The stochastic SI model with recruitment and deaths I. comparison with the closed SIS model

Abstract

We compare the stochastic and deterministic version of an SI model with recuitment, background deaths, and deaths due to the disease. For the stochastic version, analysis of the mean number of susceptibles, m, and infecteds, m, and of the means of conditioned on nonextinction of the infection, m* and m*, shows that (1) if R0 [les] 1, the disease dies out monotonically for the deterministic and stochastic models, and (2) if R0 > 1, the disease dies out early with a probability close to (1/R0)a, where a is the number of infecteds introduced, or m rises to a peak and then dies out slowly. For small populations N, the peak is an obvious maximum. If N [ges] 100, the peak in m is hidden in a long, nearly stationary plateau and m* is close to the deterministic endemic level for a large range of parameter values. The analytical results are illustrated with simulations. The results for the SI model are motivated by and compared with the corresponding results for the closed SIS model.