  The epidemic process on a graph is considered for which infectious contacts
occur at rate which depends on whether a susceptible is infected for the first
time or not. We show that the Vasershtein coupling extends if and only if
secondary infections occur at rate which is greater than that of initial ones.
Nonetheless we show that, with respect to the probability of occurrence of an
infinite epidemic, the said proviso may be dropped regarding the totally
asymmetric process in one dimension, thus settling in the affirmative this
special case of the conjecture for arbitrary graphs due to [Stacey (2003), {\em
Ann. Appl. Probab.} {\bf 13}, 669-690].
