SIR model
The SIR model describes an epidemic in which susceptible individuals can become infectious and, after recovery or removal, do not return to the susceptible class during the period being modelled.
Historical note. The classical SIR framework originates from the epidemic model developed by William O. Kermack and Anderson G. McKendrick in 1927.
\[S(t)=\text{susceptible},\qquad I(t)=\text{infectious},\qquad R(t)=\text{recovered or removed}.\]For a closed population with no births or deaths included separately,
\[S(t)+I(t)+R(t)=N.\]Why does the model make sense?
The model represents two biological processes: infection moves people from \(S\) to \(I\), and recovery or removal from infectiousness moves people from \(I\) to \(R\).
Under homogeneous mixing, the infectious fraction of the population is \(I/N\). If \(\beta\) is the effective transmission-rate parameter, one susceptible individual experiences infection at rate \(\beta I/N\). With \(S\) susceptible individuals, the total infection rate is
\[\beta\frac{SI}{N}.\]If each infectious individual leaves the infectious class at rate \(\gamma\), then the total recovery/removal rate is
\[\gamma I.\]Building the equations from the diagram
The susceptible class has only an outflow, infection:
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}}.\]The infectious class has an inflow from new infections and an outflow from recovery:
\[\boxed{\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I}.\]The recovered/removed class receives the individuals leaving \(I\):
\[\boxed{\frac{dR}{dt}=\gamma I}.\]What behaviour does the SIR model predict?
At first, if transmission into \(I\) is faster than recovery out of \(I\), the infectious population grows. As infection spreads, \(S(t)\) decreases. This reduces the infection term \(\beta SI/N\). Eventually there may be too few susceptible individuals for new infections to replace those recovering, and \(I(t)\) begins to fall.
This explains the familiar epidemic pattern: growth, a peak, and decline. Importantly, the decline can occur even while susceptible people remain; the epidemic does not need to infect everyone before it starts falling.
The epidemic threshold
From the infectious equation,
\[\frac{dI}{dt}=\left(\beta\frac{S}{N}-\gamma\right)I.\]At the beginning of an outbreak in a nearly fully susceptible population, \(S/N\approx1\), so
\[\frac{dI}{dt}\approx(\beta-\gamma)I.\]The basic reproduction number for this simple model is
\[\boxed{R_0=\frac{\beta}{\gamma}}.\]If \(R_0>1\), a small introduction can initially grow in a nearly fully susceptible population. If \(R_0<1\), infections decline from the start.
When is an SIR model useful?
The SIR structure is useful when individuals who recover can reasonably be treated as protected from reinfection for the time period being studied. It is a classical approximation for acute immunising infections. Measles is a standard example because infection usually produces long-lasting immunity.
SIR can also be useful as a simplified model when the details of an exposed stage are not important for the particular question.
When is it not appropriate?
If infection has an important latent period between becoming infected and becoming infectious, an SEIR model can represent that stage explicitly. If immunity is absent, short-lived, or repeatedly lost, SIS or SIRS-type models may be more appropriate. Additional compartments may also be needed when age, vaccination, hospitalisation, births and deaths, or other biological mechanisms are central to the question.