**Project leaders:** Professors Rick Durrett and Matt Junge**Project manager:** Zoe Huang**Team members:** Melody Jiang, Remy Kassem, Grayson York, and Brandon Zhao

**Project summary: **

We investigated critical values for the contact process on various infinite trees this summer. The contact process is a process based on the spread of disease where nodes can be either infected or susceptible. Infected sites become susceptible at a constant rate and susceptible sites become infected based on some infection rate and how many infected neighbors they have.The first critical value is the smallest infection rate such that the infection always survives, and the second critical value is the smallest infection rate such that the infection always survives near where the infection started.

It has already been shown that these two critical values differ for a large class of trees. We worked to find tighter bounds for the critical values for certain trees. Specifically, on the (1,n) tree, where nodes alternate between having 2 and n+1 neighbors (that is, 1 or n children and one parent), we were able to prove a tight asymptotic bound for the second critical value.

This project has culminated in the following paper, available on arXiv:

Yufeng Jiang, Remy Kassem, Grayson York, Brandon Zhao, Xiangying Huang, Matthew Junge, and Rick Durrett: **The contact process on periodic trees**