Dynamics of structured populations (08w5031)

Objectives

A standard method in the mathematical modeling of populations and their interactions is to model the populations as whole entities and derive equations for all relevant classes. This method typically leads to systems of ordinary or partial differential equations. While their exploration continues, the qualitative behavior of these models and their relevance to biological applications is fairly well understood. Ongoing biological research provides us with more detailed insight into the structure of populations and the individual behavior of its members. To include these biological details into more refined models is a chance and a challenge.

For example, the larval stages in mosquitoes can be included into a new model for West-Nile Virus, the social order in wolf packs can be included into models for homerange formation, or the cell reproduction cycle can be included into tumor growth and treatment models. Typical structure variables are age, size, stage, function, spatial location, genotype, or phenotype. A combination of one
or more of these structure variables into a model leads to systems of
reaction-diffusion equations, integro-difference and integro-differential
equations, integral equations, matrix models, and hyperbolic and parabolic partial differential equations. The methods for analyzing these structured population models are based on linear algebra, dynamical system theory, and the
theory of ordinary and partial differential equations.

During this workshop we focus on three main topics of structured
population modeling:

(i) Spatial Models
(ii) Epidemiology of Infectious Diseases
(iii) Biomedical Modelling

(i) Spatial Models: Since the 1970's, the classical models for
spatial spread of populations in the form of reaction-diffusion
equations have been studied widely and applied successfully to many different
questions in biology. However, their applicability to certain biological
systems is limited and new approaches are needed. Alternative models in the form of hyperbolic transport equations or parabolic/hyperbolic Fokker-Planck
equations structure the population into velocity classes and describe how
individuals change velocity. These models have received much attention, and their relationship to the classical reaction-diffusion models is fairly
well understood through scaling and moment closure methods. New applications of transport models to cancer invasion will be discussed under biomedical applications.

Another recent development in modeling spatial spread of populations is to incorporate different compartments for different modes of movement or even a compartment for non-moving individuals. These models capture biologically diverse phenomena such as seeds being transported by wind and by birds, animals and cells stopping their movement to reproduce, and proteins in the cell nucleus undergoing binding-unbinding events with immobile cell structures. Mathematically speaking, these structured models come in the form of integro-differential or integro-difference equations (with mixed dispersal kernels), integral equations, and as systems of coupled partial and ordinary
differential equations. It is known that dividing the population into
compartments with different movement characteristics can significantly
alter the models predictions, e.g., regarding
invasion speeds and stability.

While there is an overwhelmingly rich qualitative mathematical theory available for reaction-diffusion equations, a similarly complete theory for these alternative structured models is only just emerging. One focus of this workshop will be on novel developments of the qualitative theory of structured population models. Advances in this area will help in the choice of the most appropriate model for different applications in biology, such as the dynamics of the immune system, the epidemic spread of an infectious disease, and the invasion of microbes.

(ii) Epidemiology: The transmission characteristics of newly emerging
diseases, such as SARS, West Nile Virus, and AIDS, are quite different
from the local periodic outbreaks of "classical" infectious diseases, such as childhood diseases, and new models and methods are discussed to describe and analyze these outbreaks. Spatial heterogeneity is an important factor to consider in the spread of many diseases, e.g., influenza. Models for realistic vaccination schedules usually require the inclusion of the population age structure, in particular if antiviral doses are limited. Infection-age is a crucial structural variable in models for the emergence and spread of antibiotic-resistant bacteria, be it in the treatment of tuberculosis or the prevention of hospital-acquired infections.

When no vaccine is available (e.g., SARS) control by isolation and/or
quarantine can be built into a model. In particular, in light of the threats of
terrorism, there is an increased need to understand and to model spatial
spread of epidemics as well as optimal vaccination strategies and control.
Among other things, this modeling requires a blending of different mathematical techniques and the embedding of these epidemic models into a more general class of structured population models, as they will be discussed at our workshop.

(iii) Biomedical Modeling: Mathematical modeling in medicine has
become a major focus of research and has contributed new insights into medical
processes on the individual, cellular and intra-cellular level. The techniques and analytical methods, originally developed in the context of population dynamics, are now being extended to medical applications. For example, transport equations play a role for the conduction of nervous impulses, hyperbolic models are used to model movement of leucocytes, reaction-diffusion equations are used to model embryonal development, to model wound healing, or angiogenesis in cancer. The relevant population structures include the cell reproduction cycle,
the circadian rhythm, the age of an organ or tissue and the stages in cell development (for example 6 stages are involved in the modeling of stem cell
differentiation into blood cells). Again, ordinary and partial differential equations and integro-differential equations are the major tools for biomedical modeling. Questions of interest include pattern formation (in development),
persistence (in cancer) and control (in diseases and in the immune system).
During this workshop we will emphasize the extension of the existing mathematical modeling methods to biomedical applications. We will focus on new questions and therapies that make use of structured population modeling,
for example taking account of the cell reproduction cycle in radiation
treatment planning.

SCHEDULING AND ORGANIZATION
At the proposed workshop, we will discuss current developments and
exchange recent ideas on the three main topics listed above. For each of
these topics, we plan to have a 1 hour introductory lecture by a
distinguished researcher. This lecture will be followed by a number of
more specific talks given by specialists in the respective fields. We
will allow plenty of time for discussions in smaller groups.

The list of potential participants (below) includes the full spectrum from advanced and experienced leaders in the field to young researchers and graduate students. The list contains already 5 graduate students and 2 Postdocs. We plan to invite the students depending on their status at the time of the workshop and we might include newcoming students. In addition we plan to use the BIRS travel fund from University of Alberta to allow graduate students as external observers in addition to the 40 participants.

The workshop will provide a unique opportunity for graduate students and PDF’s to discuss their work with international experts in this field.