Two problems related to prescribed curvature measures
Yong Huang Lu Xu
Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. Moreover, we promote Ivochkina's $C^2$ estimates for prescribed curvature equation in [12,13].
keywords: curvature equations. curvature measure Hypersurfaces
A mathematical model of HTLV-I infection with two time delays
Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values $R_0$, the corresponding reproductive number of a viral infection, and $R_1$, the corresponding reproductive number of a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing" effects from the intracellular delay with those ``destabilizing" influences from immune delay.
keywords: Hopf bifurcation. Epidemic threshold HTLV-I infection Lyapunov functional time delay
A delayed HIV-1 model with virus waning term
Bing Li Yuming Chen Xuejuan Lu Shengqiang Liu
In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD$8^+$ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number $R_0$. If $R_0<1$, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if $R_0>1$, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when $R_0>1$ is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.
keywords: delay permanence. stability CTLs HIV-1 infection virus waning immune response
An SEI infection model incorporating media impact
Xuejuan Lu Shaokai Wang Shengqiang Liu Jia Li

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number $\Re_0$ for the model, and show that the modeled disease dies out regardless of initial infections when $\Re_0 < 1$, and becomes uniformly persistently endemic if $\Re_0>1$. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided $\Re_0 $ is greater than and near one. However, if $\Re_0 $ is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger $\Re_0$, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.

keywords: Media impact Hopf bifurcation stability infectious disease
Long time strong convergence to Bose-Einstein distribution for low temperature
Xuguang Lu

We study the long time behavior of measure-valued isotropic solutions $F_t$ of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions $F_t$ that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution $F_t$ as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then $F_t$ converges strongly to equilibrium as $t$ tends to infinity. We also propose a new condition on the initial datum $F_0$ such that a corresponding solution $F_t$ converges strongly to equilibrium.

keywords: Boltzmann equation Bose-Einstein particles low temperature long time strong convergence

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