# Epidemic process

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

$$ F ( t ; x , y ) = {\mathsf E} x ^ {\mu _ {1} ( t) } y ^ {\mu _ {2} ( t) } $$

satisfies the differential equation

$$ \frac{\partial F }{\partial t } = \lambda ( x ^ {2} - x y ) \frac{\partial ^ {2} F }{\partial x \partial y } + \mu ( 1 - x ) \frac{\partial F }{\partial x } . $$

#### Comments

#### References

[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.*URL: http://encyclopediaofmath.org/index.php?title=Epidemic_process&oldid=46834