Michael Y. Li

A Geometric Approach to Global-Stability Problems

A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the

Global stability for the SEIR model in epidemiology

C^1

A graph-theoretic approach to the method of global Lyapunov functions

closing lemma of Pugh. The classical result of Lyapunov is obtained as a special case.

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH VERTICAL TRANSMISSION

The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.

Global Dynamics of an In-host Viral Model with Intracellular Delay

A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.

Impact of Intracellular Delays and Target-Cell Dynamics on In Vivo Viral Infections

We study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission. The total host population is assumed to have constant density and the incidence term is of the bilinear mass-action form. We prove that the global dynamics are completely determined by the basic reproduction number R0 (p,q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If

Global stability of an SEIS epidemic model with recruitment and a varying total population size.

R_0(p,q)\le 1,

Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion

the disease-free equilibrium is globally stable and the disease always dies out. If R0 (p,q)>1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the vertical transmission to the basic reproduction number is also analyzed.

Global Dynamics of a General Class of Multistage Models for Infectious Diseases

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R0 for the viral infection, and establish that the global dynamics are completely determined by the values of R0. If R0≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R0>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R0>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.

Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression

The dynamics of an in-host model with general form of target-cell dynamics, nonlinear incidence, and distributed delay are investigated. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV. We derive the basic reproduction number