Uniform almost-sure fluid limit of the SI model (${\sup }_{t\le T}|X_n(t)-x(t)|\to 0\text{ almost surely}$)

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 limitThe 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 $n$.