Kermack-McKendrick Model

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.

REFERENCES:

Anderson, R. M. and May, R. M. "Population Biology of Infectious Diseases: Part I." Nature 280, 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. A 115, 700-721, 1927.

Wolfram Research, Inc. "Kermack-McKendrick Disease Model." http://library.wolfram.com/webMathematica/Biology/Epidemic.jsp.