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In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

Hepatitis B is an infectious disease worthy of attention. Considering the incubation period, psychological inhibition factor, vaccine, limited medical resources and horizontal transmission, an SIRS model is proposed to describe hepatitis B transmission dynamics. In order to describe the behavior changes caused by people's psychological changes, the non-monotonic incidence rate is adopted in the model. We use the saturated treatment rate to describe the limited medical resources. Mathematical analysis shows the existence conditions of the equilibria, forward or backward bifurcation, Hopf bifurcation and the Bogdanov–Takens bifurcation. During the observation of the case data of hepatitis B in China, it is found that there are mainly three features, periodic outbreaks, aperiodic outbreaks, and periodic outbreaks turns to aperiodic outbreaks. According to the above features, we select three different representative regions, Jiangxi, Zhejiang province and Beijing, and then use our model to fit the actual monthly hepatitis B case data. The basic reproduction numbers that we estimated are 1.7712, 1.4805 and 1.4132, respectively. The results of data fitting are consistent with those of theoretical analysis. According to the sensitivity analysis of ℛ 0 , we conclude that reducing contact, increasing treatment rate, strengthening vaccination and revaccinating can effectively prevent and control the prevalence of hepatitis B. [ABSTRACT FROM AUTHOR]