Graphical representation of the contact process

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