Probability Seminar

Zoe HuangDuke University
Contact process on trees

Monday, April 12, 2021 - 4:00pm

The contact process describes an epidemic model where each infected individual recovers at rate 1 and infects its healthy neighbors at rate $\lambda$. We investigate the contact process on inhomogeneous trees including periodic trees and Galton-Watson trees. For the contact process on periodic trees we prove sharp asymptotics for the critical values and the existence of an intermediate phase. For the contact process on Galton-Watson trees, we show that 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. 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 $\lambda_2$. This talk is based on joint work with Rick Durrett.


Meeting ID: 941 9483 5917
Passcode: prob