In this paper we extend the classical SIS epidemic model from a deterministic framework to a stochastic one, and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals I(t). We then prove that this SDE has a unique global positive solution I(t) and establish conditions for extinction and persistence of I(t). We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real life diseases.

One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness (see e.g. [5], [10], [11], [13], [14], [15], [17], [19] and [21]). The method of Lyapunov functions is one of the most powerful techniques in the study of stability and boundedness. So far, most of the results in this area do not only require the Lyapunov functions in different modes have the same feature (e.g. polynomials with the same degree) but also that the diffusion operator in different modes be bounded by the same type of functions. These requirements are restrictive and often cannot be met by those hybrid SDDEs that have different nonlinear structures in different modes. To study the stability and boundedness of such hybrid SDDEs, we will in this paper use different types of Lyapunov functions (e.g. polynomials with different degrees) for different modes. Moreover, the condition on the diffusion operator is relaxed significantly

Recently, Mao (2013) discusses the mean-square exponential stabilization of continuous-time hybrid stochastic differential equations by feedback controls based on discrete-time state observations. Mao (2013) also obtains an upper bound on the duration τ between two consecutive state observations. However, it is due to the general technique used there that the bound on τ is not very sharp. In this paper, we will consider a couple of important classes of hybrid SDEs. Making full use of their special features, we will be able to establish a better bound on τ . Our new theory enables us to observe the system state less frequently (so costs less) but still to be able to design the feedback control based on the discrete-time state observations to stabilize the given hybrid SDEs in the sense of mean-square exponential stability

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There are lots of papers on the delay dependent stability criteria for differential delay equations (DDEs), stochastic differential delay equations (SDDEs) and hybrid SDDEs. A common feature of these existing criteria is that they can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In other words, there is so far no delay-dependent stability criterion on nonlinear equations without the linear growth condition (we will refer to such equations as highly nonlinear ones). This paper is the first to establish delay dependent criteria for highly nonlinear hybrid SDDEs. It is therefore a breakthrough in the stability study of highly nonlinear hybrid SDDE