I was first interested in respiratory control as a student, and did a
little work on the Grodins model. Nothing came of it till Giri
Kalamangalam came to Oxford and did a D.Phil. on the
subject. Essentially, the Grodins model consists of straightforward
but complicated first order ordinary differential equations for blood,
brain, etc. concentrations of oxygen and carbon dioxide, with a number
of (variable) delays due to blood transport between the
compartments. Various simplifications are possible, and the model
reduces, more or less, to various versions of the delay recruitment
model discussed by Mackey and Glass, for example. Consequently,
oscillations and chaos can occur, and the interesting mathematical
problem is to understand the relationship between the chaos in the
differential equation and the chaos in the underlying map.
Pest management
Like many people, I find the Ludwig-Jones-Holling model of spruce
budworm infestations to be an instructive pedagogical example of how
relaxation oscillations occur in `realistic' models. What is less
clear is how they work in the original Jones `site' model, which is in
essence a very complicated difference equation. David Hassell was an
M.Sc. student in 1996 whose dissertation analysed the site model, and
a paper resulting from this has now been published. While it is
possible to understand analytically how the outbreak and subsequent
collapse occur, it is less obvious how the forest recovery operates,
and this appears to involve the tree age structure in a complicated
way.
Medical data analysis
There are a lot of problems in medicine where data can be obtained,
but its interpretation is not easy. Examples include fetal heart rate
monitoring, where the fetal heart rate is studied for signs of
distress. During labour, the interpretation of these signals is
apparently rather subjective, and an objective evaluation would be
welcome. Non-invasive measurements of blood pressure and heart rate in
intensive care medicine are used to monitor baroreceptor activity. The
basic problem in this situation is that from measurements of two
coupled oscillators, each of them chaotic, one wants to ascertain the
degree of coupling. These and other similar problems are related to
nonlinear time series analysis, and I am interested in developing
practical techniques which can be used to address them.
Immunology
The basic problem that interested me was how the immune system could
contrive to eradicate completely a bacterial infection (for
example). Insofar as the basic interaction is between populations of B
or T cells and the antigen, then one doesn't expect a continuum model
to allow the antigen population to reach the extremely low levels
necessary to attain extinction via stochastic processes, at least if
the antigen has an intrinsic growth rate. It turns out that the delay
in the production of B cells, for example, allows such extremely small
levels to be attained. This is because large delay t causes
minimum antigen densities of order exp(-exp(O(t))) to be
obtained. This was in a model due to Dibrov and co-workers, and my
interest is in extending that concept from B cells to other parts of
the immune system.
Blood cell production
The various kinds of blood cells (red, white, platelets)
are produced in the bone marrow from pluripotent stem cells. There are
a number of diseases, including the leukaemias, in which blood cell
counts oscillate, and it is thought that these oscillations are caused
by an instability in the feedback control system. The models used to
describe these systems invariably involve delays, both in the cell
cycle and in the process of differentiation, but it is sometimes
possible to gain analytical insight, and in turn this may lead to new
ways to understand the disease mechanism.
Relevant publications
Fowler, A.C. 1977 Convective diffusion of an enzyme
reaction. SIAM J. Appl.
Math. 33, 289-297.
Fowler, A.C. 1981 Approximate solution of a model of
biological immune
responses incorporating delay. J. Math. Biol. 13, 23-45.