Introduction
The over-arching concept,
which forms the basic requirement for the analytical methods, is the
relationship between social and political coherence and the experience
of pandemic disease. Hence there is a need to address four axiomatic
questions asked in all cases: What has happened? What might
happen? What could happen? What should happen?
The main question, then,
for analytical methods is as follows:
What are the appropriate modes of analysis to grip these diverse
data and make them tractable for effective policy response?
The analytical methods
presented will explore what may happen when social and political systems
are moving towards trigger regions. Simple mathematical constructs
provide a formal description of the control factors to help us to
understand the impact of various intervention options. Three main
topics will introduce the mathematical background for the discussion:
Catastrophe Theory (to explain the characteristics and control drivers
of a system as it approaches a threshold region), Epidemiological
modelling (in particular looking towards reasons why standard epidemiological
models appear not to be valid for HIV), Social networks and Triage
(to discuss the epidemiological implications of sex-networks and the
political implications of having to apply Triage principles).
1.
Cusps, thresholds and catastrophe theory
Firstly let me demonstrate
the significance of the cusp and the underlying control surface on
which it sits. The control surface is described by two main dynamics:
Imagine a ruler held horizontally
and under pressure so that it forms an arch. If a gradually increasing
weight is applied to the middle of the arched ruler there will be
a limiting point at which the ruler suddenly gives way. If this limiting
point is plotted on a two-dimensional graph where the two axes are
the size of the weight and the degree of horizontal pressure, then
we can build up a locus of points over all the feasible values of
inward pressure and load weight. This locus of "collapsing points"
is described by a cusp on the pressure-load control plane. The "collapsing
points" are points of minimum energy for the loaded and pressured
system and the key issue is that within the cusp region there are
two ruler positions that have minimum energy (the system is bi-stable).
Epidemiological models
tend to use two main parameters: critical threshold and virulence;
these will be discussed in detail later but it is important to understand
that, in general, thresholds are not fixed and we need to have a formal
model to help us to understand why they move.
We can use the Zeeman
catastrophe machine [Poston & Stewart] to describe states of a
system so that implications and consequences of change or intervention
in that system can be formally explored. The catastrophe machine (see
Annex) shows the nature of change in a system under "tight"
control against change when the system is under "loose"
control. How should we define tight and loose in this context of political
control and means for intervention?
The model in Woodcock
and Davis uses catastrophe theory to show the interactions between
popular involvement and political control. Their examples (pp 135-146
[Woodcock]) illustrate that catastrophe theory has been applied successfully
in the political landscape.
2.
Socio-dynamics and epidemiology
There are many types of
socio-dynamics models [Gass, Cobb] mostly based on systems dynamics
techniques [for example Blanche and Stella]. These techniques depend
on well-defined equations to describe the feed-back and feed forward
relationships between system entities. Gass incorporates catastrophe
folds into his differential equations to introduce discontinuities
into the dynamics of the system. These techniques use influence diagrams
to describe the system under study. There are several other useful
proprietary influence diagram techniques (such as Analytica).
Several different systems
and sub-systems must be described in terms of their dynamics: epidemiological
(HIV spread rates and characteristics), political control, religious
and social influences, etc. Epidemiologists use threshold/diffusion
models to determine the spreading characteristics of diseases. Each
individual has a likelihood for contracting the disease (an individual
threshold) dependent on their lifestyle and values. Diffusion models
use these thresholds in combination with a measure of virulence and
a notion of random contact between individuals (that allows derivation
of a critical threshold for the network of individuals) to determine
a spreading rate. If the spreading rate is less than the critical
threshold for a given time (as was the case for SARS) then the disease
gradually dies out. The diffusion model holds for most diseases because
the of the assumption that contact networks are essentially random.
The contact networks for HIV however are something other than random.
3. Social networks and
Triage
[Barabasi] makes the case
that the spread of HIV resembles more the spread of computer viruses
than biological viruses such as SARS. Socio-sexual networks are similar
in structure to the Internet in that they are scale-free networks.
The distribution curve for the number of links between different nodes
in a scale-free network follows an exponential curve (similar to the
Pareto 80/20 distribution). The number of nodes with exactly K links
follows a power law (with a unique exponent that usually varies between
2 and 3). (To visualize the physical differences between random and
scale-free networks envisage a representation of a road network between
US cities and an airline-route network - this then shows the significance
of "hubs")
Sexual networks tend to
be scale-free and the defining characteristic of the "sex-web"
is a hub node (formally defined by Roman in her PhD study of Stockholm
students); for example, Gaetan Dugas and Wilt Chamberlain who had
sexual contact with on average 250 partners per year. Given that sexual
contact essentially follows a Pareto curve perhaps targeted intervention
should be aimed at the top-20 group of individuals?
There are clear implications
for treatment, education, control and vaccine administration. Due
to limited resources and time it seems imperative that we employ the
principles of Triage. Who should be targeted and how should it be
done? The main problem is that the situation, due to its criticality,
demands triage measures. There is one other important dimension of
the situation, though, that makes it difficult to establish triage
rules and that is unpredictability. When we have criticality and unpredictability
it is perhaps wise to work to reduce both before trying to intervene
directly using triage measures. New models of epidemiology, specific
to HIV, will help reduce an aspect of unpredictability. It will also
point to the rates of change that need to be and can be slowed and
this will help to adjust the time available (one of the critical resources).
So initially the thinking
should perhaps move away from individual targeting, towards broader
ways of reducing the triage-inducing pressures by working to reduce
social stigma and elevating responsibility for AIDS to a global level
whilst managing fear and maintaining respect and support (moral and
financial). This then brings us back to cusps. The understanding of
the control parameters keeping us within the cusp region gives us
an indication of what should happen; one answer being to extend the
range of intervention measures (by changing global constraints and
values) that could happen.
References
Barabasi A, Linked:
the new science of networks, Perseus 2002
See also: Z.Dezso & A-L.Barabasi Halting Viruses in Scale-free
Networks, Nov 2003.
Gass N, Forecasting
Crisis and Conflict: Entropies of Geopolitical Regions, Research
Note 4/94 Department of National Defence, Ottawa Canada, August 1994
Cobb L and Thrall R, Mathematical
Frontiers in Social political sciences, Chapter 2: Stochastic Differential
equations pp 53-57 Stochastic Epidemic Theory, AAAS Washington
DC, 1981.
Woodcock AER and Davis
M, Catastrophe Theory: a revolutionary new way of understanding
how things change, Penguin Books 1978. Chapter 8: Applications
in politics and public opinion.
T.Poston and I.Stewart,
Catastrophe Theory and its Applications, Pitman 1978
ANNEX
The Catastrophe Machine
The diagram of a Catastrophe machine (adapted from [Poston&Stewart]
Figure 7.3) illustrates how a controller can effect change in a non-linear
system. The Catastrophe machine comprises a disk mounted on a vertical
board and controlled and constrained by elastic bands. The observable
system is the disk that is free to rotate about a central point fixed
to the mounting board. Two lengths of elastic are attached to the
disk at one point (B) on its circumference. The lower piece of elastic
is attached to the mounting board at a point (A) directly below the
fixed point of the disk. The end of the other piece of elastic (C)
is free to move.
Simple experimentation
illustrates the difficulties of modelling complex non-linear systems
that are constrained and have a well-defined degree of external control.
When the external controller moves point C in a horizontal direction
he changes the state of the observable system by causing it to rotate.
The vertical position
of the point C is determined by the degree of control required over
the system. When the elastic is stretched so that point C lies at
the top extremity of the mounting board, the disc rotates smoothly
but the amount of lateral movement/ disc rotation is very restricted.
When the elastic is slack and point C is at the lowest point of the
control surface near to the top edge of the disc, again the movement
is smooth but limited. If the external controller wants to achieve
a big change in the observable system, he has to move the point C
within the area of the cusp on the control surface. At the mid-line
of the cusp on the control surface, the external controller can achieve
maximum movement of the disc (i.e. 180 degrees rotation) but this
system change involves a large catastrophic jump.
The main lesson to be
learnt from using such simple physical models is that a full and detailed
mathematical understanding of the underlying control surface is essential.
If a high degree of control is required to implement a large system
change then a mathematical description of the underlying control surface
will help predict the system behaviour. Manipulation of the degrees
of freedom and the dimensions of the control space may help to find
ways of effecting change that will avoid catastrophic discontinuities.
Our models must be built so that we can gain understanding of system
behaviour under all conditions of external control, so that we can
predict the dynamics of system change.
