= Uniform almost-sure fluid limit of the SI model
{title2=$\sup_{t\leq T}|X_n(t)-x(t)|\to0\text{ almost surely}$}
The stochastic <SI model> with susceptible fraction $X_n$, initial limit $a$, and total infection rate $n\lambda X_n(1-X_n)$ has deterministic limit
$$
x(t)=\frac{ae^{-\lambda t}}{1-a+ae^{-\lambda t}}.
$$
The <Poisson time-change representation of a Markov chain> writes $X_n$ as its integrated drift minus a centered Poisson error. On $[0,T]$ its clock is at most $n\lambda T/4$. The <Poisson maximal concentration bound> makes every fixed error tolerance have summable probabilities in $n$, so the <Borel-Cantelli lemma> gives uniform error decay <almost surely>. The <Gronwall inequality> transfers that decay to $X_n-x$, because the drift is <Lipschitz continuous>. The assertion is for fixed finite intervals and does not claim approximation over times growing with $n$.
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