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Fix $T>0$ and put $K=\lambda T/4$. Since $0\le X_n\le 1$, the clock in part (a) is at most $nK$ on $[0,T]$. The Poisson maximal concentration bound gives, for every $\delta >0$,Here $h(u)>0$ for $u>0$, so the probabilities are summable in $n$. The Borel-Cantelli lemma, first for positive rational $\delta$ and then for all $\delta$, yields ${\sup }_{t\le T}|{\epsilon }_n(t)|\to 0$ almost surely. No independence between the processes for different population sizes is required.

The deterministic SI model solves the logistic equation $x^{\prime }=-\lambda x(1-x)$ with $x(0)=a$, givingOn $[0,1]$, $b$ is Lipschitz continuous with constant $\lambda$. Subtracting the two integral equations givesThe Gronwall inequality impliesThis is the uniform almost-sure fluid limit of the SI model. One can take the intersection of the probability-one events for integer $T$ to obtain the conclusion on every finite interval. Use population sizes with $an$ integral; if the initial susceptible count is rounded, the extra initial error is at most $1/n$ and the same Gronwall inequality proof applies.

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 $n$.

For a nonnegative martingale, the displayed maximal probability bound follows by stopping at the first crossing of $a$. The stopped expectation is at most its initial expectation, and on the crossing event its value is at least $a$. Bound the stopping time first and pass to the limit using nonnegativity. This form is useful for the Poisson maximal concentration bound.