June  8, 2023


Short Thematic Program on Delay Differential Equations
May 2015
Organizing Committee
Odo Diekmann (Utrecht)
Sue Ann Campbell (Waterloo)
Stephen Gourley (Surrey)
Yuliya Kyrychko (Sussex)

Eckehard Schöll (TU Berlin)
Michael Mackey (McGill)
Hans-Otto Walther (Giessen)
Glenn Webb (Vanderbilt)
Jianhong Wu (York)



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


[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.