MBE
A note for the global stability of a delay differential equation of hepatitis B virus infection
Bao-Zhu Guo Li-Ming Cai
The global stability for a delayed HIV-1 infection model is investigated. It is shown that the global dynamics of the system can be completely determined by the reproduction number, and the chronic infected equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [S. A. Gourley,Y. Kuang and J.D.Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2(2008), 140-153].
keywords: HBV model time delay. Lyapunov functional global stability
DCDS-B
Global stability for a heroin model with two distributed delays
Bin Fang Xue-Zhi Li Maia Martcheva Li-Ming Cai
In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one.
keywords: distributed delay Heroin model Lyapunov function global stability. basic reproduction number
DCDS-B
Global stability of the dengue disease transmission models
Jing-Jing Xiang Juan Wang Li-Ming Cai
In this paper, we further investigate the global stability of the dengue transmission models. By using persistence theory, it is showed that the disease of system uniformly persists when the basic reproduction number is larger than unity. By constructing suitable Lyapunov function methods and LaSalle Invariance Principle, we show that the unique endemic equilibrium of the model is always globally asymptotically stable as long as it exists.
keywords: basic reproduction number Lyapunov function global stability. Mass action incidence

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