TITLE:
Asymptotics and Well-Posedness of the Derived Distribution Density in a Study of Biovariability
AUTHORS:
Hongyun Wang, Wesley A. Burgei, Hong Zhou
KEYWORDS:
Distribution of Individual Injury Susceptibility in a Crowd, Biovariability, Asymptotic Approximation, Pade Approximation
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.6,
June
28,
2018
ABSTRACT: In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearing
loss injury among subjects in a crowd with a wide spectrum of heterogeneous
individual injury susceptibility due to biovariability. The injury risk of a
crowd is defined as the average fraction of injured. We examined mathematically
the injury risk of a crowd vs the number of acoustic impulses the crowd
is exposed to, under the assumption that all impulses act independently in
causing injury regardless of whether one is preceded by another. We concluded
that the observed dose-response relation can be explained solely on
the basis of biovariability in the form of heterogeneous susceptibility. We derived
an analytical solution for the distribution density of injury susceptibility,
as a power series expansion in terms of scaled log individual non-injury
probability. While theoretically the power series converges for all argument
values, in practical computations with IEEE double precision, at large argument
values, the numerical accuracy of the power series summation is completely
wiped out by the accumulation of round-off errors. In this study, we
derive a general asymptotic approximation at large argument values, for the
distribution density. The combination of the power series and the asymptotics
provides a practical numerical tool for computing the distribution density.
We then use this tool to verify numerically that the distribution obtained
in our previous theoretical study is indeed a proper density. In addition, we
will also develop a very efficient and accurate Pade approximation for the
distribution density.