Similar mathematical models can be used to analyze the spread of any number of other diseases. In fact, mathematical models are often used by epidemiologists to test the effectiveness of different control measures for stemming the threat of diseases, as well as to predict how a given disease will spread throughout a community. Mathematical models can even be used to look at the behavior of epidemics of the past for historical or sociological analyses. A few recent examples of the application of these types of models have been selected:
In 2001, Britain’s agricultural community was paralyzed by an
explosive epidemic of foot and mouth disease
(article).
Foot and mouth is a highly
contagious viral disease that infects cows and pigs, characterized by
blisters on the mouth and feet or hooves. Two main control strategies
were attempted: vaccination and culling.
Mathematical epidemiologists attempted to model how well each of these
methods would work in garnering a strangle-hold on the epidemic. In order
to compare these methods, models were created comparing spatial contact
patterns between farms. The most interesting aspect of the models used
in this study was the stochastic dependence of infection rates on the
distance between infectious and susceptibles.
It was found by Ferguson et al that culling and vaccination were
effective in stemming the spread of the disease, and that culling was
more effective than vaccination. For maximum effectiveness, all livestock
on the infected farm must be slaughtered within 24 hours, as well as the
livestock of neighboring farms within 48 hours.
Factoid! Did you know that livestock infected with
foot and mouth disease can transmit the disease by wind in the form of
plumes of virus contained within droplets over distances up to 60 km
over land, or 250 km over water?
(Ref 4)
Several mathematicians have made attempts to determine the best
response to the re-introduction of smallpox into a population due to
bio-terrorism. Two vocal mathematicians in the field are Halloran and
Kaplan. They disagree over the merits of pre-event versus post-event mass
vaccinations. The reasoning for this disagreement is that the Halloran’s
model predicts a significantly faster rate of infection than Kaplan’s,
leading to the conclusion that a post-event vaccination strategy would
not be fast enough to contain the outbreak.
The major difference between the two’s models is that Halloran’s
is a discrete individual model, whereas Kaplan’s is a more traditional
continuous population model. For the past 30 years, there has been debate
over which type of model more accurately predicts epidemiological (and other)
events. Proponents of the discrete individual model believe that the model
more accurately reflects the behavior of smallpox among small populations
by modeling individuals and that this advantage can be extended to larger
populations. Opponents of this model believe that it does not scale well,
and that the model cannot predict the behavior of a disease in a large
population.
(Ref 5)