Epidemics on critical random graphs: limits and continuum descriptions

Abstract

Understanding how diseases spread through populations is vital for mitigation efforts. For any disease at hand, the specifics of how a disease spreads through a community depends on many factors: how the disease is transmitted, how contagious the disease is, how long are individuals in the community contagious, what is the community structure of the population, are there any mitigation efforts taken, among many others. One way to sidestep the complications from studying a particularly disease at hand is to develop mathematical tools that work for understanding the behavior of epidemics on random community structures with relatively simple disease transmission mechanism. This is the topic of the present thesis. More precisely, we model the community structure by a random graph Gn on n vertices. We make the simplifying assumption that each individual in the population (represented by a vertex in the graph) who becomes infected is infected for just a single day and is then forever cured of the disease. However, during that day the infected individual transmits thedisease to each of their susceptible neighbors. We are interested in the large n behavior of Zn(t) = #{infected vertices in Gn on day t}, t = 0, 1, · · · . It turns out, the behavior of the epidemic over time corresponds to understanding certain aspects of the geometry of the connected components of the random graph. For many random graphs models, in particular the Erd˝os-R´enyi and configuration models which we consider, the geometry of the connected components undergoes a quite dramatic change depending on a parameter R0. The parameter R0 is the average number of susceptible individuals infected by a single infected individual. The connected components, and hence the number of individuals infected in the epidemic, exhibit very different properties based on if R0 < 1, R0 = 1 or R0 > 1. When R0 < 1 one typically sees that the number of individuals infected is at most logarithmic in the population size n and when R0 > 1 one typically sees that the largest possible epidemic infects order n many individuals. The situation is quite different when R0 = 1 where one often sees that each of the largest possible outbreaks when just a single individual is infected will affect Θ(n^γ) many individuals for some γ ∈ (0, 1) depending only on the random graph and not on the individuals selected. This is where we focus our attention and such random graphs Gn are called critical. A popular approach for studying critical random graphs has emerged over the past few decades. It turns out that for many models of critical random random graphs (see Chapter 3 for more details and references) the metric space structure of the largest connected components of critical random graphs has some limiting description in terms of a continuumrandom graph. Unfortunately, the standard techniques used to describe the metric space structure do not immediately imply that the process Zn has some limiting description as n → ∞. In this thesis we show that the standard approach for proving convergence of the connected components of a critical random graph to a limiting metric space, to say G , is essentially enough to show that the process Zn has some large n scaling limit, to say Z. This then gives a precise connection between properties of the metric space G and propertiesof the limiting process Z. For example, understanding compactness of the metric space G becomes understanding whether or not Z has compact support.