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.
Highlights
In the partner model each of N individuals can be susceptible or infected and in a partnership or not
We describe some sample path estimates, a diffusion limit theorem, and results on drift and diffusivity of functions of continuous time Markov chains that are used throughout the paper
√rare log set N, scales like the reciprocal of its we find the time to reach level stationary probability x scales roughly like and letting x be N p for constant a p, which is the time scale on which we control ztN
Summary
In the partner model each of N individuals can be susceptible or infected and in a partnership or not. As in the proof of Theorem 6, to show τ0(xN ) ⇒ τ0(x), it is enough to show that for each ǫ > 0 we can find δ > 0, so that if xN (t) ≤ δ xN (t + ǫ) = 0 with probability at least 1 − ǫ This is easy to do once we note that Xt is dominated by the critical branching process Xt in which each particle splits in two, or dies, each at rate one. The corresponding result for the partner model follows the same three steps, but is more complicated because the process is four dimensional and there are two different time scales. The latter sections are devoted to proofs of the lemmas
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