The contact process on Galton-Watson trees

The contact process on Galton-Watson trees

Probability Seminar

Zoe Huang (Duke)

Thursday, December 5, 2019 -
4:15pm to 5:15pm
Location: 
125 Hanes Hall UNC

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.

Last updated: 2020/04/02 - 2:34pm