SI model
The SI model is the simplest model of an infection spreading through a population. It divides a closed population into only two groups:
\[S(t)=\text{number of susceptible individuals},\]\[I(t)=\text{number of infectious individuals}.\]There is no arrow leaving \(I\): once an individual becomes infected, the SI model treats that individual as remaining infected for the rest of the period being studied.
\[S(t)+I(t)=N.\]Why does the model make sense?
An infection can spread only when susceptible and infectious individuals come into contact. If there are many infectious individuals, susceptible individuals have more opportunities to encounter infection. If there are very few susceptible individuals left, there are fewer people available to become infected.
Under homogeneous mixing, the fraction of the population that is infectious is
\[\frac{I}{N}.\]If \(\beta\) represents the effective transmission rate, then the infection rate among all susceptible individuals is
\[\beta S\frac{I}{N}=\beta\frac{SI}{N}.\]Each new infection removes one individual from \(S\) and adds that same individual to \(I\). Therefore,
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}},\qquad \boxed{\frac{dI}{dt}=\beta\frac{SI}{N}}.\]What behaviour does it predict?
As long as both \(S>0\) and \(I>0\),
\[\frac{dI}{dt}=\beta\frac{SI}{N}>0,\]so the number infected increases. Because the model has no recovery or removal process, infected individuals accumulate. Eventually, in the idealised model, the susceptible population approaches zero and almost the whole population becomes infected.
When is an SI model useful?
The SI model is appropriate when infection can reasonably be treated as effectively permanent over the time period of interest. A standard biological example is an untreated lifelong infection, such as HIV infection before effective treatment is represented in the model: an infected person does not naturally return to the susceptible class.
It can also be useful as a deliberately simplified first model when the main question is how transmission alone changes the susceptible and infected populations, or when recovery occurs on a much longer time scale than the period being studied.
When is it not appropriate?
The SI model is not a good description of infections in which recovery is an important part of the dynamics. For infections such as influenza or COVID-19, individuals normally leave the infectious state after a finite infectious period. Ignoring that process would make the SI model predict that infectious individuals remain infectious indefinitely and would therefore overstate the long-term infectious population.
For such diseases, an SIR or SEIR-type model is usually more appropriate. An SEIR model is particularly useful when an exposed or latent period between infection and infectiousness needs to be represented.