Abstract:
In this paper, we compute the non-detection probability of a randomly moving target by a stationary or moving searcher in a square search region. We find that when the searcher is stationary, the decay rate of the non-detection probability achieves the maximum value when the searcher is fixed at the center of the square search region; when both the searcher and the target diffuse with significant diffusion coefficients, the decay rate of the non-detection probability only depends on the sum of the diffusion coefficients of the target and searcher. When the searcher moves along prescribed deterministic tracks, our study shows that the fastest decay of the non-detection probability is achieved when the searcher scans horizontally and vertically.

Abstract:
We consider the problem of searching for a target that moves between a hiding area and an operating area over multiple fixed routes. The search is carried out with one or more cookie-cutter sensors, which can detect the target instantly once the target comes within the detection radius of the sensor. In the hiding area, the target is shielded from being detected. The residence times of the target, respectively, in the hiding area and in the operating area, are exponentially distributed. These dwell times are mathematically described by Markov transition rates. The decision of which route the target will take on each travel to and back from the operating area is governed by a probability distribution. We study the mathematical formulation of this search problem and analytically solve for the mean time to detection. Based on the mean time to capture, we evaluate the performance of placing the searcher(s) to monitor various travel route(s) or to scan the operating area. The optimal search design is the one that minimizes the mean time to detection. We find that in many situations the optimal search design is not the one suggested by the straightforward intuition. Our analytical results can provide operational guidances to homeland security, military, and law enforcement applications.

We revisit one of the classical search problems in which a diffusing target encounters a stationary searcher. Under the condition that the searcher’s detection region is much smaller than the search region in which the target roams diffusively, we carry out an asymptotic analysis to derive the decay rate of the non-detection probability. We consider two different geometries of the search region: a disk and a square, respectively. We construct a unified asymptotic expression valid for both of these two cases. The unified asymptotic expression shows that the decay rate of the non-detection probability, to the leading order, is proportional to the diffusion constant, is inversely proportional to the search region, and is inversely proportional to the logarithm of the ratio of the search region to the searcher’s detection region. Furthermore, the second term in the unified asymptotic expansion indicates that the decay rate of the non-detection probability for a square region is slightly smaller than that for a disk region of the same area. We also demonstrate that the asymptotic results are in good agreement with numerical solutions.

Abstract:
We study the problem of detecting a target that moves between a hiding area and an operating area over multiple fixed routes. The research is carried out with one or more cookie-cutter sensors with stochastic intermission, which turn on and off stochastically governed by an on-rate and an off-rate. A cookie-cutter sensor, when it is on, can detect the target instantly once the target comes within the detection radius of the sensor. In the hiding area, the target is shielded from being detected. The residence times of the target, respectively, in the hiding area and in the operating area, are exponentially distributed and are governed by rates of transitions between the two areas. On each travel between the two areas and in each travel direction, the target selects a route randomly according to a probability distribution. Previously, we analyzed the simple case where the sensors have no intermission (i.e., they stay on all the time). In the current study, the sensors are stochastically intermittent and are synchronized (i.e., they turn on or off simultaneously). This happens when all sensors are affected by the same environmental factors. We derive asymptotic expansions for the mean time to detection when the on-rate and off-rate of the sensors are large in comparison with the rates of the target traveling between the two areas. Based on the mean time to detection, we evaluate the performance of placing the sensor(s) to monitor various travel route(s) or to scan the operating area.

Abstract:
We study the problem of a diffusing particle confined in a large
sphere in the n-dimensional space being absorbed into a small sphere at the
center. We first non-dimensionalize the problem using the radius of large
confining sphere as the spatial scale and the square of the spatial scale
divided by the diffusion coefficient as the time scale. The non-dimensional
normalized absorption rate is the product of the physical absorption rate and
the time scale. We derive asymptotic expansions for the normalized absorption
rate using the inverse iteration method. The small parameter in the asymptotic
expansions is the ratio of the small sphere radius to the large sphere radius.
In particular, we observe that, to the leading order, the normalized absorption
rate is proportional to the (n － 2)-th power of the small parameter for .

Abstract:
In this manuscript, we consider the case where a Brownian particle is subject to a static periodic potential and is driven by a constant force. We derive analytic formulas for the average velocity and the effective diffusion.

Abstract:
Hearing loss is a common military health problem and it is closely related to exposures to impulse noises from blast explosions and weapon firings. In a study based on test data of chinchillas and scaled to humans (Military Medicine, 181: 59-69), an empirical injury model was constructed for exposure to multiple sound impulses of equal intensity. Building upon the empirical injury model, we conduct a mathematical study of the hearing loss injury caused by multiple impulses of non-uniform intensities. We adopt the theoretical framework of viewing individual sound exposures as separate injury causing events, and in that framework, we examine synergy for causing injury (fatigue) or negative synergy (immunity) or independence among a sequence of doses. Starting with the empirical logistic dose-response relation and the empirical dose combination rule, we show that for causing injury, a sequence of sound exposure events are not independent of each other. The phenomenological effect of a preceding event on the subsequent event is always immunity. We extend the empirical dose combination rule, which is applicable only in the case of homogeneous impulses of equal intensity, to accommodate the general case of multiple heterogeneous sound exposures with non-uniform intensities. In addition to studying and extending the empirical dose combination rule, we also explore the dose combination rule for the hypothetical case of independent events, and compare it with the empirical one. We measure the effect of immunity quantitatively using the immunity factor defined as the percentage of decrease in injury probability attributed to the sound exposure in the preceding event. Our main findings on the immunity factor are: 1) the immunity factor is primarily a function of the difference in SELA (A- weighted sound exposure level) between the two sound exposure events; it is virtually independent of the magnitude of the two SELA values as long as the difference is fixed; 2) the immunity factor increases monotonically from 0 to 100% as the first dose is varied from being significantly below the second dose, to being moderately above the second dose. The extended dose-response formulation developed in this study provides a theoretical framework for assessing the injury risk in realistic situations.

Abstract:
We consider the hearing loss injury among subjects in a crowd with a wide spectrum of individual intrinsic injury probabilities due to biovariability. For multiple acoustic impulses, the observed injury risk of a crowd vs the effective combined dose follows the logistic dose-response relation. The injury risk of a crowd is the average fraction of injured. The injury risk was measured in experiments as follows: each subject is individually exposed to a sequence of acoustic impulses of a given intensity and the injury is recorded; results of multiple individual subjects were assembled into data sets to mimic the response of a crowd. The effective combined dose was adjusted by varying the number of impulses in the sequence. The most prominent feature observed in experiments is that the injury risk of the crowd caused by multiple impulses is significantly less than the value predicted based on assumption that all impulses act independently in causing injury and all subjects in the crowd are statistically identical. Previously, in the case where all subjects are statistically identical (i.e., no biovariability), we interpreted the observed injury risk caused by multiple impulses in terms of the immunity effects of preceding impulses on subsequent impulses. In this study, we focus on the case where all sound exposure events act independently in causing injury regardless of whether one is preceded by another (i.e., no immunity effect). Instead, we explore the possibility of interpreting the observed logistic dose-response relation in the framework of biovariability of the crowd. Here biovariability means that subjects in the crowd have their own individual injury probabilities. That is, some subjects are biologically less or more susceptible to hearing loss injury than others. We derive analytically the distribution of individual injury probability that produces the observed logistic dose-response relation. For several parameter values, we prove that the derived distribution is mathematically a proper density function. We further study the asymptotic approximations for the density function and discuss their significance in practical numerical computation with finite precision arithmetic. Our mathematical analysis implies that the observed logistic dose-response relation can be theoretically explained in the framework of biovariability in the absence of immunity effect.

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.

Abstract:
We study a general framework for assessing the injury probability corresponding
to an input dose quantity. In many applications, the true value of
input dose may not be directly measurable. Instead, the input dose is estimated
from measurable/controllable quantities via numerical simulations
using assumed representative parameter values. We aim at developing a simple
modeling framework for accommodating all uncertainties, including the
discrepancy between the estimated input dose and the true input dose. We
first interpret the widely used logistic dose-injury model as the result of dose
propagation uncertainty from input dose to target dose at the active site for
injury where the binary outcome is completely determined by the target dose.
We specify the symmetric logistic dose-injury function using two shape parameters:
the median injury dose and the 10 - 90 percentile width. We relate
the two shape parameters of injury function to the mean and standard deviation
of the dose propagation uncertainty. We find 1) a larger total uncertainty
will spread more the dose-response function, increasing the 10 - 90 percentile
width and 2) a systematic over-estimate of the input dose will shift the injury
probability toward the right along the estimated input dose. This framework
provides a way of revising an established injury model for a particular test
population to predict the injury model for a new population with different
distributions of parameters that affect the dose propagation and dose estimation.
In addition to modeling dose propagation uncertainty, we propose a
new 3-parameter model to include the skewness of injury function. The proposed
3-parameter function form is based on shifted log-normal distribution
of dose propagation uncertainty and is approximately invariant when other
uncertainties are added. The proposed 3-parameter function form provides a
framework for extending skewed injury model from a test population to a
target population in application.