Approximating the Sum of Lognormal Distributions to Enhance Models of Inhalational Anthrax
William R. Hogan, MD, MS, (Department of Biomedical Informatics, University of Pittsburgh), firstname.lastname@example.org, and
Garrick L. Wallstrom, PhD, (Department of Biomedical Informatics, University of Pittsburgh), email@example.com
In many biological defense applications, it is useful to model the incubation period of disease. Numerous authors have simulated inhalational anthrax, either to analyze response policies or to evaluate outbreak detection systems [1-4]. Each of these authors used a mathematical description of the incubation period of inhalational anthrax, and at least two authors have written entire papers devoted to the topic [5, 6].
However, many authors omit from their models of inhalational anthrax the time from symptom onset to the time an individual presents for medical care. We refer to this interval as the visit delay. One possible reason for excluding visit delay is that it is expected to be much shorter than the incubation period for anthrax. A second reason may be the mathematical difficulties involved with computing the sum of the distributions used to model incubation period and visit delay.
By far the most common distribution that researchers use to model the incubation period of anthrax--and infectious diseases in general--is the lognormal distribution. The reason is that it is simple and it describes adequately the distribution of incubation periods for a wide variety of diseases [7, 8]. We therefore chose to model visit delay using the lognormal distribution as well. However, if we model incubation period and visit delay with lognormal distributions, then the distribution over the interval from exposure to presentation for medical care is the sum of two lognormal distributions, for which no closed form solution exists. This situation is not a problem for simulating individual cases of anthrax: simply generate a random variate from the lognormal distribution for the incubation period and a random variate from the lognormal distribution for visit delay and sum them to obtain the time the individual presents for medical care. However, to compute efficiently summary statistics over an entire exposed population as in the model of Wein et al. , for example, it is more convenient to use a single lognormal distribution, because numerical solutions to the convolution of two lognormal distributions, or a lognormal distribution with some other distribution for visit delay, is likely to make the overall computation inefficient or impractical.
We used data about visit delay for inhalational anthrax from a review of cases of inhalational anthrax by Holty . We estimated the mu and sigma parameters of the lognormal distribution this data and maximum-likelihood estimation, with resulting values of 1.015 and 0.737, respectively.
Wu et al.  developed a method to approximate the distribution of the sum of two lognormal variables with a single lognormal distribution in the context of cellular phone technology. We apply this method to the incubation period and visit delay of inhalational anthrax, and provide a solution that anyone can use to quickly derive a single lognormal distribution for the incubation period plus visit delay. Our method also accounts for shorter incubation periods with higher doses of spores (per Wilkenings description of how various anthrax models incorporate this relationship ).
The method of Wu et al. begins by assuming that the two lognormal random variables are independent, and thus the moment-generating function of their sum is simply the product of two lognormal moment-generating functions. Their method then approximates the moment-generating functions with their respective Gauss-Hermite representations. These approximations set up a system of equations whose solutions yield the mu and sigma parameters of the single lognormal distribution that approximates the distribution of the sum of the two lognormal variables.
In this work, we used the A1 model of inhalational anthrax of Wilkening  and our lognormal distribution fit to Holtys data. We assume that the dose of inhaled spores does not influence visit delay, and thus we assume a single distribution for visit delay over all spore doses. Note that the method of Wu et al. is general and we could apply it to other models of anthrax and models of other diseases.
We computed the mu and sigma parameters of the lognormal for the incubation period for log10(dose inhaled spores) = 1, 2, 3, 4, 5, 6, and 7 using the Wilkening A1 model, and input each of these distributions with the single distribution for visit delay into the procedure of Wu et al.. The end result was seven lognormal distributions representing the sum of the incubation period and visit delay at seven doses of inhaled spores. We computed the Hellinger distances between these lognormal approximations and the true convolutions and found that the Hellinger distances ranged from 0.0103 to 0.1021. We then used these 7 mu and sigma points and natural cubic spline interpolation to fit a curve to these mu and sigma parameters as a function of log10(spore dose), allowing one to easily compute a single lognormal distribution for the total time from spore exposure to the time of presentation to the healthcare system, for any given dose of inhaled spores.
In conclusion, our method for approximating the interval from exposure until the time an individual seeks medical care as the sum of two lognormal distributions--one for the incubation period and one for visit delay--is convenient, computationally efficient, and generates excellent approximations. Researchers in the future can build anthrax models that account for visit delay as well as incubation period without incurring the additional computational cost of modeling visit delay as a separate lognormal distribution. Moreover, the method is general applicable to lognormal incubation periods and visit delays of other diseases of interest.
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