Hal L. Smith

An introduction to delay differential equations with applications to the life sciences

1 Introduction.-The Simplest Delay Equation.-Delayed Negative Feedback: A Warm-Up.- Existence of Solutions.- Linear Systems and Linearization.- Semidynamical Systems and Delay Equations.- Hopf Bifurcation.- Distributed Delay Equations and the Linear Chain Trick.- Phage and Bacteria in a Chemostat.-References.- Index.

VIRUS DYNAMICS: A GLOBAL ANALYSIS*

Exploiting the fact that standard models of within-host viral infections of target cell populations by HIV, developed by Perelson and Nelson [SIAM Rev., 41 (1999), pp. 3--44] and Nowak and May [Vir...

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH VERTICAL TRANSMISSION

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

The Poincare-Bendixson theorem for monotone cyclic feedback systems

R_0(p,q)\le 1,

Systems of ordinary differential equations which generate an order preserving flow. A survey of results

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.

Monotone Semiflows Generated by Functional Differential Equations

AbstractWe prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inRn of the form

Planar competitive and cooperative difference equations

x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).

Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems

We apply our results to a variety of models of biological systems.