SI model

The SI model divides a closed population into susceptible and permanently infectious compartments. Under frequency-dependent homogeneous mixing,and conservation of population gives $S+I=N$.

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 $n$.

If $S(0)=N\theta /(1+\theta )$ and $I(0)=N/(1+\theta )$, thenAfter centering time at the incidence peak $t_{\ast }={\beta }^{-1}\log \theta$, the incidence curve is $N\beta /[4{\cosh }^2\{\beta (t-t_{\ast })/2\}]$.

The plant-disease modelhas total population $N=S+I$ and disease prevalence $\theta =I/N$ satisfyingThus infection does not alter total-population growth. The invasion threshold at the carrying capacity $N=1$ is $\beta =1$.

Removing susceptible and infected plants at the same per-capita rate $k$ changes the reduced equations toFor $0\le k<1$, the surviving population has carrying capacity $1-k$, and infection invades precisely when $\beta (1-k)>1$.