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Diffusion limit for the partner model at the critical value. (English) Zbl 1402.60123
Summary: 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. In [Ann. Appl. Probab. 26, No. 3, 1297–1328 (2016; Zbl 1345.60115)], the last author et al. found the critical value and studied the subcritical and supercritical regimes. In [ibid. 26, No. 5, 2824–2859 (2016; Zbl 1353.60086)], the last author 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.

MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F17 Functional limit theorems; invariance principles
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