| Hepatitis B virus(HBV)infection is a major healthy problem worldwide, therefore, the analysis of the spread of HBV is more and more important. In this paper, we formulate three models from the macro and micro perspective respectively and analyze the stability of the system near the equilibriums.Firstly, we study the macro HBV model with vaccination and vertical transmission according to the propagation rules and current prevention. We de?ne the basic reproductive number R0. We prove that disease-free equilibrium is global asymptotically stable when R0< 1, the disease will die out. We prove that endemic equilibrium is local asymptotically stable when R0> 1. We make some numerical simulation to verify the conclusion.Secondly, we introduce an improved hepatitis B virus(HBV) model with drug therapy.We take logistic hepatitis growth and standard incidence. Besides, infection liver cells may be return to the uninfected state by loss all covalently closed circular DNA(cccDNA) from their nucleus. Therefore, we take into account the cure of infected cells. The basic reproductive number R0 determines the extinction and the persistence of virus infection. When R0< 1,the disease-free equilibrium is global asymptotically stable and the infection becomes extinct eventually; When R0> 1, using the theory of stability and bifurcation, we obtain that the unique endemic equilibrium exists and a Hopf bifurcation will occur under certain conditions.Numerical simulations verify the above theory.Finally, we investigate a HBV model with logistic hepatitis growth and time delay. We rede?ne the basic reproductive number R0. When R0< 1, we prove that the disease-free equilibrium is global asymptotically stable for any delay. When R0> 1, it is shown that there exists threshold value τ0. When τ < τ0, endemic equilibrium is local asymptotically stable under some conditions; when τ > τ0, endemic equilibrium loses its stability; whenτ = τ0, a Hopf bifurcation occurs. We make some sensitivity analysis. |
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