# Rick Durrett : Diffusion limit for the partner model at the critical value

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and disease transmission only occurs within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds) then the critical model survives for time \(O(N^{1/2})\). Here we improve that result by proving the convergence of \(i_N(t)=I(tN^{1/2})/N^{1/2}\) to a limiting diffusion. We do this by showing that in the first O(1), this four dimensional process collapses to two dimensions: the number of *SI* and *II* partnerships are constant multiples of the the number of infected singles \(I_t\). The other variable \(Y_t\), the total number of singles, behaves like an Ornstein-Uhlenbeck process on a time scale O(1) and averages out of the limit theorem for \(i_N(t)\).
This is joint work with Anirban Basak and Eric Foxall.

**Category**: Probability**Duration**: 01:14:48**Date**: September 22, 2016 at 4:25 PM**Views**: 104-
**Tags:**seminar, Probability Seminar

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