Ja. Jacquez et Cp. Simon, THE STOCHASTIC SI MODEL WITH RECRUITMENT AND DEATHS .1. COMPARISON WITH THE CLOSED SIS MODEL, Mathematical biosciences, 117(1-2), 1993, pp. 77-125
We compare the stochastic and deterministic versions of an SI model wi
th recruitment, background deaths, and deaths due to the disease. For
the stochastic version, analysis of the mean number of susceptibles, m
(X), and infecteds, m(Y), and of the means conditioned on nonextinctio
n of the infection, m(X) and m(Y)*, shows that (1) if R0 less-than-or
-equal-to 1, the disease dies out monotonically for the deterministic
and stochastic models, and (2) if R0 > 1, the disease dies out early w
ith a probability close to (1/R0)a, where a is the number of infecteds
introduced, or m(Y) rises to a peak and then dies out slowly. For sma
ll populations N, the peak is an obvious maximum. If N greater-than-or
-equal-to 100, the peak in m(Y) is hidden in a long, nearly stationary
plateau and m(Y) is close to the deterministic endemic level for a l
arge range of parameter values. The analytical results are illustrated
with simulations. The results for the SI model are motivated by and c
ompared with the corresponding results for the closed SIS model.