Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.     

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.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

Asymptotic dynamics of a deterministic and stochastic predator–prey model with disease in the prey species

In this paper, we discuss a predator–prey model with the Beddington–DeAngelis functional response of predators and a disease in the prey species. At first we study permanence and global stability of a positive equilibrium for the deterministic version of the model. Then we include a stochastic perturbation of the white noise type. We analyse the influence of this stochastic perturbation on the systems and prove that the positive equilibrium is also globally asymptotically stable in this case. The key point of our analysis is to choose appropriate Lyapunov functionals. We point out the differences between the deterministic and stochastic versions of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

具有接种的确定性和随机SIS流行病模型的全局稳定性

We study the stability of endemic equilibriums of the deterministic and stochastic SIS epidemic models with vaccination. The deterministic SIS epidemic model with vaccination was proposed by Li and Ma(2004), for which some sufficient conditions for the global stability of the endemic equilibrium were given in some earlier works. In this paper, we first prove by Lyapunov function method that the endemic equilibrium of the deterministic model is globally asymptotically stable whenever the basic reproduction number is larger than one. For the stochastic version, we obtain some sufficient conditions for the global stability of the endemic equilibrium by constructing a class of different Lyapunov functions.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

A Notion of Stochastic Input-to-State Stability and Its Application to Stability of Cascaded Stochastic Nonlinear Systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.     

Stochastic stability analysis for joint process driven and networked hybrid systems

The stochastic stability and impulsive noise disturbance attenuation in a class of joint process driven and networked hybrid systems with coupling delays (JPDNHSwD) has been investigated. In particular, there are two separable processes monitoring the networked hybrid systems. One drives inherent network structures and properties, the other induces random variations in the control law. Continuous dynamics and control laws in networked subsystems and couplings among subsystems change as events occur stochastically in a spatio-temporal fashion. When an event occurs, the continuous state variables may jump from one value to another. Using the stochastic Lyapunov functional approach, sufficient conditions on the existence of a remote time-delay feedback controller which ensures stochastic stability for this class of JPDNHSwD are obtained. The derived conditions are expressed in terms of solutions of LMIs. An illustrative example of a dynamical network driven by two Markovian processes is used to demonstrate the satisfactory control performance.     

Global stability of two-group SIR model with random perturbation

In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings.     

Almost sure exponential stability of stochastic reaction diffusion systems

The Lyapunov direct method, as the most effective measure of studying stability theory for ordinary differential systems and stochastic ordinary differential systems, has not been generalized to research concerning stochastic partial differential systems owing to the emptiness of the corresponding Ito differential formula. The goal of this paper is just employing the Lyapunov direct method to investigate the stability of Ito stochastic reaction diffusion systems, including asymptotical stability in probability and almost sure exponential stability. The obtained results extend the conclusions of [X.X. Liao, X.R. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Analysis and Applications 14 (2) (1996) 165-185; X.X. Liao, S.Z. Yang, S.J. Cheng, Y.L. Fu, Stability of general neural networks with reaction-diffusion, Science in China (F) 44 (5) (2001) 389-395].     

Dynamics of stochastic 2D Navier-Stokes equations

In this paper, we study the dynamics of a two-dimensional stochastic Navier-Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2D Navier-Stokes equation generate a perfect and locally compacting C1,1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier-Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.     

Stability in Terms of Two Measures for Stochastic Differential Equations with Markovian Switching

Abstract

In this paper, we investigate the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Our new theory can not only be used to show a given system to be stochastically stable in the classical sense, but can also be used to deal with some situations where the classical stability theory is not applicable.     

Orbits in a stochastic Goodwin–Lotka–Volterra model

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka–Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study a stochastic version, with Brownian noise introduced via a heterogeneous productivity factor. Existence conditions for a solution to the system are provided. We prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurrent domains. Numerical insights are provided.     

Stochastic piecewise affine control with application to pitch control of helicopter

In this paper, at first the stability condition which gives an upper stochastic bound for a class of Stochastic Hybrid Systems (SHS) with deterministic jumps is derived. Here, additive noise signals are considered that do not vanish at equilibrium points. The presented theorem gives an upper bound for the second stochastic moment or variance of the system trajectories. Then, the linear case of SHS is investigated to show the application of the theorem. For the linear case of such stochastic hybrid systems, the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Then utilizing the stability theorem, an output feedback controller design procedure is proposed which requires the Bilinear Matrix Inequalities (BMI) to be solved. Next, the pitch dynamics of a helicopter is approximated with a set of linear stochastic systems, and the proposed controller is designed for the approximated model and implemented on the main nonlinear system to demonstrate the effectiveness of the proposed theorem and the control design method.     

具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性

研究了一类具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性.利用时变区间矩阵的分解技术、矩阵的Kronecker积的性质和Lyapunov函数法,得到了该系统均方鲁棒稳定的几个充分性条件.通过一个数值例子说明了所得的这些充分性条件的有效性和实用性.     

Asymptotic behavior for second order stochastic evolution equations with memory

In this paper, a general second order integro-differential evolution equation with memory driven by multiplicative noise is considered. We prove the existence of global mild solution and asymptotic stability of the zero solution using Lyapunov function techniques. Moreover, we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations.