# Diffusion limit for the partner model at the critical value

## TL;DR

This paper proves that the scaled infection process in a critical partner model converges to a diffusion process, providing a detailed understanding of the epidemic's extinction time at the critical threshold.

## Contribution

It establishes the convergence of the scaled infection process to a diffusion and analyzes the extinction time in the critical regime, improving previous results.

## Key findings

- The infection process converges to a limiting diffusion.
- The extinction time has a well-defined limit.
- The process collapses to two dimensions rapidly.

## Abstract

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring 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 $I_0$) the extinction time in the critical model is of order $\sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(\sqrt{N}t)/\sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, 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. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $\sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $\tau_N$ is the extinction time of $i_N(t)$ (on the $\sqrt{N}$ time scale) then $\tau_N$ has a limit.