First we look at some simple models in which we assume a homogeneous population and random contacts. Remarkably, these illustrate some of the basic ideas of the epidemiology of infectious diseases. Figure 4 shows connectivity diagrams for two types of diseases.
Figure 4
An SIS disease is one in which susceptibles (S) become
infected (I) and recover without immunity and so are
again susceptible. X are the susceptibles and Y
are the infecteds and infectious individuals. The
connectivity diagram shown is for a constant, closed
population without deaths. The rate of recovery per
infected is 
, a constant. However, the rate
coefficient for infection, called the force of
infection, 
is not a constant but is a function
of X and Y.
The other model is for an SIR disease in which a
susceptible (S) becomes infected (I) and infecteds that
recover are immune (R). This model is a bit more
complicated in that there is a constant recruitment of
new susceptibles at rate U and there is a background
mortality rate coefficient, 
, which is the same for
susceptibles, infecteds and immunes, i.e. there are no
extra deaths due to the disease. The force of infection,
, is a function of
X,
Y and
Z; we shall develop that shortly. We shall use the SIR
model to show how one writes the equations and then use
it to develop some of the most important ideas in the
epidemiology of infectious diseases.
