We consider evoSIR, a variant of the SIR model, on Erdős-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate ρ and rewire to a randomly chosen individual. We compute the critical infection rate λ c and the probability of a large epidemic by showing that they are the same for the delSIR model in which S − I connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at λ c .

A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values λ 1 and λ 2 for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will continue the study of the periodic case. Two significant new results give sharp asymptotics for the critical value λ 2 of (1, n) trees, λ 2 (n) ∼ 0.5(log n)/n, and generalize that result to the (a 1 , . . . , a k , n) tree when max i a i ≤ n 1−ǫ and a 1 • • • a k = n b . We also give results for (a, b, c) trees. Our results in this case improve those found by Pemantle. However, the values come from solving cubic equations, so the explicit formulas are not pretty, but it is surprising that they depend only on a + b + c and abc.

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Red and blue particles are placed in equal proportion throughout either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We study the expected number of steps needed to extinguish every particle. In particular, we compare this quantity to the one-type setting, and study the effect of asymmetric particle speeds.