Probability Seminar: Zoe Huang, Duke

Zoe Huang
Duke

The contact process on Galton-Watson trees

Abstract: The contact process describes an epidemic model where each

infected individual recovers at rate 1 and infects its healthy neighbors

at rate $\lambda$. We show that for the contact process on Galton-Watson

trees, when the offspring distribution (i) is subexponential the

critical value for local survival $\lambda_2=0$ and (ii) when it is

Geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much

smaller than previous estimates. This is based on an improved (and in a

sense sharp) understanding of the survival time of the contact process

on star graphs.  Recently it is proved by Bhamidi, Nam, Nguyen and Sly

(2019) that when the offspring distribution of the Galton-Watson tree

has exponential tail, the first critical value $\lambda_1$ of the

contact process is strictly positive. We prove that if the contact

process survives then the number of infected sites grows exponentially

fast. As a consequence we show that the contact process dies out at the

critical value $\lambda_1$ and does not survive strongly at $\lambda_2$.

Based on joint work with Rick Durrett.

Refreshments will be served at 3:45 in the 3rd floor lounge of Hanes Hall