### Abstract and Introduction

We present a stochastic model for the spread of smallpox after a small number of index cases are introduced into a susceptible population. The model describes a branching process for the spread of the infection and the effects of intervention measures. We discuss scenarios in which ring vaccination of direct contacts of infected persons is sufficient to contain an epidemic. Ring vaccination can be successful if infectious cases are rapidly diagnosed. However, because of the inherent stochastic nature of epidemic outbreaks, both the size and duration of contained outbreaks are highly variable. Intervention requirements depend on the basic reproduction number (*R _{0} *), for which different estimates exist. When faced with the decision of whether to rely on ring vaccination, the public health community should be aware that an epidemic might take time to subside even for an eventually successful intervention strategy.

Recently, concerns about a bioterror attack with the smallpox virus or other infectious disease agents have risen.^{[1,2]} While new vaccines with fewer adverse consequences are being developed,^{[3]} the existing vaccines, which have potential side effects and may be lethal, are the only vaccines available.^{[4,5]} In the United States during the recent voluntary smallpox vaccination program, a limited number of healthcare workers volunteered for vaccination because of the risks associated with vaccination and the low risk for infection.^{[6]} If an outbreak occurs, vaccination strategies include ring vaccination around diagnosed cases of smallpox or a mass vaccination to begin as soon as the first cases are diagnosed. Without natural smallpox infections, practical experience with ring vaccination against smallpox cannot be gained; accounts of the vaccination programs that eradicated smallpox in the 1970s are the only source of information.^{[7]} Combined with information collected during the last decades of smallpox circulation, mathematical modeling offers a tool to explore various vaccination scenarios if an outbreak occurs.^{[8,9,10,11,12,13,14,15]}

We investigated which conditions are the best for effective use of ring vaccination, a strategy in which direct contacts of diagnosed cases are identified and vaccinated. We also investigated whether monitoring contacts contributes to the success of ring vaccination. We used a stochastic model that distinguished between close and casual contacts to explore the variability in the number of infected persons during an outbreak, and the time until the outbreak is over. We derived expressions for the basic reproduction number (*R _{0} *) and the effective reproduction number (

*R*). We investigated how effectiveness of ring vaccination depends on the time until diagnosis of a symptomatic case, the time to identify and vaccinate contacts in the close contact and casual contact ring, and the vaccination coverage required to contain an epidemic.

_{υ}Emerging Infectious Diseases. 2004;10(5) © 2004 Centers for Disease Control and Prevention (CDC)

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