# 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

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