4.1: The SI Model
The simplest model of an infectious disease categorizes people as either susceptible or infective \((S I)\). One can imagine that susceptible people are healthy and infective people are sick. A susceptible person can become infective by contact with an infective. Here, and in all subsequent models, we assume that the population under study is well mixed so that every person has equal probability of coming into contact with every other person. This is a major approximation. For example, while the population of Amoy Gardens could be considered well mixed during the SARS epidemic because of shared water pipes and elevators, the population of Hong Kong as a whole could not because of the larger geographical distances, and the limited travel of many people outside the neighborhoods where they live.
We derive the governing differential equation for the SI model by considering the number of people that become infective during time \(\Delta t\). Let \(\beta \Delta t\) be the probability that a random infective person infects a random susceptible person during time \(\Delta t\). Then with \(S\) susceptible and \(I\) infective people, the expected number of newly infected people in the total population during time \(\Delta t\) is \(\beta \Delta t S I\). Thus,
\[I(t+\Delta t)=I(t)+\beta \Delta t S(t) I(t) \nonumber \]
and in the limit \(\Delta t \rightarrow 0\),
\[\dfrac{d I}{d t}=\beta S I \nonumber \]
We diagram (4.1.1) as
\[S \stackrel{\beta S I}{\longrightarrow} I . \nonumber \]
Later, diagrams will make it easier to construct more complicated systems of equations. We now assume a constant population size \(N\), neglecting births and deaths, so that \(S+I=N\). We can eliminate \(S\) from (4.1.1) and rewrite the equation as
\[\dfrac{d I}{d t}=\beta N I\left(1-\dfrac{I}{N}\right) \nonumber \]
which can be recognized as a logistic equation, with growth rate \(\beta N\) and carrying capacity \(N\). Therefore \(I \rightarrow N\) as \(t \rightarrow \infty\) and the entire population will become infective.