Diffusion limit for the partner model at the critical value.

*(English)*Zbl 1402.60123Summary: 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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\textit{A. Basak} et al., Electron. J. Probab. 23, Paper No. 102, 42 p. (2018; Zbl 1402.60123)

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