## Electronic Journal of Probability

### Diffusion limit for the partner model at the critical value

#### 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 [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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 102, 42 pp.

Dates
Accepted: 2 October 2018
First available in Project Euclid: 12 October 2018

https://projecteuclid.org/euclid.ejp/1539309901

Digital Object Identifier
doi:10.1214/18-EJP229

Mathematical Reviews number (MathSciNet)
MR3870445

Zentralblatt MATH identifier
06970407

#### Citation

Basak, Anirban; Durrett, Rick; Foxall, Eric. Diffusion limit for the partner model at the critical value. Electron. J. Probab. 23 (2018), paper no. 102, 42 pp. doi:10.1214/18-EJP229. https://projecteuclid.org/euclid.ejp/1539309901

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