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

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

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