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
(1)
(2)
(3)
Since the three derivatives sum to 0, the total population is conserved,
(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.
, 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.
and 
from observations of 
and 
is a coefficient-identification inverse
problem (Marinov et al. 2014, Marinov and Marinova 2020).
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.
