Mathematics Colloquium: Matthew Wascher, "A phase transition for the contact process with avoidance on Z, Z_n, and the star graph"

Matthew Wascher, of the Ohio State University, will be delivering this week's mathematics colloquium.

Abstract: The (classical) contact process, or SIS epidemic, is a model for the spread of disease through a population. We model the population with a graph, where vertices represent individuals and directed edges represent potential pathways for infection to spread. At any given time, each vertex is either "infected" or "healthy," and given an infection rate parameter λ, the process evolves according to the following dynamics. Each infected vertex infects each of its out-neighbors at rate λ. Simultaneously, each infected vertex become healthy at rate 1. This model is known to exhibit a phase transition in λ for many graphs, such as the nearest-neighbor lattice Z. This means that there exists a value λ_c such that the long term survival behavior of the epidemic when λ<λ_c differs from this behavior when λ>λ_c.

We consider a modified contact process that we call the contact process with
avoidance. The process retains the infection and recovery dynamics of the
classical contact process, but in addition each healthy vertex can avoid each
of its infected neighbors at rate α by turning off the directed edge from
that infected neighbor to itself until the infected neighbor recovers. This
model presents a challenge because, unlike the classical contact process (α=0), it has not been shown to be an attractive particle system. We study the
survival dynamics of this model on the nearest-neighbor lattice Z, the cycle
Z_n, and the star graph. On Z, we show there is a phase transition in λ
between almost sure extinction and positive probability of survival. On Z_n, we
show that as the number of vertices n→∞, there is a phase transition
between log and exponential survival time in the size of the graph. On the star
graph, we show that as n→∞ the survival time is polynomial in n for
all values of λ and α. This contrasts with the classical contact
process where the survival time on the star graph is exponential in n for all
values of λ.