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*Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay*, Discrete and Continuous Dynamical Systems - Series B, 12(2009), 511--524] presented a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. By combining the analysis of the characteristic equation and the Lyapunov-LaSalle method, they obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. In the present paper, we show that the global dynamics are fully determined for $\Re_1<1<\Re_0$ and $\Re_1>1$ (Theorem 4.2 and Theorem 4.3) without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov-LaSalle invariance principle.

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

There is a growing interest to understand impacts of latent infection age and infection age on viral infection dynamics by using ordinary and partial differential equations. On one hand, activation of latently infected cells needs specificity antigen, and latently infected CD4+ T cells are often heterogeneous, which depends on how frequently they encountered antigens, how much time they need to be preferentially activated and quickly removed from the reservoir. On the other hand, infection age plays an important role in modeling the death rate and virus production rate of infected cells. By rigorous analysis for the model, this paper is devoted to the global dynamics of an HIV infection model subject to latency age and infection age from theoretical point of view, where the model formulation, basic reproduction number computation, and rigorous mathematical analysis, such as relative compactness and persistence of the solution semiflow, and existence of a global attractor are involved. By constructing Lyapunov functions, the global dynamics of a threshold type is established. The method developed here is applicable to broader contexts of investigating viral infection subject to age structure.

In this paper, we consider the dynamics of a generalized three-dimensional Hénon map. Necessary and sufficient conditions on the existence and stability of the fixed points of this system are established. By applying the center manifold theorem and bifurcation theory, we show that the system has the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation under certain conditions. Numerical simulations are presented to not only show the consistence between examples and our theoretical analysis, but also exhibit complexity and interesting dynamical behaviors, including period-10, -13, -14, -16, -17, -20, and -34 orbits, quasi-periodic orbits, chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors. These results demonstrate relatively rich dynamical behaviors of the three-dimensional Hénon map.

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

*On latencies in malaria infections and their impact on the disease dynamics*, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.

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