One of the most useful tools in analyzing the spread of disease is a system of
evolutionary equations that reflects the dynamics among three distinct categories
of a population: those susceptible (S) to a disease, those infected (I) with
it, and those recovered (R) from it. This SIR model is applicable to a range of
diseases, from smallpox to the flu. To predict the impact of a particular disease it
is crucial to determine certain parameters associated with it, such as the average
number of people that a typical infected person will infect. Researchers estimate
these parameters by applying statistical methods to gathered data, which aren.t
complete because, for example, some cases aren.t reported. Armed with reliable
models, mathematicians help public health officials battle the complex, rapidly
changing world of modern disease.

Today.s models are more sophisticated than those of even a few years ago. They
incorporate information such as contact periods that vary with age (young people
have contact with one another for a longer period of time than do adults from
different households), instead of assuming equal contact periods for everyone. The capacity to treat variability makes it possible to predict the effectiveness of targeted vaccination strategies to combat the flu, for instance. Some models now use graph theory and matrices to represent networks of social interactions, which are important
in understanding how far and how fast a given disease will spread.

For More Information: Mathematical Models in Population Biology and Epidemiology, Fred Brauer and Carlos Castillo-Chavez.