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

• Published in: Mathematical biosciences 1991-12, Vol.107 (2), p.161-186
• Main Authors: Jacquez, John A., O'Neill, Philip
• Format: Article
• Language: English
• Subjects: Computer Simulation; Disease Outbreaks - statistics & numerical data; Humans; Infection - epidemiology; Models, Biological; Stochastic Processes
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/0025-5564(91)90003-2

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 βcXY / N. If an infective is introduced into a population of susceptibles, the basic reproduction number, R 0, plays a fundamental role for both, though the threshold results differ somewhat. For the deterministic model, no epidemic can occur if R 0 ⩽ 1 and an epidemic occurs if R 0 > 1. For the stochastic model we find that on average, no epidemic will occur if R 0 ⩽ 1. If R 0 > 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 R 0 increases above 1.