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Epidemic process

From Encyclopedia of Mathematics - Reading time: 1 min


A random process (cf. Stochastic process) that serves as a mathematical model of the spread of some epidemy. One of the simplest such models can be described as a continuous-time Markov process whose states at the moment $ t $ are the number $ \mu _ {1} ( t) $ of sick persons and the number $ \mu _ {2} ( t) $ of exposed persons. If $ \mu _ {1} ( t) = m $ and $ \mu _ {2} ( t) = n $, then at the time $ t $, $ t + \Delta t $, $ \Delta t \rightarrow 0 $, the transition probability is determined as follows: $ ( m , n ) \rightarrow ( m + 1 , n - 1 ) $ with probability $ \lambda _ {mn} \Delta = O ( \Delta t ) $; $ ( m , n ) \rightarrow ( m - 1 , n ) $ with probability $ \mu m \Delta t + O ( \Delta t ) $. In this case the generating function

F(t;x,y)=Exμ1(t)yμ2(t)

satisfies the differential equation

Ft=λ(x2xy)2Fxy+μ(1x)Fx.

Comments[edit]

References[edit]

[a1] N.T.J. Bailey, "The mathematical theory of infections diseases and its applications" , Hafner (1975)
[a2] D. Ludwig, "Stochastic population theories" , Springer (1974)

How to Cite This Entry: Epidemic process (Encyclopedia of Mathematics) | Licensed under CC BY-SA 3.0. Source: https://encyclopediaofmath.org/wiki/Epidemic_process
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