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Nonlinear Dynamics in Biology and Medicine / Dynamique non linéaire en physiologie et en médecine
(Org: Shigui Ruan, Dalhousie University)

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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 yield

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 to

(i) non-invasively locate and quantify arrhythmogenic substrate in cardiac patients,
(ii) assess the potential effects of anti-arrhythmic drug interventions, and
(iii) predict the consequences of related cardiac therapies.

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 [1].

References

[1]

John C. Clements, Jukka Nenonen, P. K. J. Li and B. Milan Horácek, Activation dynamics in anisotropic cardiac tissue via de-coupling. 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 cold' cases.

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 environment

x¢i (t) = xi [ai(t) - bi(t) xi(t)] + n å j=1  æ è exp-gij tij dij (t-tij) xj(t-tij) - dji (t) xi (t) ö ø ,     (i=1,2,...,n)

where x_i (i=1,2,...,n) denotes species x in patch i. a_i(t), b_i(t) and d_ij(t) are all continuous functions. a_i(t) is the intrinsic growth rate of species x in patch i; b_i(t) represents the self-inhibition coefficient of species x in patch i. d_ij(t) is the dispersal coefficient from patch j to patch i (d_ii(t)=0), where t_ij ³ 0 represents a constant dispersal time and g_ij is the death rate for the species during dispersion from patch j to patch i.

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 isolation

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.