First introduced and studied by Harris (1974), the Contact Process (CP) is a stochastic process usually used to model the spread of an epidemic in a population. We model population by a graph, where vertices represent the individuals and edges the possible pathways for infection to spread. Each individual can be "healthy" or "infected". The process evolves according to the following dynamic. Each infected vertex become healthy at rate 1. Simultaneously each infected vertex infects all of its neighbours with rate λ.
In this talk, we will give some general and basic properties of the CP, and we will focus on the following interesting and natural question: does the infection become extinct or not ? We will show that for many graphs, we can exhibit a (at least one) phase transition.