


Mathematical Biology
Org: Gail Wolkowicz (McMaster) [PDF]
 JULIEN ARINO, University of Manitoba, Winnipeg, Manitoba
Effect of the introduction of refractory vectors in a
vectorborne disease
[PDF] 
A model for a vectorborne disease is considered, in which some
vectors are refractory to infection by the pathogen. The model
describes two populations of vectors: wild vectors can be infected by
the pathogen, while refractory vectors cannot. However,
refractoriness comes at the cost of reduced fitness. Interbreeding
between wild and refractory vectors can produce both types of vectors,
as described in the model by a somewhat complicated demography. This
model for vectors is then coupled to a simple SIR model for hosts, and
the dynamics of the whole system is studied. We show that in the
absence of disease, refractory vectors can become established in the
population. When the disease is present, this has the effect of
lowering the basic reproduction number, rendering the disease easier
to control.
 ELENA BRAVERMAN, University of Calgary, 2500 University Drive NW, Calgary,
AB, T2N 1N4
On Nicholson's Blowflies and Other Models with a Distributed
Delay
[PDF] 
We consider the Nicholson's blowflies equation with a distributed
delay
N¢(t) = dN(t) + p 
ó õ

t
h(t)

N(s) e^{a N(s)} d_{s}R(t,s), t ³ 0, 

and obtain existence, positiveness and permanence results for
solutions with positive initial conditions. In the range of
parameters p,d, where the relevant equation with a constant
delay is locally asymptotically stable (see the paper by Michael Li
et al., we prove that the solution is globally stable, as far
as h(t) tends to infinity for t ® ¥. We also consider the
general equation with several distributed delays
x¢(t) + 
m å
k=1

r_{k}(t) 
ó õ

t
¥

f_{k} 
æ è

x(s) 
ö ø

d_{s} R_{k} (t,s) = 0, 

which includes equations with several delays and integrodifferential
equations as special cases and obtain some additional results for this
equation, like linearized oscillation theorems. The results are
applied to logistic, LasotaWazewska and Nicholson's blowflies
equations with a distributed delay.
In addition, the "Mean Value Theorem" is proved which claims that
any solution of an equation with a distributed delay also satisfies
the linear equation with a variable concentrated delay.
 SUE ANN CAMPBELL, University of Waterloo
Synchronization, Multistability and Clustering: how useful
are predictions from phase models?
[PDF] 
We consider a model of a network of hippocampal interneurons based on
the work of Wang and Buzsaki. We construct a phase model
representation of the network, and show that this model can give
reasonably accurate quantitative information, such as the size of
basins of attraction and the maximum heterogeneity permissible in the
inherent frequencies of the neurons before synchrony is lost. We show
that predictions of existence and stability of the synchronous
solution from a two cell network carry over to Ncell networks,
either exactly or in the limit of large N.
This is joint work with Jeff Chadwick.
 YUMING CHEN, Wilfrid Laurier University
Global Attractivity of a Positive Periodic Solution of a
Delayed Periodic Respiration Model
[PDF] 
In this talk we consider a delayed periodic respiration model. First,
using the method of coincidence degree, we establish the existence of
a periodic solution. Then, we give several sufficient conditions for
the global attractivity of this periodic solution.
This is a joint work with L. Huang.
 TROY DAY, Queen's University, Jeffery Hall, Kingston, ON, K7L 3N6,
Canada
Modeling the emergence and control of infectious diseases
[PDF] 
Mathematical models are increasingly being used to understand and
predict the spread of infectious diseases. Often deterministic models
suffice for tracking the dynamics of wellestablished diseases, but
the initial stages in the emergence of new infectious diseases are
often marked by considerable stochasticity. This stochasticity enters
in at least two important ways. First, many new diseases arise from
crossspecies transmission, and some degree of evolutionary adaptation
is often required before the pathogen can spread. Stochasticity in
the evolutionary process in terms of which mutations arise and reach
fixation plays a key role is whether or not such diseases take hold.
Second, once a pathogen has adapted and is beginning to spread,
medical interventions (e.g., quarantine) will typically be imposed.
The relatively small initial number of cases during this phase also
means, however, that stochasticity will play an important role in the
extent to which such interventions are effective. I will present some
recent modeling results treating both of these phenomena, with reference
to the emergence of pandemic influenza as well as the 2003 outbreak of
SARS.
 HERB FREEDMAN, University of Alberta, Department of Mathematical &
Statistical Sciences, Edmonton, AB, T6G 2G1
Mathematical Models of Cancer Treatment Using Competition
[PDF] 
This talk contains a brief survey of modelling the growth and
treatment of cancer using the competition paradigm for interaction
with normal cells. The models consist of systems of ordinary
differential equations. Treatment techniques include radiotherapy,
chemotherapy and immunotherapy.
 MICHAEL LI, University of Alberta, Department of Mathematical and
Statistical Sciences, Edmonton, AB, T6G 2G1
Global Stability in Multigroup Epidemic Models
[PDF] 
For a class of ngroup epidemic models of SEIR type, the uniqueness
and global stability of the endemic equilibrium, when the basic
reproduction number is greater than 1, is proved using a global
Lyapunov function.
 XINZHI LIU, University of Waterloo, Department of Applied Mathematics,
Waterloo, Ontario, N2L 3G1, Canada
Study on HIV transmission among intravenous drug users
[PDF] 
The intravenousdrugusing community has been strongly affected by the
HIV virus, and can further contribute to the transmission of AIDS in
the nondrugusing community. It has also been indicated that IVdrug
users provide a major link between the heterosexual and the homosexual
population, thus affecting HIV prevalence in many different population
classes. As a result, methods to reduce or remove HIV infections
among IVdrug users need to be considered. In this talk, a model for
HIV transmission among intravenous drug users is studied. Several
control methods are introduced and analyzed. In each case, a
stability analysis of the diseasefree equilibrium point may provide
an estimation of the effectiveness of the control method.
 CONNELL MCCLUSKEY, Wilfrid Laurier University
A Study of Mimicry in Poison Arrow Frogs
[PDF] 
In the Ecuadorean rainforest, there is an interesting case of mimicry
among three species of frogs; one species (E. parvulus) is quite
poisonous, one is slightly poisonous (E. bilinguis), and the other is
nonpoisonous (A. zaparo). The two poisonous species look similar
(though not identical), and have overlapping habitats. Throughout the
same geographical region, the nonpoisonous frog species mimics the
poisonous frogs, displaying colouring that is practically
indistinguishable from whichever poisonous species happens to occupy
the region. In the overlap zone, where both poisonous species are
present, the nonpoisonous species accurately mimics only one of the
poisonous species (and therefore is an imperfect mimic for the other
poisonous species).
One would expect the mimic to copy either the more abundant or more
noxious of the two poisonous frogs. In this case, the more abundant
poisonous frog is also the more noxious. Surprisingly, the mimic
copies the less frequent and less poisonous species. It was
previously hypothesized that the mechanism driving this unexpected
result is differential generalization of learning among the predators.
In particular, the more intensely negative the experience of capturing
a poisonous frog, the more likely a predator is to avoid not just that
species of frog, but also frogs that look similar. We have modelled
this species interaction using a predatorprey system of ordinary
differential equations, varying the degree to which the predators
generalize their learned avoidance to include the imperfect mimics.
This is joint work with Jeff Orchard at the University of Waterloo.
 STEPHANIE PORTET, University of Manitoba, Winnipeg, Manitoba
Dynamics of in vivo intermediate filament organization
[PDF] 
The cytoskeleton is a complex arrangement of structural proteins
organized in networks: microfilaments, intermediate filaments and
microtubules. Each network has specific properties and organization
as well as particular roles in the cell. The organization of a
cytoskeletal network is the main determinant of its cellular function.
In this work, the organization of the intermediate filament network is
studied. The model describes the dynamics of four structural states
of the intermediate filament material: soluble proteins, particles
(precursors of filaments), and short and long filaments. Assembly
processes are considered, taking into account the formation and growth
of particles, the elongation of particles into short filaments, and
the integration and solubilization of filaments. Different hypotheses
are tested by mathematical and numerical analysis.
 ROBERT SMITH, University of Ottawa, 585 King Edward, Ottawa, ON
Predicting the potential impact of a cyctotoxic Tlymphocyte
HIV vaccine: how often should you vaccinate and how strong
should the vaccine be?
[PDF] 
To stimulate the immune system's natural defences, a HIV vaccination
program consisting of regular boosts of cytotoxic Tlymphocytes (CTLs)
has been proposed. We develop a mathematical model to describe such a
vaccination program, where the strength of the vaccine and the
vaccination intervals are constant. We apply the theory of impulsive
differential equations to show that the model has an orbitally
asymptotically stable periodic orbit. We show that, on this orbit, it
is possible to determine vaccine strength and vaccination intervals so
that the number of infected CD4^{+} T cells remains below a maximal
threshold. We also show that the outcome is more sensitive to changes
in the vaccine strength than the vaccination interval and illustrate
the results with numerical simulations.
 JAMES WATMOUGH, University of New Brunswick
The final size of an epidemic
[PDF] 
The early disease transmission model of Kermack and McKendrick
established two main results that are still at the core of most
disease transmission models today: the basic reproduction number,
R_{o}, as a threshold for disease spread in a population;
and the final size of an epidemic. As models become more complex, the
relationship between disease spread, final size and R_{o}
are not as clear; yet R_{o} remains the main object of
study when comparing control measures.
In this talk I review the final size relation for a simple epidemic
model and discuss its form in more complex models for treatment and
control of influenza and HIV.
 JIANHONG WU, York
Progress in modeling pandemic influenza and bird flu
[PDF] 
We summarize our recent progress in modeling pandemic influenza and
bird flu, using a variety of deterministic dynamical models. We show
how model analysis and simulations are useful to evaluate different
control strategies and to understand the mechanisms for the current
spread patterns of the H5N1 avian flu.
 HUAIPING ZHU, York
Stability and Oscillations for SIR Epidemiological Models
[PDF] 
SIR compartmental epidemiological models have been widely used to
study the transmission dynamics of certain infectious diseases. In
this talk, I shall discuss the stability and oscillations of such
three dimensional models with a general nonlinear incidence function.
 XINGFU ZOU, University of Western Ontario
Can the cutburn strategy eradicate a woodboring beetle
infestation?
[PDF] 
We propose a mathematical model for an infestation of a wooded area by
a beetle species in which the larva develop deep in the wood of living
trees. Due to the difficulties of detection, we presume that only a
certain proportion of infested trees will be detected and that
detection, if it happens, will occur only after some delay which could
be long. An infested tree once detected is immediately cut down and
burned. The model is stage structured and contains a second time
delay, the development time of the beetle from egg to adult. There is
a delicate interplay between the two time delays due to the
possibility in one case for a larva to mature even in a tree destined
for destruction. We present conditions sufficient for infestation
eradication and discuss the significance of the conditions
particularly in terms of the proportion of infested trees that need to
be detected and removed. If the infestation is successfully
eradicated there are always a number of trees that completely escape
infestation and we compute lower bounds and an approximation for this
number. Finally, we present the results of some numerical
simulations.
This is a joint work with Stephen Gourley.

