Nonlinear Dynamics in Biology and Medicine / Dynamique non linéaire en physiologie et en médecine
(Org: Shigui Ruan, Dalhousie University)
- MOSTAFA ADIMY, University of Pau, avenue de l'Université, 64000 Pau, France
A singular transport system describing a proliferating
maturity structured cell population
In this work, we investigate a system of two nonlinear partial
differential equations, arising from a model of cellular proliferation
which describes the production of blood cells in the bone marrow. Due
to cellular replication, the two partial differential equations
exhibit a retardation of the maturation variable and a temporal delay
depending on this maturity. Our aim is to prove that the behavior of
primitive cells influences the global behavior of the population.
- JULIEN ARINO, Department of Mathematics, McMaster University, 1280 Main
Street West, Hamilton, ON L8S 4K1
Competitor-mediated coexistence in a chemostat with variable
We consider a very general model of growth in the chemostat, where we
suppose that the conversion between uptaken nutrient and cellular
growth is variable and depends on the substrate concentration. We
investigate some of the properties of this system when competition
between several species is considered. In particular, we show
numerically that competitor-mediated coexistence is possible, and
present some of the very complex behaviors exhibited by this system.
- JACQUES BÉLAIR, Université de Montréal, Montréal, Quebec H3C 3J7
Bifurcations in a model of the production of white blood cells
A model for the dynamics of the production of white blood cells is
derived and analysed. The model takes the form of a system of two
delayed-differential equations with two discrete time lags. We
identify the steady states and determine their stability. As the
parameter values in the equations are allowed to vary, an equilibrium
undergoes supercritical Hopf bifurcations, as well as saddle-node
bifurcations of limit cycles. Care is taken to relate the bifurcations
to the biological parameters inducing them. In particular, an increase
in the apoptosis rate of either stem cells or white blood cell
precursors is shown to be related to oscillations in the total number
of circulating cells.
Joint work with Samuel Bernard and Michael Mackey.
- BERND BLASIUS, University of Potsdam, Institute of Physics, Am Neuen
Palais 10, D-14469 Potsdam, Germany
Synchronization of epidemic outbreaks in networks of cities
Recurrent epidemics of infectious childhood diseases such as measles
are a major health problem and have been subject to extensive
theoretical research. Here we develop a theory for the dynamics of
epidemic outbreaks and their synchronization in a network of coupled
cities. Each city is described by a seasonally forced SEIR model. The
model generates chaotic dynamics with annual and biennial dynamics in
excellent agreement with long-term data sets. A new qualitative
criterion based on the attractor topology is developed to distinguish
between major outbreaks and epidemic fade-outs. This information is
coded into a symbolic dynamics. We are able to deduce a one
dimensional first return map of the chaotic SEIR equations, which upon
iteration is able to generate the symbolic sequence of major
outbreaks. The synchronization of epidemic outbreaks in a network of
cities is defined as measure-based on the symbolic dynamics. This is
applied to real data sets and numerical simulation results for
different network topologies.
- FRED BRAUER, University of British Columbia, Vancouver, BC
Infection-age dependent disease models
Disease transmission models with infectivity depending on the time
since becoming infected were first formulated by W. O. Kermack and
A. G. McKendrick [Proc. Roy. Soc. Ser. A 115(1927),
700-721]. However, variable infectivity was ignored until models for
AIDS were developed by H. R. Thieme and C. Castillo-Chavez [SIAM
J. Appl. Math. 53(1993), 1447 1479; Mathematical and Statistical
Approaches to AIDS Epidemiology (C. Castillo-Chavez, ed.), Springer
Verlag, 1989, 157-176]. We extend these models to models which
include density-dependent demographics and possible recovery from
infection. The central question is whether variable infectivity can
cause instability of the endemic equilibrium.
- SUE ANN CAMPBELL, University of Waterloo, Waterloo, ON N2L 3G1
Delayed Coupling Between Two Neural Network Loops
Coupled loops with time delays are common in physiological systems
such as neural networks. A Hopfield-type network is studied that
consists of a pair of one-way loops each with three neurons with
two-way coupling (of either excitatory or inhibitory type) between a
single neuron of each loop. Time delays are introduced in the
connections between the loops, and the effects of coupling strengths
and delays on the network dynamics are investigated. It is shown that
these effects depend strongly on whether the coupling is symmetric (of
the same type in both directions) or asymmetric (inhibitory in one
direction, excitatory in the other).
- JOHN CLEMENTS, Dalhousie University
Simulation and localization of cardiac dysfunction: from
modeling the dynamics to clinical applications
The objective of this research is to derive anatomically and
physiologically accurate mathematical models of electrical activation
in the human heart. These comprehensive simulation models will be used
My talk will focus on the mathematical aspects of this long-term
collaborative project (with B. M. Horacek, M. Gardner, J. Fitz-Clarke
et al.) in cardiac electrophysiology .
(i) non-invasively locate and quantify arrhythmogenic
substrate in cardiac patients,
(ii) assess the potential effects of anti-arrhythmic drug
(iii) predict the consequences of related cardiac therapies.
John C. Clements, Jukka Nenonen, P. K. J. Li and B. Milan
Activation dynamics in anisotropic cardiac tissue via
Annals of Biomedical Engineering, to appear July, 2004.
- TROY DAY, Queen's University, Kingston, Ontario
Modeling the Effectiveness of Quarantine Strategies
In this talk I will present some results exploring the effectiveness
of quarantine strategies for newly emerging infectious diseases. Two
issues will be addressed. In the first, I will use some simple
probabilistic models to determine how quarantine durations should be
set to minimize the risk that infected individuals are released back
into the community. In the second I will ask whether quarantine is
even a useful method for controlling emerging infectious
diseases. This will be done through a comparison of the effectiveness
of patient isolation in the presence and in the absence of quarantine,
by using simple deterministic and stochastic models.
- LEAH KESHET, University Of British Columbia, Dept. of Mathematics,
Vancouver, BC V6T 1Z2
Clonal selection of T cells in autoimmune diabetes
I briefly survey what is known about the pathogenesis of autoimmune
(Type 1) diabetes, and how immune cells (CD8+ T cells) are triggered
to proliferate and destroy the pancreatic cells (beta-cells) that
produce insulin. I then summarize work done in my group on modeling a
type of immunization (peptide therapy) procedure, and why caution has
to be used in its application. This is joint work with A. F. M. Maree
(Utrecht) and P. Santamaria (Calgary) and is funded by MITACS.
- ANDREW EDWARDS, Dalhousie University
Using state-space models to investigate why some fish
populations increase under industrial fishing
Industrial fishing has reduced the biomass of large predatory fish to
about 10% of pre-fishing levels. But not all species exhibit
a monotonic decline in abundance. For example, populations of
Atlantic sailfish Istiophorus albicans often increase
threefold before eventually being fished down to low
levels. We construct nonlinear population models to understand
these dynamics and to test various ecological hypotheses. We
use state-space models in a Bayesian framework, which allows
us to incorporate both observation error (the data are not
precise) and process uncertainty (models are not exact
representations of the real world). We utilise Markov Chain
Monte Carlo (MCMC) methods, using the free software WinBUGS.
- RANDY ELLIS, Queen's University, Kingston, Ontario
Computer-Assisted Surgery for Bone Deformities
Human limbs deform because of many causes, including genetic
predisposition, malnutrition, metabolic processes, diseases such as
arthritis, and poor healing following fracture. Over the past seven
years we have treated over 200 patients suffering from bone
deformities, using custom software to plan and intraoperatively guide
surgeons on complex reconstructive procedures.
This talk will present the principles of our work and clinical
examples. It combines kinematics, dynamics, computer graphics,
visualization, and 3D tracking to give surgeons unprecendented
abilities to treat complex orthopedic conditions.
- HERB FREEDMAN, University of Alberta
ODE Models of Cancer Treatment
This talk will describe some recent work in modelling various forms of
cancer treatment for different cancers. We think of cancer and normal
cells as competing for bodily resources. Cancers at one site, several
sites and throughout (such as leukemia) are considered. Chemotherapy,
immunotherapy and radiation therapy models are described.
- STEPHEN GOURLEY, Surrey
- ABBA GUMEL, University of Manitoba
Modelling the Impact of Some Anti-HIV Control Strategies
Models for assessing control strategies against the spread of HIV
infection in a community as well as in vivo will be presented.
The impact of the anti-HIV strategies in formulating an effective
public health policy against HIV infection will be addressed. This is
a collection of joint work with some members of the Mathematical
Biology Team at the University of Manitoba.
- ALUN LLOYD, North Carolina State University, Department of Mathematics,
Raleigh, NC 27695, USA
Drug Resistance in Acute Viral Infections
A wide range of viral infections, such as HIV or influenza, can now be
treated using antiviral drugs. Since viruses can evolve rapidly, the
emergence and spread of drug resistant virus strains is a major
concern. We shall describe within and between host models that can
help indicate settings in which resistance is more or less likely to
be problematic. In particular, we shall discuss the potential for the
emergence of resistance in the context of human rhinovirus infection,
an acute infection that is responsible for a large fraction of `common
- CONNELL McCLUSKEY, McMaster University, Hamilton, Ontario
Global Results for a Chemostat with Two Species and Two Resources
We study a chemostat with two species feeding on two perfectly
substitutable resources. The rate at which each species consumes each
resource is assumed to be linear, and the growth yield ratios are
assumed to be constant. Under certain conditions on the model
parameters, Lyapunov functions can be used to demonstrate that there
is a globally asymptotically stable equilibrium. Using the techniques
of Li and Muldowney, the global behaviour can be determined for a
larger subset of the parameter space. In particular, the global
behaviour can be resolved for some cases for which the positive
equilibrium is a saddle.
Joint work with Gail Wolkowicz and Mary Ballyk.
- SHIGUI RUAN, Dalhousie/Miami
- YASUHIRO TAKEUCHI, Shizuoka University
Permanence of Dispersal Population Models
We consider the following single-species time delayed system in patchy
where xi (i=1,2,...,n) denotes species x in patch i.
ai(t), bi(t) and dij(t) are all continuous functions.
ai(t) is the intrinsic growth rate of species x in patch i;
bi(t) represents the self-inhibition coefficient of species x in
patch i. dij(t) is the dispersal coefficient from patch j to
patch i (dii(t)=0), where tij ³ 0 represents a
constant dispersal time and gij is the death rate for the
species during dispersion from patch j to patch i.
x¢i (t) = xi [ai(t) - bi(t) xi(t)] + ||
exp-gij tij dij (t-tij) xj(t-tij) - dji (t) xi (t) ||
Our results show that at least one "food-rich" patch ensures
permanence for the total system.
- PAULINE VAN DEN DRIESSCHE, Dept. Math. & Stats, University of Victoria, Victoria,
BC V8W 3P4
Dispersal in Predator-Prey Systems
Distributions of dispersal times are incorporated into Lotka-Volterra
models. These are formulated as integro-differential equations that
describe predator-prey dynamics and dispersal between habitat
patches. If one species disperses (predators are often more mobile
than their prey), then dispersal almost always stabilizes the
equilibrium. The exception occurs when every trip has exactly the
same duration, thus the travel time distribution is a delta
function. In this case of discrete delay, there is a set of parameter
values for which the method used is inconclusive.
Joint work with Michael Neubert and Petra Klepac, Woods Hole
Oceanographic Institute, USA.
- JAMES WATMOUGH, University of New Brunswick, Fredericton, NB
Multiple setting disease transmission models: quarantine and
Models for disease transmission in heterogeneous populations typically
divide the population into several homogeneous compartments.
Incidence of the disease is then due to contacts both within and
between compartments. Nold (Math. Biosci. 52(1980), 227-240)
proposed the following three models for mixing: proportional mixing,
where contacts are made in proportion to the number of individuals in
each compartment; restricted mixing, where contacts are strictly
within compartments; and preferred mixing, a combination of the
previous two. The models consist of a system of differential
equations, with nonlinearities arising from bilinear (mass action)
incidence terms and the coupling between compartments. We extend
Nold's model to the case where contact are made in several settings.
A quarantine/isolation model for the transmission of SARS-CoV is given
as an example.
- JIANHONG WU, York University, Toronto, Ontario M3J 1P3
Modeling Delay and Diffusion via Hyperbolic-Parabolic Equations
Some recent progress in the modeling and analysis of delayed spatial
diffusion in structured populations will be presented. Model
derivations will be discussed, and results on wave solutions, global
attractors and synchronization will be reported.
- YICANG ZHOU, Science College, Xi'an Jiaotong University, Xi'an 710049,
P. R. China
SARS Prediction in China
During the transmission period of Severe Acute Respiratory Syndrome
(SARS) we have predicted its future spread on the basis of
epidemiological models and statistic data. The results were released
in May 21, 2003, and the prediction matches the statistical date well.
The simple SIR model is used for the prediction. The main attention
is paid to the parameter estimation. An easy method is given to
determine the transmission rate. The transmission rate is chosen as
a time dependent parameter and has the shape of exponential curve to
reflect the effect of various control measures. A prediction software
is designed for the people who work in the public health departments.
After the daily data of reported SARS case are input, the prediction
curve can be given automatically. Few parameters are also introduced
to show the influence of the start time and the stringency of the
control measure on the transmission. Factors to be interweaved in
epidemic modelling are mentioned.
- HUAIPING ZHU, York University
Modelling the West Nile virus among birds and mosquitoes
By considering the mosquitos as with or without WNv and birds as
infected and uninfected, I will introduce a set of differential equations
to model the transmission of WNv among mosquitos and birds. Some
analytical and numerical results will be presented.