New Results for the Two-Stage Contact Process

Abstract

In this paper, we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999). We also fill in the details of an incomplete proof.

Duality and complete convergence for multi-type additive growth models

“…Taking the model on Z d with nearest-neighbour interactions, the conditions of Theorem 4 are satisfied, so complete convergence holds. This was shown in [4], with the proof given along the same lines but specifically for the two-stage contact process.…”

“…This is a natural generalization of the contact process introduced in [7] and further studied in [4] in which (viewing the contact process as a model of population growth) there is an…”

“…The term 'growth model' is borrowed from [3], though we work here in continuous rather than discrete time, and with systems that in general may have more than one type of active particle. Examples of additive growth models are the two-stage contact process in [7] and [4], the household model studied in [14], and a spatial analogue of any multi-type branching process (see [10] for an introduction to these processes). On the other hand, the multi-type contact process from [11] is not an additive growth model because it is not additive.…”

Two limit theorems for the high-dimensional two-stage contact process

“…Proposition 2.1. (Foxall, 2015) If λ c and λ c are defined as in Equations (2.1) and (2.2) respectively, then λ c = λ c .…”

“…In Krone (1999), a duality relationship between the two-stage contact process and a 'on-off' process is given. Several important open questions are proposed at the end of Krone (1999), some of which are answered in Foxall (2015) authored by Foxall. For instance, it is shown in Foxall (2015) that the complete convergence theorem holds for the two-stage contact process, i.e., the process converges weakly to a convex combination of two invariant distributions δ 0 and ν, where δ 0 is concentrated on the state where all the vertices are in state 0 and ν is the upper invariant law of the process, the definition of which we will review in the next section.…”