The Kermack-McKendrick model is an SIR model for the number of people infected with a contagious illness in a closed population over time.
It was proposed to explain the rapid rise and fall in the number of infected patients
observed in epidemics such as the plague (London 1665-1666, Bombay 1906) and cholera
(London 1865). It assumes that the population size is fixed (i.e., no births, deaths
due to disease, or deaths by natural causes), incubation period of the infectious
agent is instantaneous, and duration of infectivity is same as length of the disease.
It also assumes a completely homogeneous population with no age, spatial, or social
structure.

The model consists of a system of three coupled nonlinear ordinary differential equations,

(1)

(2)

(3)

where
is time,
is the number of susceptible people, is the number of people infected, is the number of people who have recovered and developed
immunity to the infection, is the infection rate, and is the recovery rate.

The key value governing the time evolution of these equations is the so-called epidemiological threshold,

(4)

Note that the choice of the notation is a bit unfortunate, since it has nothing to do with .
is defined as the number of secondary infections caused by a single primary infection;
in other words, it determines the number of people infected by contact with a single
infected person before his death or recovery.

When ,
each person who contracts the disease will infect fewer than one person before dying
or recovering, so the outbreak will peter out (). When , each person who gets the disease will infect more
than one person, so the epidemic will spread (). is probably the single most important quantity in epidemiology.
Note that the result derived above, applies only to the basic Kermack-McKendrick
model, with alternative SIR models having different
formulas for
and hence for .

The Kermack-McKendrick model was brought back to prominence after decades of neglect by Anderson and May (1979). More complicated versions of the Kermack-McKendrick model that better reflect the actual biology of a given disease are often used.

Anderson, R. M. and May, R. M. "Population Biology of Infectious Diseases: Part I." Nature280, 361-367,
1979.Jones, D. S. and Sleeman, B. D. Ch. 14 in Differential
Equations and Mathematical Biology. London: Allen & Unwin, 1983.Kermack,
W. O. and McKendrick, A. G. "A Contribution to the Mathematical Theory
of Epidemics." Proc. Roy. Soc. Lond. A115, 700-721, 1927.Wolfram
Research, Inc. "Kermack-McKendrick Disease Model." http://library.wolfram.com/webMathematica/Biology/Epidemic.jsp.