Biology Reference
In-Depth Information
of possible outcomes is unlikely to be revealed by description and
observation alone. Gaining an understanding of multivariate systems
with complex dynamics requires the use of mathematics.
Recently, the reticence to use mathematics in biology and medicine
has changed, and in some disciplines, such as the epidemiology of
infectious agents, mathematics has become a tool of choice. It is used, for
example, in defining control options and their potential impact for
pandemics of directly transmitted pathogens such as those created by the
HIV, SARS, and influenza A viruses.
18
e
25
Concomitantly, the subject of
mathematical model formulation and analysis has entered most grad-
uate programs on the epidemiology of infectious agents at research-
intensive universities.
Understanding Observed Patterns of Infection
In the case of soil-transmitted helminths such as Ascaris,mathemat-
ical models have many uses. At the simplest level, they aim to explore
how observed pattern is influenced by various biological and epide-
miological processes. The simplest example is that of the relationship
between two key epidemiological measures, the prevalence, p
(measuring the fraction of the population infected), and the average
intensity of infection (worm load or an indirect measure such as eggs
per gram of feces), M. Both are summary statistics of the frequency
distribution of parasite numbers per host (
Figure 9.1
). Statistical
procedures for fitting well-understood probability models such as the
Poisson distribution to observed data reveal that the Poisson model
(which is based on the assumption that each event occurs at random,
and independent of other events) is a poor mirror of observation. For the
Poisson, the mean equals the variance in value. Observed distributions
have variances greatly in excess of the mean, denoting high aggregation
of worms in the human population. The negative binomial model which
has two parameters, the mean M and a parameter k which varies
inversely with the degree of parasite aggregation within the host pop-
ulation, fits much better for most studies where worm numbers are
recorded by expulsion chemotherapy. This probability model reveals
that the relationship between prevalence and intensity is highly
nonlinear with definition:
Þ
k
p
¼ 1 ð1 þ
M
=
k
(9.1)
As illustrated in
Figure 9.2
, large changes in the mean M may lead to
only small changes in the prevalence p, when the degree of aggregation is
high (k
1). This simple observation, based on a well-known probability
model, has important practical implications. Where community-based
chemotherapy reduces average worm loads by a large degree, this may
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