Cameron
Browne, University of Ottawa
Pulse Vaccination in a Polio Meta-population Model
Poliomyelitis infections continue to arise in certain regions of Africa
and the Middle East, despite the significant progress made toward the goal
of eradication. There are several factors which contribute to the difficulty
of eradication, including: areas of high transmission and low vaccination
coverage, mobile populations, seasonality, and persistence of the virus
in the environment. One of the main strategies utilized to fight Polio over
the past 40 years has pulse vaccination.
In order to better understand this control strategy, we consider an impulsive
$SIR$-type metapopulation model with seasonality, an environmental reservoir,
and arbitrary pulse vaccination schedules in each patch. A basic reproduction
number, $\mathcal R_0$, is defined and proved to be a global threshold for
the system. Numerical calculations show the importance of, both, synchronizing
the pulse vaccinations between the patches and the timing of the pulses
with respect to the seasonality. A stochastic version of the model is also
considered. It is found that pulse vaccination has a major advantage over
a continuous vaccination strategy in terms of the probability of eradication.
We also explore how the coupling between the patches and the level of environmental
transmission affect the results.
Sue Ann Campbell, University of Waterloo
A Plankton Model with Delayed Nutrient Recycling
We consider a three compartment (nutrient-phytoplankton-zooplankton) model
with nutrient recycling. When there is no time delay the model has a conservation
law and may be reduced to an equivalent two dimensional model. We consider
how the conservation law is affected by the presence of time delay (both
discrete and distributed) in the nutrient recycling. We study the stability
and bifurcations of equilibria when the total nutrient in the system is
used as the bifurcation parameter. This is joint work with Matt Kloosterman
and Francis Poulin.
Dong Eui Chang, University of Waterloo
Damping-Induced Self-Recovery Phenomenon In Mechanical Systems With An
Unactuated Cyclic Variable
The falling cat problem has been very popular since Kane published a paper
on this topic in 1969. A cat, after released upside down, executes a 180-degree
reorientation, all the while having a zero angular momentum. It makes use
of the conservation of angular momentum that is induced by rotational symmetry
in the dynamics. But if there is an external force that breaks the symmetry,
then the angular momentum will not be conserved any more.
Recently, we have discovered an exciting nonlinear phenomenon in mechanical
systems with one unactuated cyclic variable on which a viscous damping force
is exerted. In this case, there arises a new conserved quantity, called
damping-added momentum, in place of the original momentum map. Using this
new conserved quantity, we show that the trajectory of the cyclic variable
asymptotically converges back to its initial value. This phenomenon, called
damping-induced self-recovery, can occur even when the damping coefficient
is not constant as long as the integral of the coefficient satisfies a certain
condition.
The self-recovery phenomenon can be observed in a simple experiment with
a rotating stool and a bicycle wheel which is a typical setup in physics
classes to demonstrate the conservation of angular momentum. Sitting on
the stool, one spins the wheel by his hand while holding it horizontally.
A reaction torque will be created to initiate the rotational motion of the
stool to the opposite direction. After some time, if the person applies
a braking force halting the wheel spin, then the stool will asymptotically
return to its original position, tracing back its past path regardless of
the number of rotations the stool has made, provided that there is a viscous
damping force on the rotation axis of the stool.
Hermann Eberl, University of Guelph
How we failed to solve an optimal control problem for a biofilm reactor
with suspended growth
We present an extension of Freter's 1983 model of a continously stirred
bioreactor with wall attachement (or equivalently: a biofilm reactor with
suspended growth), accounting for substrate gradients in the microbial depositions.
This is achieved by linking a CSTR model with a Wanner-Gujer like biofilm
model. In the general multi-species, multi-substrate case this leads to
a free boundary value problem for a coupled hyperbolic-parabolic problem
which is not easily accessible to analytical techniques. However, in the
simpler single species, single substrate case we can formally rewrite the
model as a system of three ODEs, the longterm behavior of which can be studied
with elementary techniques. In the second part of the talk we study the
problem of controling the reactor such that a given amount of substrate
is degraded as much as possible in as short a time as possible. This is
joint work with Alma Masic (Lund University).
Xu Fei, Wilfrid Laurier University
Interspecific strategic effects of mobility in predator-prey systems
In this talk, we will investigate the dynamics of a predator-prey system
with the assumption that both prey and predators use game theory-based strategies
to maximize their per capita population growth rates. The predators adjust
their strategies in order to catch more prey per unit time, while the prey,
on the other hand, adjust their reactions to minimize the chances of being
caught. Numerical simulation results indicate that, for some parameter values,
the system has chaotic behavior. Our investigation reveals the relationship
between the game theory-based reactions of prey and predators, and their
population changes.
Bernhard Lani-Wayda, Mathematisches Institut der Universit\"at Giessen
Four-dimensional \v{S}il'nikov-type dynamics in x'(t) = -\alpha \cdot x(t-d(x_t))
It was shown by Hans-Otto Walther that the state-dependent delay function
$d$ in the `linear-looking' equation $ x'(t) = -\alpha \cdot x(t-d(x_t))
$ can be chosen such that it produces a solution homoclinic to a saddle.
Chosing proper subsets from the domain of a return map, and using the fixed
point index, we prove that the return map is semi-conjugate to the shift
in two symbols. The situation is an infinite-dimensional analogue to the
one described by \v{S}il'nikov (1967) in dimension 4, and our methods are
much inspired by the work of W\'{o}jczik and Zgliczy\'{n}ski on covering
relations. Joint work in progress with Hans-Otto Walther.
Jean-Philippe Lessard, Université Laval
Rigorous Computations for Infinite Dimensional Problems
Studying and proving the existence of solutions of nonlinear dynamical
systems using standard analytic techniques is a challenging problem. In
particular, this problem is even more challenging for partial differential
equations, variational problems or functional delay equations which are
naturally defined on infinite dimensional function spaces. As a consequence
of these challenges and with the recent availability of powerful computers
and sophisticated software, numerical simulations quickly became one of
the primary tool used by scientists to conjecture the behaviour of the dynamics
of the above mentioned nonlinear equations. A standard approach adopted
by mathematicians is to get insights from numerical simulations to formulate
new conjectures, and then attempt to prove the conjectures using pure mathematical
techniques only. As one shall argue, this strong dichotomy need not exist
in the context of dynamical systems, as the strength of numerical analysis
and functional analysis can be combined to prove, in a direct computational
way, existence of solutions of infinite dimensional dynamical systems. The
goal of this talk is to present such rigorous numerical methods to the context
of proving the existence of steady states, time periodic solutions, traveling
waves and connecting orbits of finite and infinite dimensional differential
equations.
F.M.G. Magpantay, York University
An Age-Structured Population Model with State-Dependent Delay
We considered an age-structured population model with distinct juvenile
and adult stages in which the two stages consume different limited food
sources. The new model involves the McKendrick PDE, a nonlinear boundary
condition due to the birth rate, and a threshold condition with state-dependent
delay due to a varying age of maturity. We present some analysis on this
model and compare it to a version with constant delay as well as other existing
population models. This is a joint work with N. Kosovali´c and J.
Wu.
Daniel Munther, York University
The ideal free strategy with weak Allee effect
This talk examines the interplay between optimal movement strategies and
the weak Allee effect within the context of two competing species in a spatially
heterogenous environment. When both species have the same populations dynamics,
previous studies identified an `ideal free' strategy which is able to exclude
any other competitor playing a
`non-ideal free' strategy. We find that if the ideal free disperser is subject
to a `weak' Allee effect, a competing species utilizing very weak or very
strong advection will still be excluded despite having superior population
dynamics. However, for intermediate advection rates, such a competitor can
invade the ideal free disperser and even drive it to extinction. Not only
do these results enhance ecological understanding of competing species,
but they provide insight into the non-linear theory of reaction-advection-diffusion
models when the usual linearization techniques offer no information.
Felix Njap, University of Waterloo
Bifurcation analysis of a model of Parkinsonian STN-GPe activity
Excessive oscillations in the beta (15-30Hz) band are seen in the basal
ganglia of patients with Parkinson’s disease. An important question
concerns the conditions under which theseoscillations can occur. The mathematical
technique of bifurcation analysis is applied to a simple model of 2 populations
to determine the critical boundaries in parameter space. These boundaries
separate regions of different dynamics. A number of bifurcations (up to
codimension 2) are found. In particular, a region of beta oscillations has
been identified in the model under “Parkinsonian conditions”.
No such region is present under “healthy conditions”.
Longxing Qi, Anhui University and York University
Modeling The Schistosomiasis on the Islets in Nanjing
A compartmental model is established for schistosomiasis infection in Qianzhou
and Zimuzhou, two islets in the center of Yangtzi River near Nanjing, P.
R. China. The model consists of five differential equations about the susceptible
and infected subpopulations of mammalian Rattus norvegicus and Oncomelania
snails. We calculate the basic reproductive number R0 and discuss the global
stability of the disease free equilibrium and the unique endemic equilibrium
when it exists. The dynamics of the model can be characterized in terms
of the basic reproductive number. The parameters in the model are estimated
based on the data from the field study of the Nanjing Institute of Parasitic
Diseases. Our analysis shows that in a natural isolated area where schistosomiasis
is endemic, killing snails is more effective than killing Rattus norvegicus
for the control of schistosomiasis.
Gunog Seo, York University
A comparison of two predator-prey models with Holling type I functional
response
In my talk, I will show dynamics of two models, a laissez-faire predator-prey
model and a Leslie-type predator-prey model, with type I functional response.
For both models, I will study stability of the equilibrium where the prey
and predator coexist by performing a linearized stability analysis and by
constructing a Lyapunov function. For the Leslie-type model, in particular,
I use a generalized Jacobian to determine how eigenvalues jump at the corner
of the functional response. I numerically show that my two models can possess
two limit cycles surrounding a stable equilibrium and that these cycles
arise through global cyclic-fold bifurcations. The Leslie-type model may
also exhibit supercritical and discontinuous Hopf bifurcations. I then present
and analyze a new functional response, built around the arctangent, that
smooths the sharp corner in a type I functional response. For the new functional
response, both models exhibit Hopf, cyclic-fold, and Bautin bifurcations.
Steve C. Walker, McMaster University
When does intraspecific variation matter to community-level dynamics?
Community ecologists typically ignore intraspecific variation, thereby
implicitly assuming that each individual of a species is identical. This
approach is reasonable, given the expenses associated with collecting enough
data to estimate differences between conspecifics. Here we incorporate intraspecific
variation into models of multispecies dynamics, in order to identify conditions
under which ignoring intraspecific variation will lead to incorrect deductions
about the dynamics of aggregate community properties (e.g. total biomass;
community-level trait variance). Our approach is based on Price's equation
from evolutionary biology and laws of total central moments from probability
theory. This approach allows us to partition the dynamical equations for
aggregate community properties into two additive components: one that ignores
intraspecific variation and one that does not. We find that ignoring intraspecific
variation can lead to dramatically incorrect deductions in many, but not
all, circumstances.
Gail S. K. Wolkowicz, McMaster University
Mathematical model of anaerobic digestion in a chemostat:effects of syntrophy
and inhibition
Anaerobic digestion is a complex naturally occurring process during which
organic matter is broken down to biogas and various byproducts in an oxygen-free
environment. It is used for waste and wastewater treatment and for production
of biogas, especially methane. A system of differential equations modelling
the interaction of microbial populations in a chemostat is used to describe
three of the four main stages of anaerobic digestion: acidogenesis, acetogenesis,
and methanogenesis. To examine the effects of the various interactions and
inhibitions, we first study an inhibition-free model and obtain results
for global stability using differential inequalities together with conservation
laws. These results are compared with the predictions for the model with
inhibition. In particular, inhibition introduces regions of bistability
and stabilizes some equilibria.
Jianhong Wu, York University
Emerging flocking behaviors of the Cucker-Smale model with delayed information
processing
This is based on the joint work with Yicheng Liu of the National University
of Defense Technology. We examine the emerging behaviors such as flocking,
herding and schooling in the Cucker-Smale model with delayed information
processing in self-organized systems. We use a fixed point theoretic argument
in weighted Banach spaces to derive sharp conditions on the influence function
to ensure unconditioning flocking.
Yanyu Xiao. York University
Can treatment effect the epidemic final size?
Antiviral treatment is one of the key pharmaceutical interventions against
many infectious diseases. This is particularly important in the absence
of preventive measures such as vaccination. However, the evolution of drug-resistance
in treated patients and its subsequent spread to the population pose significant
impediments to the containment of disease epidemics using treatment. Previous
models of population dynamics of influenza infection have shown that in
the presence of resistance, the epidemic final size (i.e., the total number
of infections throughout the epidemic) is affected by the treatment rate.
These models, through simulation experiments, illustrate the existence of
an optimal treatment rate, not necessarily the highest possible rate, for
minimizing the epidemic final size. However, the conditions for the existence
of such an optimal treatment rate have never been found. Here, we provide
these conditions for a general class of models covering previous literature,
and investigate the combination effect of treatment and transmissibility
of the resistant strain on the epidemic final size. For the first time,
we obtain the final size relations for an epidemic model with two strains
of a disease (i.e., drug sensitive and drug resistance). We discuss the
general model with specific functional forms of resistance emergence, and
illustrate the theoretical findings with numerical simulations.
Hossien Zivari-Piran, York University
Numerical qualitative analysis of a large-scale model for measles spread
In this talk I describe the dynamic of a model of measles infection developed
by Heffernan and Keeling (2008). The combined immunological and epidemiological
model includes waning immunity and vaccination and is formulated as a large-scale
system of ODEs. I show how the dynamics of this system changes as the parameters
controlling waning time and vaccination level vary. The relevant local and
global bifurcations are investigated using special numerical methods for
large-scale systems, leading to a novel approach for explaining the mechanisms
underlying the oscillatory dynamic of measles.
Xingfu Zou, University of Western Ontario
Can multiple Malaria species co-persist? ----Dynamics of multiple malaria
species
There are several species of malaria protozoa spreading in different regions.
On the other hand, the world becomes more highly connected by travel than
ever before. This raises a natural concern of possible epidemics caused
by multiple species of malaria parasites in one region. In this study, we
use mathematical models to explore such a possibility. Firstly, we propose
a model to govern the within-host dynamics of two species. Analysis of thismodel
practically excludes the possibility of co-persistence (or super-infection)
of the two species in one host. Then we move on to set up another model
to describe the dynamics of disease transmission between human and mosquito
populations without the co-infection class (using the results for the within-host
model). By analyzing this model, we find that co-endemics of both species
in a single region is possible within certain range of model parameters.
This is a joint work with Yanyu Xiao.
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