Contact tracing in stochastic and deterministic epidemic models Johannes M uller a, * , Mirjam Kretzschmar b , Klaus Dietz c a Biomathematik, Universit atT ubingen, Auf der Morgenstelle 10, D-72076 T ubingen, Germany b Department of Infectious Diseases Epidemiology, RIVM, P.O. Box 1, 3720 BA Bilthoven, The Netherlands c Department of Medical Biometry, University of T ubingen, T ubingen, Germany Received 21 May 1999; received in revised form 9 November 1999; accepted 18 November 1999 Abstract We consider a simple unstructured individual based stochastic epidemic model with contact tracing. Even in the onset of the epidemic, contact tracing implies that infected individuals do not act independent of each other. Nevertheless, it is possible to analyze the embedded non-stationary Galton±Watson process. Based upon this analysis, threshold theorems and also the probability for major outbreaks can be derived. Furthermore, it is possible to obtain a deterministic model that approximates the stochastic process, and in this way, to determine the prevalence of disease in the quasi-stationary state and to investigate the dynamics of the epidemic. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Contact tracing; Epidemic models; Galton±Watson process; Threshold theorem; Quasi-stationary state 1. Introduction A large part of mathematical epidemiology is concerned with the investigation of mechanisms and ecacy of control strategies against infectious diseases. Many types of control measures ± such as vaccination or screening ± are implemented at the population level and take little account of the impact of contact structure on the individual level. Often, only core groups are taken into consideration, e.g. commercial sex workers are intensely screened for sexually transmitted diseases (STDs). These kinds of control strategies are meanwhile quite well understood [1]. However, it is also possible to implement control measures at the individual level under con- sideration of the current contact structure and the history of previous contacts. For example, Mathematical Biosciences 164 (2000) 39±64 www.elsevier.com/locate/mbs * Corresponding author. Tel.: +49-7071 297 6843; fax: +49-7071 294 322. E-mail address: johannes.mueller@uni-tuebingen.de (J. Mu È ller). 0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII:S0025-5564(99)00061-9