The S-I-R model was originally developed to model epidemics, not
phone apps. That's why it's called S-I-R. $S$ stands for Susceptible,
meaning people who are healthy but might become sick. $I$ stands for
Infected, and $R$ stands for Recovered.
The S-I-R Model of Disease Spread
Key points:
$I=I(t)$
represents the number of infected individuals at time $t$.
$S=S(t)$ represents the number of susceptible individuals at time $t$.
$R = R(t)$ represents the number of recovered individuals at time $t$,
who are now immune to the disease.
The Model:
\begin{eqnarray}
S' & = & -aSI \cr
I' &=& aSI - bI \cr
R' & = & bI
\end{eqnarray}
These equations are identical to the equations for market penetration.
The differences are:
The coefficient $b$ is called the recovery coefficient instead
of the attrition coefficient. As before, it is the reciprocal of the
average time that an individual spends in state $I$. That is, it is the
reciprocal of the average length of the disease.
The model
is the same for epidemics as for marketing, but the interpretation is very
different. In marketing a product, we want $I$ to be as big as possible, but in
managing an epidemic we want $I$ to be as small as possible.