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