Chris Rorres

Quantitative methods of identifying the key nodes in the illegal wildlife trade network

Significance Despite advances in technology and general descriptive work, current approaches at reducing the illegal wildlife trade are failing. We take a more analytical approach to identify the key countries involved in the illegal wildlife trade network by using a new database of illegal wildlife trade reports, HealthMap Wildlife Trade, to identify (i) the key exporter, intermediary, and importer countries and (ii) the countries where enforcement activities and educational campaigns might most effectively disrupt the networks. Identifying these key countries can provide useful information on how to allocate resources to combat the illegal trade in wildlife, a major focus for conservation and public health agendas. Innovative approaches are needed to combat the illegal trade in wildlife. Here, we used network analysis and a new database, HealthMap Wildlife Trade, to identify the key nodes (countries) that support the illegal wildlife trade. We identified key exporters and importers from the number of shipments a country sent and received and from the number of connections a country had to other countries over a given time period. We used flow betweenness centrality measurements to identify key intermediary countries. We found the set of nodes whose removal from the network would cause the maximum disruption to the network. Selecting six nodes would fragment 89.5% of the network for elephants, 92.3% for rhinoceros, and 98.1% for tigers. We then found sets of nodes that would best disseminate an educational message via direct connections through the network. We would need to select 18 nodes to reach 100% of the elephant trade network, 16 nodes for rhinoceros, and 10 for tigers. Although the choice of locations for interventions should be customized for the animal and the goal of the intervention, China was the most frequently selected country for network fragmentation and information dissemination. Identification of key countries will help strategize illegal wildlife trade interventions.

Completing Book II of Archimedes’s On Floating Bodies

ConclusionOne need only glance at Archimedes’s Proposition 8 above to see thatOn Floating Bodies is several orders of magnitude more sophisticated than anything else found in ancient mathematics. It ranks with Newton’sPrincipia Mathematica as a work in which basic physical laws are both formulated and accompanied by superb applications.

Distributed M-ary hypothesis testing with binary local decisions

Parallel distributed detection schemes for M-ary hypothesis testing often assume that for each observation the local detector transmits at least log2 M bits to a data fusion center (DFC). However, it is possible for less than log2 M bits to be available, and in this study we consider 1-bit local detectors with M > 2. We develop conditions for asymptotic detection of the correct hypothesis by the DFC, formulate the optimal decision rules for the DFC, and derive expressions for the performance of the system. Local detector design is demonstrated in examples, using genetic algorithm search for local decision thresholds. We also provide an intuitive geometric interpretation for the partitioning of the observations into decision regions. The interpretation is presented in terms of the joint probability of the local decisions and the hypotheses. 2003 Elsevier B.V. All rights reserved.

Estimating the kernel parameters of premises-based stochastic models of farmed animal infectious disease epidemics using limited, incomplete, or ongoing data

Three different estimators are presented for the types of parameters present in mathematical models of animal epidemics. The estimators make use of the data collected during an epidemic, which may be limited, incomplete, or under collection on an ongoing basis. When data are being collected on an ongoing basis, the estimated parameters can be used to evaluate putative control strategies. These estimators were tested using simulated epidemics based on a spatial, discrete-time, gravity-type, stochastic mathematical model containing two parameters. Target epidemics were simulated with the model and the three estimators were implemented using various combinations of collected data to independently determine the two parameters.

Ongoing Estimation of the Epidemic Parameters of a Stochastic, Spatial, Discrete-Time Model for a 1983–84 Avian Influenza Epidemic

SUMMARY. We formulate a stochastic, spatial, discrete-time model of viral “Susceptible, Exposed, Infectious, Recovered” animal epidemics and apply it to an avian influenza epidemic in Pennsylvania in 1983–84. Using weekly data for the number of newly infectious cases collected during the epidemic, we find estimates for the latent period of the virus and the values of two parameters within the transmission kernel of the model. These data are then jackknifed on a progressive weekly basis to show how our estimates can be applied to an ongoing epidemic to generate continually improving values of certain epidemic parameters.

Stochastic modeling of animal epidemics using data collected over three different spatial scales

A stochastic, spatial, discrete-time, SEIR model of avian influenza epidemics among poultry farms in Pennsylvania is formulated. Using three different spatial scales wherein all the birds within a single farm, ZIP code, or county are clustered into a single point, we obtain three different views of the epidemics. For each spatial scale, two parameters within the viral-transmission kernel of the model are estimated using simulated epidemic data. We show that simulated epidemics modeled using data collected on the farm and ZIP-code levels behave similar to the actual underlying epidemics, but this is not true using data collected on the county level. Such analyses of data collected on different spatial scales are useful in formulating intervention strategies to control an ongoing epidemic (e.g., vaccination schedules and culling policies).

Archimedes the Mathematician

Although Archimedes’ fame among the general populace in antiquity was based on his military machines, inventions, and legends, he earned his true immortality through his mathematical works. Here I discuss some of Archimedes’ mathematical ideas that are being used and applied twenty-three centuries after his lifetime.

Contact tracing for the control of infectious disease epidemics: Chronic Wasting Disease in deer farms

Contact tracing is a crucial component of the control of many infectious diseases, but is an arduous and time consuming process. Procedures that increase the efficiency of contact tracing increase the chance that effective controls can be implemented sooner and thus reduce the magnitude of the epidemic. We illustrate a procedure using Graph Theory in the context of infectious disease epidemics of farmed animals in which the epidemics are driven mainly by the shipment of animals between farms. Specifically, we created a directed graph of the recorded shipments of deer between deer farms in Pennsylvania over a timeframe and asked how the properties of the graph could be exploited to make contact tracing more efficient should Chronic Wasting Disease (a prion disease of deer) be discovered in one of the farms. We show that the presence of a large strongly connected component in the graph has a significant impact on the number of contacts that can arise.

Archimedes' Count of Homer's Cattle of the Sun

In the first few lines of The Odyssey, Homer foretells how Odysseus’ crew “perished through their own sheer folly in eating the cattle of the Sun-god Hyperion”. These Cattle of the Sun grazed near the Sicilian town of Taormina (Tauromenion to its ancient Greek settlers) and, although endlessly warned not to, Odysseus’ crew slaughtered some of them for food. For this sacrilege Zeus tossed them from their ship to their deaths with his thunderbolts, leaving Odysseus to continue his odyssey alone. In describing the sacred cattle, Homer indirectly gives their count by writing that they comprised seven herds containing fifty cattle each (Book XII: “Of oxen fifty head in every herd feed, and their herds are seven”), leaving it to the reader to determine the total number of cattle. Centuries later this simple multiplication problem was the inspiration for Archimedes’ famous “Cattle Problem”, whose first line is: “If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily”. Archimedes, who lived in the Sicilian-Greek city-state of Syracuse, 85 kilometers south of Taormina, would have been very familiar with Homer’s tale. In his problem, Archimedes challenges his colleague Eratosthenes to compute the number of the Cattle of the Sun having a larger and more complicated composition than the one described by Homer. Archimedes’ problem is so complicated that the total number of cattle contains 206,545 digits. In this article I describe the origins of this problem in antiquity, its rediscovery in the eighteenth century, and the attempts since then to solve it. Its complete resolution had to await the computer age, since before then someone estimated that it would take the work of “a thousand men for a thousand years” to determine the exact solution. Attempts at its solution fueled the field of Diophantine Analysis – the analysis of problems whose solutions are restricted to whole numbers – and, in particular, the study of the so-called Pell Equation. Today a notebook computer using sophisticated algorithms can generate the number of cattle in seconds, taking more time to print out the number than to actually compute it. The amount of intellectual activity that has surrounded this problem over 23 centuries suggests the validity of Voltaire’s remark, “There was more imagination in the head of Archimedes than in that of Homer”.