A note on monotonicity of spatial epidemic models

TL;DR

This paper investigates the conditions under which the monotonicity property holds in spatial epidemic models, particularly focusing on the extension of the Vasershtein coupling and its implications for infinite epidemic probabilities.

Contribution

It establishes that the Vasershtein coupling extends only when secondary infection rates exceed initial infection rates, and confirms monotonicity for a one-dimensional asymmetric process, settling a special case of a conjecture.

Findings

Vasershtein coupling extension depends on infection rates

Monotonicity holds in one-dimensional asymmetric models

Affirms a special case of Stacey's conjecture

Abstract

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].