Threshold Parameters for Epidemics in Different Community Settings NIELS G. BECKER zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE AND ABBAS BAHRAMPOUR zyxwvutsrqponmlkjihgfedcbaZYXWVUT School of Statistics, La Trobe University Bundoora, Australia AND KLAUS DIETZ Institut fir M edizinische Biometrie, Universitiit Tiibingen, Tiibingen, Germany Received 11 May 1994; revised 6 October 1994 ABSTRACT Threshold parameters of epidemic models play a central role in the assessment of proposed control strategies for infectious diseases. They have been determined for numerous standard epidemic models. This paper points out, with several examples, that threshold parameters depend on the social setting of the community and the variations in the behavior of the members of the community. Specifically, communi- ties are considered in which individuals have fixed patterns of behavior or random patterns of behavior as well as communities of households with fixed or random patterns of behavior. A threshold parameter is computed for each of the different settings. Some comparisons are made to provide insights into the effects that social settings and behavior changes have on the threshold parameters. 1. INTRODUCTION Epidemic models tend to be oversimplified. In real communities individuals reside in households, spending some time with their families and some at work or school, and they display variations in their behavior over time. Mathematical epidemic models tend to become unmanageable when such factors are incorporated, so some researchers turn to simulation models to describe epidemics in real communities [lo, 11, 131. Here some analytical results are obtained for epidemic models that acknowledge some of these social settings. We focus on epidemic threshold parameters, which play a key role when epidemic theory is used to provide guidance for the control of infectious diseases. We define an epidemic threshold parameter to be a quantity such that outbreaks of epidemics and persistence of endemic infection in a community are possible only when its value is greater that unity. For epidemics in a large community of uniformly mixing homoge- neous individuals, the basic reproduction number R, is defined as the MATHEMATICAL BIOSCIENCES 129:189- 208 (1995) 0 Elsevier Science Inc., 1995 0025.5564/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 00255564(94)ooo61-4