Estimates of Outbreak Risk from New Introductions of Ebola with Immediate and Delayed Transmission Control

Identifying incoming patients can have a larger risk-reduction effect than efforts to reduce transmissions from identified patients.


Additional Methods, Equations, and Results
To fit the negative binomial model to each dataset, we used a method-of-moments estimator, which calculates R and k values that produce the exact mean and variance exhibited by the data. To estimate 90% confidence intervals, we ran 1 million nonparametric bootstrap resamples of each dataset, with replacement, and recalculated the R and k estimates for each resample. Then we used a bias-corrected percentile method (1) to construct the confidence intervals. We used the 1-sample Kolmogorov-Smirnov test, adapted for discrete variables (2), to assess goodness of fit, and the null hypothesis that each dataset was generated from the given negative binomial distribution was not rejected (P >0.6 in all cases).
To model outbreaks stemming from case introductions, we first assumed a branching process in which the number of transmissions from each infected person is independent and identically distributed according to a discrete probability distribution governed by a probabilitygenerating function (pgf) f(s). The probability, pnz, that n independent infected persons produce a total of z transmissions, is the z th coefficient of the power series representation of [ ( )] , which can be extracted by calculating We are interested in the probability that a branching process that goes extinct (a minor outbreak or stuttering chain) includes a given total number of cases, X, over all generations, including the initial case(s) in the total. When there is a single initial case, this value is governed by a pgf, g(s), satisfying the following equation: Solving for the coefficients of the power series representation of g(s) can be achieved by using a Lagrange expansion, which results in the following (3): Probability distributions of this form have been named basic Lagrangian distributions (4).
When the number of initial cases is a random variable with pgf f0(s), then further Lagrange expansion results can be used to obtain the result, In the case that the number of initial cases is fixed at n, we have f0(s) = s n , leading to a delta Lagrangian distribution (reverting to basic when n = 1); otherwise, we have a general Lagrangian distribution (4).
Example distributions have been generated by substituting pgf's f(s) and f0(s) of several different discrete probability distributions into the above equations (3). When f(s) is the pgf of the negative binomial distribution with mean R and dispersion parameter k: where Γ represents the gamma function. Here, pnz is the probability distribution for the number of transmissions z from the initial patient(s) only. The distribution for the total number of patients x over an entire stuttering chain starting with n initial patients is where we have taken the "neg. binomial-delta" formula in Consul and Shenton (3) and replaced their parameterization of the negative binomial distribution with the one above.
Blumberg and Lloyd-Smith (5) derived an equivalent result, although only for the scenario n = 1, without using the Lagrange expansion technique. The result above shows that a generalization of their approach would yield the following relationship: ( , ) = , − ( , ).
This equation gives the intuition that an outbreak with x total patients means that those x patients produced a total of exactly xn transmissions, and the probability of that occurring from x independent patients must be adjusted by the fraction (n / x) to account for the fact that the transmissions must occur in an order that produces a valid transmission chain (5). This result was also described by Becker (6), who derived the final size distribution when the offspring distribution is expressed as a generalized power series distribution, of which the negative binomial distribution is a special case.
Next, we consider a scenario in which n initially infected persons transmit according to a negative binomial distribution with parameters (R0, k0) and any and all subsequent persons transmit according to a different negative binomial distribution with parameters (Rc, kc). The probability rnx of an outbreak of total size x (including the n initial patients) is for the case n = 1, but we found the above expression to be more convenient for calculations.
To calculate the probability of x or more transmissions, we evaluated