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Several types of differential equations, such as functional
differential equation, age-structured models, transport equations,
reaction-diffusion equations, and partial differential equations
with delay, can be formulated as abstract Cauchy problems with
non-dense domain. This monograph provides a self-contained and
comprehensive presentation of the fundamental theory of non-densely
defined semilinear Cauchy problems and their applications. Starting
from the classical Hille-Yosida theorem, semigroup method, and
spectral theory, this monograph introduces the abstract Cauchy
problems with non-dense domain, integrated semigroups, the
existence of integrated solutions, positivity of solutions,
Lipschitz perturbation, differentiability of solutions with respect
to the state variable, and time differentiability of solutions.
Combining the functional analysis method and bifurcation approach
in dynamical systems, then the nonlinear dynamics such as the
stability of equilibria, center manifold theory, Hopf bifurcation,
and normal form theory are established for abstract Cauchy problems
with non-dense domain. Finally applications to functional
differential equations, age-structured models, and parabolic
equations are presented. This monograph will be very valuable for
graduate students and researchers in the fields of abstract Cauchy
problems, infinite dimensional dynamical systems, and their
applications in biological, chemical, medical, and physical
problems.
In this new century mankind faces ever more challenging
environmental and
publichealthproblems,suchaspollution,invasionbyexoticspecies,theem-
gence of new diseases or the emergence of diseases into new regions
(West Nile
virus,SARS,Anthrax,etc.),andtheresurgenceofexistingdiseases(in?uenza,
malaria, TB, HIV/AIDS, etc.). Mathematical models have been
successfully used to study many biological, epidemiological and
medical problems, and nonlinear and complex dynamics have been
observed in all of those contexts. Mathematical studies have helped
us not only to better understand these problems but also to ?nd
solutions in some cases, such as the prediction and control of SARS
outbreaks, understanding HIV infection, and the investi- tion of
antibiotic-resistant infections in hospitals.
Structuredpopulationmodelsdistinguishindividualsfromoneanother-
cording to characteristics such as age, size, location, status, and
movement, to determine the birth, growth and death rates,
interaction with each other and with environment, infectivity, etc.
The goal of structured population models is to understand how these
characteristics a?ect the dynamics of these models and thus the
outcomes and consequences of the biological and epidemiolo- cal
processes. There is a very large and growing body of literature on
these topics. This book deals with the recent and important
advances in the study of structured population models in biology
and epidemiology. There are six chapters in this book, written by
leading researchers in these areas.
This book provides an introduction to the theory of ordinary
differential equations and its applications to population dynamics.
Part I focuses on linear systems. Beginning with some modeling
background, it considers existence, uniqueness, stability of
solution, positivity, and the Perron-Frobenius theorem and its
consequences. Part II is devoted to nonlinear systems, with
material on the semiflow property, positivity, the existence of
invariant sub-regions, the Linearized Stability Principle, the
Hartman-Grobman Theorem, and monotone semiflow. Part III opens up
new perspectives for the understanding of infectious diseases by
applying the theoretical results to COVID-19, combining data and
epidemic models. Throughout the book the material is illustrated by
numerical examples and their MATLAB codes are provided. Bridging an
interdisciplinary gap, the book will be valuable to graduate and
advanced undergraduate students studying mathematics and population
dynamics.
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