Critical value for contact processes on clusters of oriented bond percolation

•We study contact processes on clusters of oriented bond percolation.•We introduce the definitions of the critical values in the annealed case and quenched case.•We show that the two critical values are equal.•We show that the critical value is close to 1/dp1/dp for large dimension dd, where pp is the probability of open.

In this paper we are concerned with contact processes on open clusters of oriented bond percolation in ZdZd, where the disease spreads along the direction of open edges. We show that the two critical values in the quenched and annealed cases are equal with probability one and are asymptotically equal to (dp)−1(dp)−1 as the dimension dd grows to infinity, where pp is the probability that the edge is ‘open’.