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2018 Diffusion limit for the partner model at the critical value
Anirban Basak, Rick Durrett, Eric Foxall
Electron. J. Probab. 23: 1-42 (2018). DOI: 10.1214/18-EJP229


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 [7] found the critical value and studied the subcritical and supercritical regimes. Recently Foxall [4] 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.


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Anirban Basak. Rick Durrett. Eric Foxall. "Diffusion limit for the partner model at the critical value." Electron. J. Probab. 23 1 - 42, 2018.


Received: 19 May 2017; Accepted: 2 October 2018; Published: 2018
First available in Project Euclid: 12 October 2018

zbMATH: 06970407
MathSciNet: MR3870445
Digital Object Identifier: 10.1214/18-EJP229

Primary: 60K35
Secondary: 60F17


Vol.23 • 2018
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