Modeling respirator penetration values with the beta distribution: an application to occupational tuberculosis transmission.

The lognormal distribution typically is used to model variability in respiratory penetration values. The lognormal model is a good descriptor where the average penetration value is low, but may be a poor descriptor where the average penetration value is high because a significant fraction of penetration values could be predicted to exceed unity. In this regard, the beta distribution offers greater flexibility than the lognormal in modeling penetration values over the physically plausible interval [0,1]. The beta distribution also is shown to be mathematically convenient for describing the risk of airborne transmission of tuberculosis among a respirator-wearing population. Infection can occur following inhalation of respirable particles, termed droplet nuclei, carrying viable Mycobacterium tuberculosis bacilli. Based on the expected number of infectious doses inhaled, the Poisson probability model traditionally is used to predict an individual's risk of infection. This article synthesizes the beta distribution, as applied to average penetration values among a respirator-wearing population, and the Poisson distribution, as applied to an individual's infection risk, to describe the population risk of M. tuberculosis infection.