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The Contact Process with Avoidance

Banff International Research Station for Mathematical Innovation and Discovery

The (classical) contact process is a stochastic process on the vertices of a graph, which is a discrete, spatial model for the spread of a disease. The state of the contact process at time t is given by an infected subset of the vertices of the graph. At rate 1, each infected vertex becomes healthy, and therefore susceptible to reinfection. At rate lambda>0, each edge between an infected vertex and a healthy vertex transmits the infection, thus infecting the healthy vertex. The contact process has been thoroughly analyzed on the integer lattices and regular trees, where it is well-known to exhibit a phase transition: for large lambda, epidemics persist, while for smaller lambda, all vertices are eventually healthy. More recently, researchers have made progress in analyzing the behavior of the contact process on (finite) complex networks, where epidemics may persist for all lambda>0 on graphs with `heavy-tailed' degree distributions. I will discuss recent progress on a version of the contact process in which the edges of the graph are also dynamic: at rate alpha, each edge from an infected vertex to a healthy vertex will deactivate; the edge will become active again when the infected vertex becomes healthy, and only active edges can transmit the infection. This emulates avoidance of infected individuals by healthy individuals. We demonstrate that the long-time qualitative behavior of this model may or may not differ from the classical contact process, depending on the underlying network topology. A technical obstacle is the lack of a certain type of monotonicity, which is present for the classical model. Based on joint work with Shirshendu Chatterjee and Matthew Wascher.

Mathematics; Probability theory and stochastic processes; Statistics

video/mp4

Author affiliation: The Ohio State University

2021-01-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77100

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/