Diffusion limit for the partner model at the critical value
© 2018, University of Washington. All rights reserved. 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 I0) the extinction time in the critical model is of order √N. Here we improve that result by proving the convergence of iN(t)=I(√Nt)/√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 √N on the original time scale and averages out of the limit theorem for iN(t). As a by-product of our proof we show that if τN is the extinction time of iN(t) (on the √N time scale) then τN has a limit.
Basak, A; Durrett, R; Foxall, E
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