Abstracts
Jiguo Cao, Simon Fraser University
Semiparametric Method for Estimating DDE Parameters from Noisy Real Data
Delay differential equations (DDEs) are widely used in ecology, physiology
and many other areas of applied science. Although the form of the DDE
model is usually proposed based on scientific understanding of the dynamic
system, parameters in the DDE model are often unknown. Thus it is of great
interest to estimate DDE parameters from noisy data. Since the DDE model
does not usually have an analytic solution, and the numeric solution requires
knowing the history of the dynamic process, the traditional likelihood
method cannot be directly applied. We propose a semiparametric method
to estimate DDE parameters. The key feature of the semiparametric method
is the use of a flexible nonparametric function to represent the dynamic
process. The nonparametric function is estimated by maximizing the DDE-defined
penalized likelihood function. Simulation studies show that the semiparametric
method gives satisfactory estimates of DDE parameters. The semiparametric
method is demonstrated by estimating a DDE model from Nicholson’s
blowfly population data.
Odo Diekmann , Utrecht University
The delay equation formulation of structured population models
The aim of the lecture is to explain how i-level models of development,
survival, reproduction and feedback via the environment (such as, e.g.,
consumption of food) lead, by simple bookkeeping considerations, to p-level
models that take the form of a coupled system of Renewal Equations and
Delay Differential Equations [4].
For ecological insights derived from such models see [1], for a detailed
elaboration of a representative example see [2]. For lots of examples
formulated at the p-level in terms of PDE, see [3]. (The connection between
the PDE formulation and the delay equation formulation is provided by
integration along characteristics.)
[1] A.M. de Roos & L. Persson
Population and community ecology of ontogenetic development , PUP , 2013
[2] O. Diekmann, M. Gyllenberg, J.A.J. Metz, S. Nakaoka, A.M. de Roos
Daphnia revisited : local stability and bifurcation theory for physiologically
structured population models explained by way of an example
Journal of Mathematical Biology (2010) 61 : 277-318
[3] http://www.iiasa.ac.at/Research/EEP/Metz2Book.html
[4] O. Diekmann, J.A.J. Metz
How to lift a model for individual behaviour to the population level?
Philosophical Transactions Royal Society B.(2010) 365 : 3523-3530
Tony Humphries, McGill University
Parameters and their values in physiological models: Cautionary tales
and outstanding problems
From the point of view of a mathematician who has dipped several toes
in the physiological waters, I will use our work on haematopoiesis to
illustrate the difficulties that I have encountered and the challenges
that exist in determining and interpreting parameters in mathematical
models in physiology.
As well as some cautionary tales on what can go wrong if you read the
literature or listen to your colleagues, I will also outline two problems
of current interest to me.
We have developed a `model of the kinetics of G-CSF, the principal cytokine,
regulating white-blood cell production, that can replicate observed dynamics
quite well, and certainly better than previous models. Unfortunately it
can do so nearly equally well with a range of parameter values, because
some quantities that would allow us to determine unique parameters are
not measured or measurable (fraction of bound receptors). The concentrations
of freely circulating G-CSF that we observe can be explained by different
parameter sets. What should we do?
Even when all the parameters in a model are somehow ``determined'' for
healthy subjects, it is interesting to study which parameters in the model
have to be varied and how when patients present dynamical diseases. There
are several diseases associated with blood cell production that result
in time-varying blood cell counts. From a dynamical systems point of view
it would be very neat to explain this by a change of parameters resulting
in a Hopf bifurcation and creation of a stable periodic orbit. However,
data can be very sparse (blood samples might be taken daily, or even less
often for non-hospitalized subjects), noisy, and not necessarily as clearly
periodic as one would like. The simplistic approach of simulating the
differential equations model and choosing parameters to minimize a least
squares fit to the data can fail miserably. The problem of then determining
suitable parameters in the model to fit the data is very challenging,
and seems to come down to careful choice of a suitable objective function
to minimize rather than the actual minimization algorithm deployed.
Michael C Mackey, McGill University
Problems I would like to see solved (but I partially failed)
In this talk I intend to discuss several interesting numerical observations
about the temporal evolution of densities of ensembles of solutions to
differential delay equations. These phenomena are, for the most part,
relatively unknown in the mathematical community and involve apparent
extensions of the concepts of ergodicity, mixing, asymptotic periodicity
and exactness (or asymptotic stability) in the ergodic theory of dynamical
systems. However, the numerical results highlight the total absence of
any mathematical framework within which we can understand and study them.
Namely what does it mean to look at the evolution of a density under the
action of delayed dynamics? What is a density? What is the measure? What
is the evolution operator?
Many problems, some tantalizing numerics, no solutions.
Jianhong Wu, York University
Approximate temperature-driven tick development delay from lab data and
weather prediction for Lyme disease risk assessment in Canada
This talk will start with an introduction to the mathematical foundation
of the Lyme tick basic reproduction number map in Canada, produced in
collaboration with Public Health Agency of Canada, Environment Canada
and York Institute for Health Research. The talk will also present a short
description of the on-going effort in producing a similar map, but for
the Lyme disease risk. The speaker will use this Lyme disease transmission
model as an example to illustrate how discussions of the four thematic
weeks and the entire thematic program may contribute to developing, refining
and analyzing appropriate structured population models, and linking model-based
simulations and projections to laboratory, surveillance and environment
data.
