Liangjian Hu

A stochastic differential equation SIS epidemic model

In this paper we extend the classical susceptible-infected-susceptible 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

Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations

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Robust Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations

. We then prove that this SDE has a unique global positive solution

Asymptotic behavior and stability of a stochastic model for AIDS transmission

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Perturbation analysis of fuzzy linear systems

and establish conditions for extinction and persistence of

Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients

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Delay dependent stability of highly nonlinear hybrid stochastic systems

. 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.

Asymptotic moment boundedness of the stochastic theta method and its application for stochastic differential equations

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.

Risk Analysis for AIDS Control Based on a Stochastic Model with Treatment Rate

One of the important issues in the study of hybrid stochastic differential delay equations (SDDEs) is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness. In the study of asymptotic properties, the robust stability has received a great deal of attention. The theory of robust stability shows how much perturbation a given stable hybrid SDDE can tolerate so that its perturbed system remains stable. Almost all results so far on the robust stability require that the underlying SDDEs be either linear or nonlinear with linear growth condition. However, little is known on the robust stability of nonlinear hybrid SDDEs without the linear growth condition, which is one of the key topics in this paper. The other key topic is the robust boundedness. The aim here is to answer the question: how much perturbation can a given asymptotically bounded hybrid SDDE tolerate so that its perturbed system remains asymptotically bounded?

Convergence rate and stability of the truncated Euler-Maruyama method for stochastic differential equations

A stochastic model for AIDS transmission is presented in this paper. The existence and uniqueness of its solution is proved, and the solution belongs to an interval satisfying the actual condition. Furthermore, the disease equilibrium can be demonstrated to be existent, and both its almost surely exponential stability and its pth moment exponential stability can be concluded. Based on parameter analysis, the first quadrant can be integrated into several parts according to the comparison between almost surely exponential stability and pth moment exponential stability. And the potential meaning of the stability is further demonstrated. Finally, based on the above theoretical analysis, some conditions for AIDS to decrease and then to die out are obtained by computer simulation.