Houbao Xu

EXPONENTIAL STABILITY OF A REPARABLE MULTI-STATE DEVICE

The exponential stability of a multi-state device is discussed in this paper. We present that the C0-semigroup generated by the system operator is quasi-compact and irreducible. It is known that 0 is a simple eigenvalue of the system operator. In combination with this, we obtain that the time-dependent solution exponentially converges to the steady-state solution, which is the positive eigenfuction corresponding to the simple eigenvalue 0.

Availability optimisation of repairable system with preventive maintenance policy

This article investigates the steady availability of the repairable system with preventive maintenance policy. By total probability formula, we describe the system as an abstract mathematical model and present the steady-state solution. Then we get the steady availability formula of the system. By analyzing the different monotonicity of the failure rate function, we discuss the well-posedness of the optimal time interval of executing the preventive maintenance. As a result, the steady availability of the system is optimized. Some numerical examples are given to show the effectiveness of the method presented in the article.

Asymptotic property of a reparable multi-state device

This paper is devoted to studying the existence, uniqueness and asymptotic stability of a multi-state devices time-dependent solution. C 0 semigroup theory is used to prove the existence of a unique non-negative solution of the device. Moveover, by analyzing the spectrum of the system operator generated by the device, this paper proves that 0 is the unique spectral point on the imaginary axis and the other spectra lie in the left half plane. As a result, the asymptotic behavior of a multi-state device is obtained.

Modelling and analysis of repairable systems with preventive maintenance

The time-dependent solution of a kind of repairable system with preventive maintenance is investigated in the paper. With total probability formula, we show that the behavior of the system can be described as a group of ordinary differential equations coupled with partial differential equations, which can be formulated as a time delay equation in an appropriate Banach space. Based on the time-delay equation, this paper presents a difference scheme as an approximating method to solve the time-dependent solution which is necessary for analyzing the instantaneous availability of the repairable system. Some numerical examples are shown to illustrate the effectiveness of this approach.

Stability on coupling SIR epidemic model with vaccination

We develop a mathematical model for the disease which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age. Applying semigroup theory and so forth, we investigate the existence of equilibria. We also discuss local stability of steady states.

GLOBAL ASYMPTOTIC STABILITY FOR THE SIV EPIDEMIC MODEL WITH IMPULSIVE VACCINATION AND INFECTION AGE

In this paper, we study the application of a pulse vaccination strategy to eradicate hepatitis B and C modeled by SIV epidemic model. Since infection age is an important factor of hepatitis progression, we incorporate the infection age into the model. In this model, we consider the infectiousness of latent individuals. We derive the condition in which eradication solution is a global attractor, this condition depends on pulse vaccination proportion p and interpulse time T. We also obtain the condition of the global asymptotic stability of the solution. The condition shows that large enough pulse vaccination proportion and relatively small interpulse time lead to the eradication of hepatitis B and C. Moreover the results of the theoretical study might be instructive to the epidemiology of HIV.

Asymptotic stability of a repairable system with imperfect switching mechanism

This paper studies the asymptotic stability of a repairable system with repair time of failed system that follows arbitrary distribution. We show that the system operator generates a positive C0-semigroup of contraction in a Banach space, therefore there exists a unique, nonnegative, and time-dependant solution. By analyzing the spectrum of system operator, we deduce that all spectra lie in the left half-plane and 0 is the unique spectral point on imaginary axis. As a result, the time-dependant solution converges to the eigenvector of system operator corresponding to eigenvalue 0.

Stability results on age-structured SIS epidemic model with coupling impulsive effect

We develop an age-structured epidemic model for malaria with impulsive effect, and consider the effect of blood transfusion and infected-vector transmission. Transmission rates depend on age. We derive the condition in which eradication solution is locally asymptotically stable. The condition shows that large enough pulse reducing proportion and relatively small interpulse time lead to the eradication of the diseases.

Availability analysis for software system with intrusion tolerance

This paper is devoted to analyzing the instantaneous availability of a typical software system with intrusion tolerance. By formulating the system with a couple of ordinary differential and partial differential equations, this paper describes the system as a time-delay partial differential equation. Based on the time-delay model, both steady-state availability and instantaneous availability are investigated. The optimal policy for preventive patch management to maximize the steady-state availability of the software system is obtained, and its related availability criterions are also presented. Employing the finite difference scheme and Trotter-Kato theorem, we converted the time-delay partial equation into a time-delay ordinary equation. As a result, the instantaneous availability of the system is derived. Some numerical results are given to show the effectiveness of the method presented in the paper.

Numerical analysis of a reparable multi-state device

A numerical scheme is formulated for approximating the dynamic behavior of a reparable multi-state device, which can be described as a distributed parameter system of coupled partial and ordinary hybrid equations. The convergence issues are established by applying the Trotter-Kato Theorem, and simulation results show the effectiveness of the proposed scheme.