An SIR model is a compartmental epidemiological model for the time evolution of a closed population divided into susceptible people 
, infectious people 
, and removed people 
. Depending on the model, removed people may have recovered
with immunity, been isolated, or died. One of the simplest SIR models is the Kermack-McKendrick
model.
A common generalization allows the transmission and removal rates to vary with time, giving
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(1)
|
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(2)
|
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(3)
|
Since the three derivatives sum to 0, the total population is conserved,
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(4)
|
Estimating the functions 
and 
from observations of 
and 
is a coefficient-identification inverse
problem (Marinov et al. 2014, Marinov and Marinova 2020).
An intervention can be modeled by replacing 
with 
, where the control 
takes values between 0 and 1. For the constant-rate SIR
model with 
,
Britton and Leskelä (2023) proved that among piecewise
continuous controls satisfying 
and 
, where 
and 
, total incidence (the fraction of initially
susceptible people eventually infected) is minimized by applying the maximum control
on one interval of length 
. The optimal starting time
is unique.
The Season 1 episode "Vector" (2005) of the television crime drama NUMB3RS features SIR models.
