Wang

A stochastic SIRS epidemic model with nonlinear incidence rate

A stochastic SIRS model with ratio-dependent incidence rate is developed.The global dynamics of the deterministic model is shown.The stochastic dynamics of the SDE model is given.The existence of a unique stationary distribution of the SDE model is displayed. In this paper, we investigate the global dynamics of a general SIRS epidemic model with a ratio-dependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the basic reproduction number R0 determines whether there is an endemic outbreak or not: if R0 1, the disease persists. For the stochastic model, we show that its related reproduction number R0S can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness and permanence/extinction. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.

Periodic behavior in a FIV model with seasonality as well as environment fluctuations

Abstract In this paper, the dynamics of a stochastic FIV model with seasonality are investigated analytically and numerically. Sufficient criteria for extinction and weak persistence of the FIV disease in the mean are established. In the case of weak persistence in the mean, there exists at least one periodic solution, which means that the susceptible and infective individuals will coexist and exhibit periodicity in the long run. Via the numerical simulations, it is also shown that the stationary distributions are governed by two parameters.

Environmental variability in a stochastic epidemic model

In this paper, we investigate the stochastic dynamics of a simple epidemic model incorporating the mean-reverting Ornstein–Uhlenbeck process analytically and numerically. We define two threshold parameters, the stochastic demographic reproduction number Rds and the stochastic basic reproduction number R0s, to utilize in identifying the stochastic extinction and persistence of the disease. We find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion: the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease, the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease. Furthermore, via numerical simulations, we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease. Our results show that mean-reverting process is a well-established way of introducing stochastic environmental noise into biologically realistic population dynamic models.

Spatiotemporal complexity in a predator--prey model with weak Allee effects

In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.

Stochastic dynamics of feline immunodeficiency virus within cat populations

Abstract In this paper, we investigate the basic features of a simple susceptible-infected (SI) epidemic model of Feline immunodeficiency virus (FIV) within cat populations in presence of multiplicative noise terms to understand the effects of environmental driving forces on the disease dynamics. The value of this study lies in two aspects. Mathematically, we propose three threshold parameters, R s h , R 1 and R 2 to utilize in identifying the stochastic extinction and persistence. In the case of stochastic persistence, we prove that there is a stationary distribution. Based on the statistical data for rural cat populations Barisey-la-Cote in France, we perform some numerical simulations to verify/extend our analytical results. Epidemiologically, we find that: (1) Large environment fluctuations can suppress the outbreak of FIV; (2) The distributions are governed by R s h ; (3) White noise perturbations of the birth rate for infectious cats (i.e., the vertical transmission) can induce the susceptible-free dynamics.

Global stability of the steady states of an epidemic model incorporating intervention strategies

In this paper, we investigate the global stability of the steady states of a general reaction-diffusion epidemiological model with infection force under intervention strategies in a spatially heterogeneous environment. We prove that the reproductoin number R0 can be played an essential role in determining whether the disease will extinct or persist: if R0< 1, there is a unique disease-free equilibrium which is globally asymptotically stable; and if R0 >1, there exists a unique endemic equilibrium which is globally asymptotically stable. Furthermore, we study the relation between R0 with the diffusion and spatial heterogeneity and find that, it seems very necessary to create a low-risk habitat for the population to effectively control the spread of the epidemic disease. This may provide some potential applications in disease control.

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

and Applied Analysis Volume 2014, Article ID 214514, 3 pages http://dx.doi.org/10.1155/2014/214514 2 Abstract and Applied Analysis the evaluation of the interventions and control efforts of the infectious disease. C. Yan and J. Jia study the local stability of the disease-free, endemic equilibria and Hopf bifurcation of a delayed SIR epidemic model with information variables and limitedmedical resources. Y. Pei et al. propose a delay SIR epidemic model with difference in immunity and successive vaccination and obtain that the basic reproduction number governs the dynamic behavior of the system. C.Huang andA. Fan study the relationship between antimicrobial resistance and the concentration of antibiotics with dynamical model of competitive population and indicate that long-term highstrength antibiotic treatment and prevention can induce the extinction of susceptible strain. M. Li and X. Liu investigate the disease dynamics of an SIR epidemic model with nonlinear incidence rate and show that the global properties of the system depend on both properties of these general functions. C. Dai et al. investigate the dynamic behavior of a viral infectionmodel with general contact rate between susceptible host cells and free virus particles and give the local stability of the equilibria. M. A. Obaid and A. M. Elaiw propose and analyze two virus infection models with antibody immune response and chronically infected cells and give the global asymptotic stability of all steady states of themodels. G. Li and G. Li consider an SIR endemic model in which the contact transmission function is related to the number of infected population and show that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. H.-F. Huo and G.-M. Qiu study the dynamics of a malaria model and show that the disease-free equilibrium is globally asymptotically stable if R 0 < 1, and the system is uniformly persistent if R 0 > 1. C. Chen and Y. Xiao propose a mathematical model to consider the effects of saturated diagnosis and vaccination on HIV/AIDS infection and find that there exists a backward bifurcationwhich suggests that the disease cannot be eradicated even if the basic reproduction number is less than unity. When the basic reproduction number is greater than unity, the system is uniformly persistent. The findings suggest that increasing vaccination rate and vaccine efficacy and enhancing interventions like reducing share injectors can greatly reduce the transmission of HIV among IDUs in Yunnan province, China. J. Cui and Z. Wu consider an SIRS model incorporating a general nonlinear contact function andfind thatwhen the basic reproductionnumberR 0 < 1, the disease-free equilibrium is locally asymptotically stable, while when R 0 > 1, there is a unique endemic equilibrium that is locally asymptotically stable. P. Bi and H. Xiao consider a tumor-immune competitionmodel with delay which consists of two-dimensional nonlinear differential equation and give the general formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension-1 and codimension-2 bifurcations, including Hopf bifurcation and BT bifurcation. L. Wang et al. study a class of discrete SIRS epidemic models with nonlinear incidence rate and find that if basic reproduction number R 0 < 1, then the disease-free equilibrium is globally asymptotically stable, and if R 0 > 1, then the model has a unique endemic equilibrium and when some additional conditions hold, the endemic equilibrium is also globally asymptotically stable. X. Zhou and X. Shi analyze a discrete-time-delay differential mathematical model that describes HIV infection of CD4T cells with drugs therapy and give the stability of the two equilibria and the existence ofHopf bifurcation at the positive equilibrium. K. Wang et al. propose a patch model for echinococcosis due to dogs migration and show that the dynamics of themodel can be completely determined byR 0 . If R 0 < 1, the disease-free equilibrium is globally asymptotically stable. When R 0 > 1, the model is permanent and endemic equilibrium is globally asymptotically stable. Seven papers are developed to discuss the stochastic dynamics of population models. S. Zhao and M. Song consider the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of a stochastic predator-prey system with BeddingtonDeAngelis functional response and stage structure. X. Ji and S. Yuan study the dynamics of a delayed stochastic model simulating wastewater treatment process and give the sufficient conditions for the stochastic stability of its positive equilibrium. F. Rao investigates an SIR epidemic model with stochastic perturbations and gives the existence of global positive solutions, stochastic boundedness, and permanence. L. Wang et al. study the stochastic dynamics of an SIRS epidemic model incorporating media coverage and find that if the intensity of noise is large, then the disease is prone to extinction, which can provide us with some useful control strategies to regulate disease dynamics. J. Zhao et al. investigate a stochastic SI epidemic model in the complex networks and show that the solution will oscillate around the disease-free equilibrium of deterministic system when R 0 ≤ 1, while it is persistent when R 0 > 1. F. Rao et al. investigate a Hassell-Varley type predator-prey model with stochastic perturbations and find some sufficient conditions for stochastically asymptotical boundedness, permanence, persistence inmean, and extinction of the solution. L. Zu et al. analyze the influence of stochastic perturbations on a singlespecies logistic model with the population’s nonlinear diffusion among n patches and give the sufficient conditions for stochastic permanence and persistence in mean, stationary distribution, and extinction. Four papers focus on the travelingwave solutions. X. Tian and R. Xu investigate a delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence and derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state. X. Wu et al. establish the existence of traveling wave solutions and small amplitude traveling wave train solutions of a reactiondiffusion system based on a predator-prey model incorporating a prey refuge and analyze the dynamic behavior of this model in the three-dimensional phase space. T. Zhang and Q. Gou consider the minimal wave speed of bacterial colonymodel with saturated functional response and give the existence and nonexistence of the traveling wave solutions. T. Zhang et al. investigate the spreading speed of a reactiondiffusion cholera model and find that there exists a traveling wave solution. Four papers study the impulsive dynamics of population models. M. Zhao and C. Dai investigate the population Abstract and Applied Analysis 3and Applied Analysis 3 dynamics of a three-species ecological system with impulsive effect and give the conditions for the system to be permanent when the number of predators released is less than some critical value. In particular, the authors find that less beneficial prey can support the predator alone when more beneficial prey goes extinct. J. Li considers a class of neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays and gives the criteria on global exponential stability. M. Zhao et al. investigate the dynamics of a Holling-Tanner predator-prey system with state-dependent impulsive effects and give the existence of periodic solution of the system with statedependent impulsive effects. Z. Luo investigates the existence of multiple positive periodic solutions of a class of impulsive functional differential equations with a parameter. Five papers consider the stationary patterns in the reaction-diffusion equations. L. Zhang focuses on the pattern formation of a ratio-dependent food chain model and finds that the model dynamics exhibits complex pattern replication. X. Lian et al. investigate the spatiotemporal dynamics of a bacterial colony model and derive the conditions for Hopf and Turing bifurcations. L. Zhang and Z. Li focus on a spatially extended Holling-type IV predator-prey model that contains some important factors, such as noise (random fluctuations), external periodic forcing, and diffusion processes, and find that noise or external periodic forcing can induce instability and enhance the oscillation of the species density, and the cooperation between noise and external periodic forces inherent to the deterministic dynamics of periodically driven models gives rise to the appearance of a rich transport phenomenology. Y. Yuan et al. investigate the disease dynamics of a reaction-diffusion epidemic model and give the conditions of the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns. Y. Wang et al. investigate a nonlinear reaction-advection-diffusion model of the interaction between nutrients and plankton and find that if the sinking velocity exceeds a certain critical value, the stable state becomes unstable and the wavelength of phytoplankton increases with the increase of sinking velocity. Three papers investigate the predation dynamics. Y. Gao and S. Liu investigate a predator-prey model with dispersal for both predator and prey among n patches and derive sufficient conditions under which the positive coexistence equilibrium of this model is unique and globally asymptotically stable if it exists. X. Feng et al. formulate and investigate a nonautonomous predator-prey model with infertility control in the prey and give the conditions for the permanence and extinction of fertility prey and infertility prey. X. Fan et al. study the global property in a delayed periodic predator-prey model with stage-