Contact process

In the contact process, infected vertices recover at rate $\mu$, while each oriented nearest-neighbour edge transmits infection at rate $\lambda$. The empty configuration is absorbing.

Place recovery marks at each vertex according to independent rate-$\mu$ Poisson processes and infection arrows on each oriented nearest-neighbour edge according to independent rate-$\lambda$ Poisson processes. A vertex is infected at time $t$ exactly when a time-directed path from an initially infected vertex reaches it while following arrows and avoiding recovery marks.

Reachability in the graphical representation gives

Time reversal of the Poisson graphical representation, together with reversal of every infection arrow, gives

The survival probability is the probability that the contact process begun from one infected vertex never reaches its absorbing empty state. Its critical infection rate is ${\lambda }_c(\mu )=\inf \{\lambda :\theta (\lambda ,\mu )>0\}$.