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 [Ann. Appl. Probab. 13 (2003) 669–690].
"A note on monotonicity of spatial epidemic models." Braz. J. Probab. Stat. 33 (3) 674 - 684, August 2019. https://doi.org/10.1214/18-BJPS406