Contact process (mathematics)

The contact process is a model of an interacting particle system. It is a continuous time Markov process with state space ${\displaystyle \{0,1\}^{S}}$, where ${\displaystyle S}$ is a finite or countable graph, usually Z${\displaystyle {}^{d}}$. The process is usually interpreted as a model for the spread of an infection: if the state of the process at a given time is ${\displaystyle \eta }$, then a site ${\displaystyle x}$ in ${\displaystyle S}$ is "infected" if ${\displaystyle \eta (x)=1}$ and healthy if ${\displaystyle \eta (x)=0}$. Infected sites become healthy at a constant rate, while healthy sites become infected at a rate proportional to the number infected neighbors. One can generalize the state space to ${\displaystyle \{0,\ldots ,\kappa \}^{S}}$, such is called the multitype contact process. It represents a model when more than one type of infection is competing for space.

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