An analytical framework for relating dose, risk, and incidence: an application to occupational tuberculosis infection

An adverse health impact is often treated as a binary variable (response vs. no response), in which case the risk of response is defined as a monotonically increasing function R of the dose received D. For a population of size N, specifying the forms of R(D) and of the probability density function (pdf) for D allows determination of the pdf for risk, and computation of the mean and variance of the distribution of incidence, where the latter parameters are denoted E[SN] and Var[SN], respectively. The distribution of SN describes uncertainty in the future incidence value. Given variability in dose (and risk) among population members, the distribution of incidence is Poisson-binomial. However, depending on the value of E[SN], the distribution of incidence is adequately approximated by a Poisson distribution with parameter mu = E[SN], or by a normal distribution with mean and variance equal to E[SN] and Var[SN]. The general analytical framework is applied to occupational infection by Mycobacterium tuberculosis (M. tb). Tuberculosis is transmitted by inhalation of 1-5 microns particles carrying viable M. tb bacilli. Infection risk has traditionally been modeled by the expression: R(D) = 1 - exp(-D), where D is the expected number of bacilli that deposit in the pulmonary region. This model assumes that the infectious dose is one bacillus. The beta pdf and the gamma pdf are shown to be reasonable and especially convenient forms for modeling the distribution of the expected cumulative dose across a large healthcare worker cohort. Use of the the analytical framework is illustrated by estimating the efficacy of different respiratory protective devices in reducing healthcare worker infection risk.