Contact processes on the integers

TL;DR

This paper studies a modified contact process on integers, exploring growth regularity, convergence rates, endpoint behavior, and monotonicity properties under various conditions.

Contribution

It provides new insights into the three state contact process, including growth regularity, convergence, endpoint behavior, and monotonicity on integer lattices.

Findings

Growth of the process is regular under reverse immunization.

Convergence rates of the process are characterized.

The right endpoint exhibits specific i.i.d. behavior.

Abstract

The three state contact process is the modification of the contact process at rate $\mu$ in which first infections occur at rate $\lambda$ instead. Chapters 2 and 3 consider the three state contact process on (graphs that have as set of sites) the integers with nearest neighbours interaction (that is, edges are placed among sites at Euclidean distance one apart). Results in Chapter 2 are meant to illustrate regularity of the growth of the process under the assumption that $\mu \geq \lambda$, that is, reverse immunization. While in Chapter 3 two results regarding the convergence rates of the process are given. Chapter 4 is concerned with the i.i.d.\ behaviour of the right endpoint of contact processes on the integers with symmetric, translation invariant interaction. Finally, Chapter 5 is concerned with two monotonicity properties of the three state contact process.