Colloquium 11/1/19 3:00pm NS212D

" A phase transition for the contact process with avoidance on Z, Zn, and the star graph" Dr. Matthew Wascher

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 lambda, the process evolves according to the following dynamics. Each infected vertex infects each of its out-neighbors at rate lambda. Simultaneously, each infected vertex become healthy at rate 1. This model is known to exhibit a phase transition in lambda for many graphs, such as the nearest-neighbor lattice Z. This means that there exists a value lambda_c such that the long term survival behavior of the epidemic when lambda < lambda_c differs from this behavior when lambda > lambda_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 alpha 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 (alpha = 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 lambda between almost sure extinction and positive probability of survival. On Z_n, we show that as the number of vertices n -> infinity, 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 -> infinity the survival time is polynomial in n for all values of lambda and alpha. This contrasts with the classical contact process where the survival time on the star graph is exponential in n for all values of lambda.

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University of Louisville, Department of Mathematics.
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