Reproduction numbers and thresholds in stochastic epidemic models. I. Homogeneous populations

We compare threshold results for the deterministic and stochastic versions of the homogeneous SI model with recruitment, death due to the disease, a background death rate, and transmission rate beta cXY/N. If an infective is introduced into a population of susceptibles, the basic reproduction number, R0, plays a fundamental role for both, though the threshold results differ somewhat. For the deterministic model, no epidemic can occur if R0 less than or equal to 1 and an epidemic occurs if R0 greater than 1. For the stochastic model we find that on average, no epidemic will occur if R0 less than or equal to 1. If R0 greater than 1, there is a finite probability, but less than 1, that an epidemic will develop and eventuate in an endemic quasi-equilibrium. However, there is also a finite probability of extinction of the infection, and the probability of extinction decreases as R0 increases above 1.