Comparison inequalities for discrete- and continuous-time infection processes

Abstract

We study comparison inequalities for contact-process-type infection dynamics on $\mathbb Z^d$ with general local transmission mechanisms. The processes considered include both discrete-time oriented-percolation models and continuous-time contact processes in which infection events may transmit to random, possibly correlated sets of neighbouring sites. Our main results apply local-to-global comparison criteria beyond stochastic domination: if one local infection rule is more likely than another to hit every non-empty test set of neighbours, then the corresponding process has larger survival probabilities at all times. We compare classical independent-infection models with exchangeable fixed-budget, all-or-nothing, and burst-type infection mechanisms, deriving orderings for example of finite-time and infinite-time survival probabilities and stopping-time distributions. In continuous time, we further obtain consequences for critical survival thresholds of exchangeable infection laws. The results show that infection processes with identical or comparable mean-field infection intensity can nevertheless be rigorously ordered at the stochastic level once spatial geometry is taken into account. Additionally, it turns out that the classical continuous-time contact process dominates a variety of related continuous-time models, whereas its most natural discrete-time version, Bernoulli oriented percolation, is not dominant with respect to related discrete-time models.