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The genealogy process is typically the most time-consuming part of-and a limiting factor in the success of-forensic genetic genealogy, which is a new approach to solving violent crimes and identifying human remains. We formulate a stochastic dynamic program that-given the list of matches and their genetic distances to the unknown target-chooses the best decision at each point in time: which match to investigate (i.e., find its ancestors and look for most recent common ancestors between the match and the target), which set of potential most recent common ancestors to descend from (i.e., find its descendants, with the goal of identifying a marriage between the maternal and paternal sides of the target's family tree), or whether to terminate the investigation. The objective is to maximize the probability of finding the target minus a cost associated with the expected size of the final family tree. We estimate the parameters of our model using data from 17 cases (eight solved, nine unsolved) from the DNA Doe Project. We assess the Proposed Strategy using simulated versions of the 17 DNA Doe Project cases, and compare it to a Benchmark Strategy that ranks matches by their genetic distance to the target and only descends from known common ancestors between a pair of matches. The Proposed Strategy solves cases ≈10 - fold faster than the Benchmark Strategy, and does so by aggressively descending from a set of potential most recent common ancestors between the target and a match even when this set has a low probability of containing the correct most recent common ancestor. Our analysis provides a mathematical foundation for improving the genealogy process in forensic genetic genealogy.

Motivated by the debate over how to deal with the huge backlog of untested sexual assault kits in the U.S.A., we construct and analyze a mathematical model that predicts the expected number of hits (i.e., a new DNA profile matches a DNA sample in the criminal database) as a function of both the proportion of the backlog that is tested and whether the victim-offender relationship is used to prioritize the kits that are tested. Refining the results in Ref. (Criminol Public Policy, 2016, 15, 555), we use data from Detroit, where government funding was used to process ≈15% of their backlog, to predict that prioritizing stranger kits over nonstranger kits leads to only a small improvement in performance (a 0.034 increase in the normalized area under the curve of the hits vs. proportion of backlog tested curve). Two rough but conservative cost-benefit analyses-one for testing the entire backlog and a marginal one for testing kits from nonstranger assaults-suggest that testing all sexual assault kits in the backlog is quite cost-effective: for example, spending ≈$1641 to test a kit averts sexual assaults costing ≈$133,484 on average.

Ballistic imaging systems can help solve crimes by comparing images of cartridge cases, which are recovered from a crime scene or test-fired from a gun, to a database of images obtained from past crime scenes. Many U.S. municipalities lack the resources to process all of their cartridge cases. Using data from Stockton, CA, we analyze two problems: how to allocate limited capacity to maximize the number of cartridge cases that generate at least one hit, and how to prioritize the cartridge cases that are processed to maximize the usefulness (i.e., obtained before the corresponding criminal case is closed) of hits. The number of hits can be significantly increased by prioritizing crime scene evidence over test-fires, and by ranking calibers by their hit probability and processing only the higher ranking calibers. We also estimate that last-come first-served increases the proportion of hits that are useful by only 0.05 relative to first-come first-served.

The April 7, 2020, Wisconsin election produced a large natural experiment to help understand the transmission risks of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). As of April 14, 2020, 1 551 711 total votes were cast (https://bit.ly/2yWPhlF),1 and 1 138 491 absentee ballots were returned as of April 21, 2020,1 suggesting that approximately 413 220 people voted in person. Waiting times in Milwaukee averaged 1.5 to 2 hours.2 Poll workers had surgical masks and latex gloves, hand sanitizer was made available to voters, isopropyl alcohol wipes were used to clean voting equipment, and painting tape and signs were used to facilitate social distancing.1 (Am J Public Health. Published online ahead of print June 18, 2020: e1-e2. doi:10.2105/AJPH.2020.305770).

Motivated by jail overcrowding and the U.S. correctional system's widespread use of risk models to aid in inmate release decisions both prior to trial (i.e., pretrial release) and near the end of their jail sentence (i.e., split sentencing), we formulate and analyze a queueing network model with two novel features: there is a continuum of customer classes corresponding to an inmate's continuous risk level p, and routing in the network is dictated by a Cox proportional hazards model, where the hazard rate associated with recidivism (i.e., committing another crime) during release is proportional to eγp for some parameter γ. We perform an exact analysis of a continuous-class M/M/c/c (i.e., Erlang B) model where preemptive priority is awarded according to the risk level p, and use it to develop approximate performance measures for a queueing network of a jail with a two-threshold policy, which dictates who is granted pretrial release and who receives a split sentence. For a slightly simplified version of the model, we derive sufficient conditions under which no inmates are offered pretrial release unless all inmates are given a split sentence.

Firearms identification imaging systems help solve crimes by comparing newly acquired images of cartridge casings or bullets to a database of images obtained from past crime scenes. We formulate an optimization problem that bases its matching decisions not only on the similarity between pairs of images, but also on the time and spatial location of each new acquisition and each database entry. The objective is to maximize the detection probability subject to a constraint on the false positive rate. We use data on all cartridge casings matches detected in Israel during 2006-2008 to estimate most of the model parameters. We estimate matching accuracy from two different studies and predict that the optimal use of extraneous information would increase the detection probability from 0.931 to 0.987 and from 0.707 to 0.844, respectively. These improvements are achieved by favoring pairs of images that are closer together in space and time.

Several aid groups have proposed strategies for allocating ready-to-use (therapeutic and supplementary) foods to children in developing countries. Analysis is needed to investigate whether there are better alternatives. We use a longitudinal dataset of 5,657 children from Bwamanda to construct a bivariate time-series model that tracks each child’s height-for-age z score (HAZ) and weight-for-height z score (WHZ) throughout the first 5 y of life. Our optimization model chooses which individual children should receive ready-to-use therapeutic or supplementary food based on a child’s sex, age, HAZ, and WHZ, to minimize the mean number of disability-adjusted life years (DALYs) per child during 6–60 mo of age [which includes childhood mortality calculated from a logistic regression and the lifelong effects of stunting (i.e., low HAZ)] subject to a budget constraint. Compared with the strategies proposed by the aid groups, which do not use HAZ information, the simple strategy arising from our analysis [which prioritizes children according to low values of a linear combination of HAZ, WHZ, and age and allocates the entire budget to therapeutic (i.e., 500 kcal/d) food for the prioritized children] reduces the number of DALYs by 9% (for the same budget) or alternatively incurs the same number of DALYs with a 61% reduction in cost. Whereas our qualitative conclusions appear to be robust, the quantitative results derived from our analysis should be treated with caution because of the lack of reliable data on the impact of supplementary food on HAZ and WHZ, the application of our model to a single cohort of children and the inclusion and exclusion errors related to imperfect food targeting.

BACKGROUND: Recent studies show that transfusing older blood may lead to increased mortality. This raises the issue of whether transfusing fresher blood can be achieved without jeopardizing blood availability. STUDY DESIGN AND METHODS: We propose a simple family of policies that is defined by a single threshold: rather than transfusing the oldest available blood that is younger than 42 days, we transfuse the oldest blood that is younger than the threshold, and if there is no blood younger than the threshold then we transfuse the youngest blood that is older than the threshold. To assess this policy, we build a simulation model using data from Stanford University Medical Center. We focus on the tradeoff between the mean age of transfused blood and the fraction of transfused blood that is imported. RESULTS: For hospitals in which the local supply is greater than demand, our policy with a threshold of 14 days leads to a decrease of 10 to 20 days in the mean age of transfused blood while increasing the fraction of imported blood to less than 0.005 (i.e., 0.5%). If the health benefits from transfusing fresher blood can be confirmed by randomized clinical trials, then conservative assumptions suggest that this policy could reduce the annual number of transfused patients who die within 1 year by 20,000. CONCLUSION: The proposed allocation policy with a threshold of 14 days could allow many US hospitals to significantly reduce the age of transfused blood, thereby possibly reducing morbidity and mortality, while having a negligible impact on supply chain operations. T he allocation of available red blood cells (RBCs) for transfusion to patients is a difficult problem: compatibility between donor and recipient must be maintained, and the age and quantity of available RBCs need to be accounted for. More specifically, while there is a tendency to use older available blood to prevent outdating, stored RBCs undergo biochemical changes that impact RBC function, and the age of transfused blood has been reported to be associated with an increased risk of infection, postoperative complications, and/or short-term and long-term mortality.

We superimpose a radiation fallout model onto a traffic flow model to assess the evacuation versus shelter-in-place decisions after the daytime ground-level detonation of a 10-kt improvised nuclear device in Washington, DC. In our model, approximately 80k people are killed by the prompt effects of blast, burn, and radiation. Of the approximately 360k survivors without access to a vehicle, 42.6k would die if they immediately self-evacuated on foot. Sheltering above ground would save several thousand of these lives and sheltering in a basement (or near the middle of a large building) would save of them. Among survivors of the prompt effects with access to a vehicle, the number of deaths depends on the fraction of people who shelter in a basement rather than self-evacuate in their vehicle: 23.1k people die if 90% shelter in a basement and 54.6k die if 10% shelter. Sheltering above ground saves approximately half as many lives as sheltering in a basement. The details related to delayed (i.e., organized) evacuation, search and rescue, decontamination, and situational awareness (via, e.g., telecommunications) have very little impact on the number of casualties. Although antibiotics and transfusion support have the potential to save approximately 10k lives (and the number of lives saved from medical care increases with the fraction of people who shelter in basements), the logistical challenge appears to be well beyond current response capabilities. Taken together, our results suggest that the government should initiate an aggressive outreach program to educate citizens and the private sector about the importance of sheltering in place in a basement for at least 12 hours after a terrorist nuclear detonation. Language: en

Motivated by the cost and disruption involved in changing from a two-finger to a ten-finger biometric system for matching U.S. visitors to a watchlist of criminals and terrorists, we investigate whether any two-finger multistage biometric strategies would fix the inadequate matching performance of poor-quality prints that plagues the U.S. Government's original two-finger, single-stage biometric system. For several multistage strategies, we solve the Stackelberg game in which the U.S. Government chooses the biometric threshold levels to maximize the detection probability subject to constraints on the false positive probability and on the mean time per visitor to perform biometric screening, and the terrorist chooses the fingerprint image quality to minimize his detection probability. The first stage of all the strategies uses the current minutiae-based fingerprint matching system, but with thresholds that depend on image quality, which in isolation achieves a detection probability of 0.771. Using face recognition (based on 2002 data) in the second stage increases the detection probability to 0.841, whereas using a slower and more thorough texture-based fingerprint matcher in the second stage leads to a detection probability of at least 0.913 and perhaps significantly higher. (Data for the texture matcher is only available for the poorest-quality prints and we assume that this is its performance for all prints.) Adding face recognition as a third stage to this latter system does not improve performance. The two-finger, two-stage strategy may be comparable in performance to the ten-finger, single-stage strategy (which has a detection probability of 0.937), is robust against gaming and poor image acquisition, requires no additional hardware, and would generate no visible changes from the original two-finger, single-stage system from a visitor's viewpoint. The uncertainty in our performance estimates needs to be better quantified, ideally with raw data on similarity scores, before recommending this strategy for implementation.

The article of record as published may be located at http://dx.doi.org/10.1287 We develop a dynamic model in which Operation Iraqi Freedom (OIF) servicemembers incur a random amount of combat stress during each month of deployement, develop posttraumatic stress disorder (PTSD) if their cumulative stress exceeds a servicemember-specific threshold, and then develop symptoms of PTSD after an additional time lag. Using Department of Defense deployment data and Mental Health Advisory Team PTSD data to calibrate the model, we predict -- because of the long time lags and the fact that some surveyed servicemembers expeerience additional combat after being surveyed -- the fraction of Amry soldiers and Marines who eventually suffer from PTSD will be approximately twice as large as in the raw survey data. We cannot put a confidence interval around this estimate, but there is considerable uncertainty (perhaps plus/minus 30%). The estimate PTSD rate translates into approximately 300,000 PTSD cases among all Army soldiers and Marines in OIF, with approximately 20,000 new cases each year the war is prolonged. The heterogeneity of threshold levels among servicemembers suggests that although multiple deployments raise an individual's risk of PTSD, in aggregate, multiple deployments lower the total number of PTSD by approximately 30% relative to a hypothetical case in which the war was fought with many more servicemembers (i.e., a draft) deploying only once. The time lag dynamics suggest that, in aggregate, reserve servicemembers show symptoms aproximately 1-2 years before active servicemembers and predict that greater than 75% of OIF servicemembers who self-reported symptoms during their second deployment were exposed to the PTSD-generating stress during their first deployment. Naval Postgraduate School, Department of Operations Research, Monterey, CA; Stanford University, Stanford, CA

This paper is the archival record of the INFORMS Philip McCord Morse Lecture delivered in 2008. It describes the author's research on four topics in homeland security and public health: preparedness and response to a bioterror anthrax attack, preparedness and response to a bioterror attack on the food supply, routes of transmission and infection control for pandemic influenza, and biometrics (e.g., fingerprint matching) to prevent terrorists from entering the country. The paper focuses on the modeling, policy recommendations, and implementation of these recommendations. The author draws lessons about policy implementation from these examples and from examples from his other homeland security work with colleagues, including a bioterror smallpox attack, preventing nuclear weapons from entering the country on a shipping container, preventing nuclear weapons from entering a city, and preventing terrorists from crossing the border between the United States and Mexico.

We develop a mathematical optimization model at the intersection of homeland security and immigration, that chooses various immigration enforcement decision variables to minimize the probability that a terrorist can successfully enter the United States across the U.S.-Mexico border. Included are a discrete choice model for the probability that a potential alien crosser will attempt to cross the U.S.-Mexico border in terms of the likelihood of success and the U.S. wage for illegal workers, a spatial model that calculates the apprehension probability as a function of the number of crossers, the number of border patrol agents, and the amount of surveillance technology on the border, a queueing model that determines the probability that an apprehended alien will be detained and removed as a function of the number of detention beds, and an equilibrium model for the illegal wage that balances the supply and demand for work and incorporates the impact of worksite enforcement. Our main result is that detention beds are the current system bottleneck (even after the large reduction in detention residence times recently achieved by expedited removal), and increases in border patrol staffing or surveillance technology would not provide any improvements without a large increase in detention capacity. Our model also predicts that surveillance technology is more cost effective than border patrol agents, which in turn are more cost effective than worksite inspectors, but these results are not robust due to the difficulty of predicting human behavior from existing data. Overall, the probability that a terrorist can successfully enter the United States is very high, and it would be extremely costly and difficult to significantly reduce it. We also investigate the alternative objective function of minimizing the flow of illegal aliens across the U.S.-Mexico border, and obtain qualitatively similar results.

Motivated by the ease with which online customers can bid simultaneously in multiple auctions, we analyze a system with two competing auctioneers and three types of bidders: those dedicated to either of the two auctions and those that participate simultaneously in both auctions. Bidding behavior is specified and proven to induce a Bayesian Nash equilibrium, and a closed-form expression for the expected revenue of each auctioneer is derived. For auctioneers selling a single item, partial pooling—i.e., the presence of some cross-auction bidders—is beneficial to both auctioneers as long as neither one dominates the market (e.g., possesses more than 60%–65% of the market share). For multi-item auctions, pooling is mutually beneficial only if both auctioneers have nearly identical ratios of bidders per items for sale; otherwise, only the auctioneer with the smaller ratio benefits from pooling. Pooling's impact on revenue decreases with the number of bidders, suggesting that popular auction sites need not be overly concerned with mitigating bidding across auctions.

We formulate and analyze an optimal stopping problem concerning a terrorist who is attempting to drive a nuclear or radiological weapon toward a target in a city center. In our model, the terrorist needs to travel through a two-dimensional lattice containing imperfect radiation sensors at some of the nodes, and decides at each node whether to detonate the bomb or proceed. We consider five different scenarios containing various informational structures and two different sensor array topologies: the sensors are placed randomly or they form an outer wall around the periphery of the city. We find that sensors can act as a deterrent in some cases, and that the government prefers the outer wall topology unless the sensors have a very low detection probability and the budget is tight (so that they are sparsely deployed).

We formulate and analyze a spatial queueing model concerning a terrorist who is attempting to drive a nuclear or radiological weapon toward a target in a city center. In our model, imperfect radiation sensors form a circular wall around the periphery of the city, and vehicles setting off sensor alarms (representing a terrorist or a nuisance alarm) arrive randomly at the perimeter of a circle (representing the wall of sensors) and drive toward the center of the circle. Interdiction vehicles, one in each wedge of the circle, chase the alarm-generating vehicles. We derive an accurate mathematical expression for the mean damage inflicted by a terrorist in this system in terms of the arrival rate of alarm-generating vehicles and the number of interdiction vehicles. Our results suggest that detection-interdiction systems using current technology are capable of mitigating the damage from a nuclear weapon made of plutonium, but not one made of uranium or a radiological weapon.

Due to lack of detention capacity (the U.S. government measures capacity by the number of detention beds), tens of thousands of apprehended illegal aliens are released into the U.S. interior each year, instead of being removed from the country. This vulnerability can be exploited by terrorist groups wanting to enter the United States. We construct a queueing model of the U.S. detention and removal operations, and derive approximate analytical expressions for key performance measures, including a simple normal approximation for the required number of beds. Due to shortcomings in the U.S. government's data collection procedures, we cannot directly estimate all of the model's parameter values. Consequently, we use the approximate analytical expressions and the 2003 U.S. government data quantifying these key performance measures to estimate several unknown parameter values. Although current funding is for approximately 21,000 detention beds, we estimate that approximately 34,500 beds are needed to remove all potential detainees (this does not include nonviolent, noncriminal Mexicans, who are returned to Mexico within several hours) based on 2003 data. The dramatic increase in the arrivals of potential detainees since 2003 suggests that approximately 50,000 beds are currently required, although the estimation of future arrival rates is very difficult due to uncertainties about the future direction of U.S. immigration policy. Our estimated bed requirements are approximately 25% higher than naive estimates that fail to account for right censoring of residence times due to some detainees being released from detention before removal to make way for higher-priority detainees.

Radiation portal monitors are starting to be deployed at overseas ports to prevent nuclear weapons from entering the U.S. in a shipping container. Current designs have containers on trucks passing through a portal monitor at approximately 10 mph, before being routed to one of several lanes at the port's front gate for a driver identification check. For a fixed cost of testing, which consists of the costs of radiation portal monitors plus offsite x-ray and possibly manual testing of containers generating a false radiation alarm that cannot be resolved by gamma-ray imaging, the neutron detection limits of the current design are compared with those of two other designs that do not affect truck congestion at the front gate. For a wide range of budgets, it is optimal to have six monitors in each lane that simultaneously test a truck while it is being processed at the front gate. This design is robust against the location (within the container) of the weapon and reduces the detection limit (relative to the curr...

We formulate and solve an optimization problem in which a terrorist is attempting to drive a nuclear weapon toward a city center, but needs to travel through an array of imperfect neutron radiation sensors that form a wall around the periphery of the city. A fleet of interdiction vehicles are available to chase, and attempt to interdict, vehicles that set off a sensor alarm. In our model, the government chooses the thickness (in terms of number of sensors) of the radiation wall, the neutron threshold in the sensors, and the number of interdiction vehicles to minimize the expected damage inflicted by a terrorist, subject to a budget constraint on sensors and interdiction vehicles. The terrorist observes the wall thickness and at each node he updates his likelihood of passing through a sensor without triggering an alarm and decides whether to proceed through the sensor or stop and detonate the bomb. Our results suggest that for an annual cost ranging from several million dollars to several tens of millions of dollars, depending upon the city's roadway topology, a single layer of sensors placed tens of miles from the city center and 10-20 dedicated interdiction vehicles could mitigate the damage from an unshielded or lightly-shielded plutonium weapon, but not from a uranium weapon or a radiological dispersal device.

Motivated by the failure of current methods to control dengue fever, we formulate a mathematical model to assess the impact on the spread of a mosquito-borne viral disease of a strategy that releases adult male insects homozygous for a dominant, repressible, lethal genetic trait. A dynamic model for the female adult mosquito population, which incorporates the competition for female mating between released mosquitoes and wild mosquitoes, density-dependent competition during the larval stage, and realization of the lethal trait either before or after the larval stage, is embedded into a susceptible–exposed–infectious–susceptible human-vector epidemic model for the spread of the disease. For the special case in which the number of released mosquitoes is maintained in a fixed proportion to the number of adult female mosquitoes at each point in time, we derive mathematical formulas for the disease eradication condition and the approximate number of released mosquitoes necessary for eradication. Numerical results using data for dengue fever suggest that the proportional policy outperforms a release policy in which the released mosquito population is held constant, and that eradication in ≈1 year is feasible for affected human populations on the order of 10 5 to 10 6 , although the logistical considerations are daunting. We also construct a policy that achieves an exponential decay in the female mosquito population; this policy releases approximately the same number of mosquitoes as the proportional policy but achieves eradication nearly twice as fast.

Background Fecal microbiota transplantation is an effective treatment for recurrent Clostridium difficile infection and is being investigated as a treatment for other microbiota-associated diseases. To facilitate these activities, an international public stool bank has been created, which screens donors and processes stools in a standardized manner. The goal of this research is to use mathematical modeling and analysis to optimize screening and donor management at the stool bank. Results Compared to the current policy of screening active donors every 60 days before releasing their quarantined stools for sale, costs can be reduced by 10.3 % by increasing the screening frequency to every 36 days. In addition, the stool production rate varies widely across donors, and using donor-specific screening, where higher producers are screened more frequently, also reduces costs, as does introducing an interim (i.e., between consecutive regular tests) stool test for just rotavirus and C. difficile. We also derive a donor release (i.e., into the system) policy that allows the supply to approximately match an exponentially increasing deterministic demand. Conclusions More frequent screening, interim screening for rotavirus and C. difficile, and donor-specific screening, where higher stool producers are screened more frequently, are all cost-reducing measures. If screening costs decrease in the future (e.g., as a result of bringing screening in house), a bottleneck for implementing some of these recommendations may be the reluctance of donors to undergo serum screening more frequently than monthly. Electronic supplementary material The online version of this article (doi:10.1186/s40168-015-0140-3) contains supplementary material, which is available to authorized users.

We develop a mathematical model to find the optimal inspection strategy for detecting a nuclear weapon (or nuclear material to make a weapon) from being smuggled into the United States in a shipping container, subject to constraints of port congestion and an overall budget. We consider an 11-layer security system consisting of shipper certification, container seals, and a targeting software system, followed by passive (neutron and gamma), active (gamma radiography), and manual testing at overseas and domestic ports. Currently implemented policies achieve a low detection probability, and improved security requires passive and active testing of trusted containers and manually opening containers that cannot be penetrated by radiography. The annual cost of achieving a high detection probability of a plutonium weapon using existing equipment in traditional ways is roughly several billion dollars if testing is done domestically, and is approximately five times higher if testing is performed overseas. Our results suggest that employing high-energy x-ray radiography and elongating the passive neutron tests at overseas ports may provide significant cost savings, and several developing technologies, radiation sensors inside containers and tamper-resistant electronic seals, should be pursued aggressively. Further effort is critically needed to develop a practical neutron interrogation scheme that reliably detects moderately shielded, highly enriched uranium.

To aid in understanding how best to respond to a bioterror anthrax attack, we analyze a system of differential equations that includes a disease progression model, a set of spatially distributed queues for distributing antibiotics, and vaccination (pre-event and/or post-event). We derive approximate expressions for the number of casualties as a function of key parameters and management levers, including the time at which the attack is detected, the number of days to distribute antibiotics, the adherence to prophylactic antibiotics, and the fraction of the population that is preimmunized. We compare a variety of public health intervention policies in the event of a hypothetical anthrax attack in a large metropolitan area. Modeling assumptions were decided by the Anthrax Modeling Working Group of the Secretary's Council on Public Health Preparedness. Our results highlight the primary importance of rapid antibiotic distribution and lead us to argue for ensuring post-attack surge capacity to rapidly produce enough anthrax vaccine for an additional 100 million people.

We developed a mathematical model of a cows-to-consumers supply chain associated with a single milk-processing facility that is the victim of a deliberate release of botulinum toxin. Because centralized storage and processing lead to substantial dilution of the toxin, a minimum amount of toxin is required for the release to do damage. Irreducible uncertainties regarding the dose–response curve prevent us from quantifying the minimum effective release. However, if terrorists can obtain enough toxin, and this may well be possible, then rapid distribution and consumption result in several hundred thousand poisoned individuals if detection from early symptomatics is not timely. Timely and specific in-process testing has the potential to eliminate the threat of this scenario at a cost of

The World Health Organization (WHO) recommends which strains of influenza to include in each year’s vaccine to countries around the globe. The current WHO strategy attempts to match the vaccine strains with the expected upcoming epidemic strains, a strategy we refer to as the follow policy. The recently proposed antigenic distance hypothesis suggests that vaccine efficacy can be enhanced by taking into account the antigenic histories of vaccinees. To assess the potential benefit of history-based vaccination, we formulate the annual vaccine-strains selection problem as a stochastic dynamic program using the theory of shape space, which maps each vaccine and epidemic strain into a point in multidimensional space. Computational results show that a near-optimal policy can be derived by approximating the entire antigenic history by a single reduced historical strain, and then solving the multiperiod problem myopically, as a series of single-period problems. The modest suboptimality of the follow policy, together with our current inability to quantitatively link the model’s objective function (a measure of cross-reactivity) with actual vaccine efficacy, leads us to recommend the continued use of the follow policy.

To aid in understanding how best to respond to a bioterror anthrax attack, we analyze a system of differential equations that includes an atmospheric release model, a spatial array of biosensors, a dose-response model, a disease progression model, and a set of spatially distributed tandem queues for distributing antibiotics and providing hospital care. We derive approximate closed-form expressions for the number of deaths as a function of key parameters and management levers, including the size of the attack, the time at which the attack is detected via symptomatic patients, the number of days to distribute antibiotics, the efficacy (both for treatment and prevention) of antibiotics, the prophylactic antibiotic distribution strategy, the prioritization of the antibiotics queue, and the detection limit, deployment density, and delay time of biosensors.

We address the problem of designing multi-item procurement auctions for a monopsonistic buyer in capacity-constrained environments. Using insights from classical auction theory, we construct an optimization-based auction mechanism (“smart market”) relying on the dynamic resolution of a linear program minimizing the buyer's cost under the suppliers' capacity constraints. Suppliers can modify their offers in response to the optimal allocation corresponding to each set of bids, giving rise to a dynamic competitive bidding process. To assist suppliers, we also develop a bidding-suggestion device based on a myopic best-response (MBR) calculation that solves an associated optimization problem. Assuming linear costs for the suppliers, we study within a game-theoretic framework the sequence of bids arising in this smart market. Under a weak behavioral assumption and some symmetry requirements, an explicit upper bound for the winning bids is established. We then formulate a complete behavioral model and solution methodology based on the MBR rationale and show that the bounds derived earlier continue to hold. We analytically derive some structural and convergence properties of the MBR dynamics in the simplest nontrivial market environment, which suggests further possible design improvements, and investigate bidding dynamics and incentive compatibility issues via numerical simulations.

To evaluate existing and alternative proposals for emergency response to a deliberate smallpox attack, we embed the key operational features of such interventions into a smallpox disease transmission model. We use probabilistic reasoning within an otherwise deterministic epidemic framework to model the 'race to trace', i.e., attempting to trace (via the infector) and vaccinate an infected person while (s)he is still vaccine-sensitive. Our model explicitly incorporates a tracing/vaccination queue, and hence can be used as a capacity planning tool. An approximate analysis of this large (16 ODE) system yields closed-form estimates for the total number of deaths and the maximum queue length. The former estimate delineates the efficacy (i.e., accuracy) and efficiency (i.e., speed) of contact tracing, while the latter estimate reveals how congestion makes the race to trace more difficult to win, thereby causing more deaths. A probabilistic analysis is also used to find an approximate closed-form expression for the total number of deaths under mass vaccination, in terms of both the basic reproductive ratio and the vaccination capacity. We also derive approximate thresholds for initially controlling the epidemic for more general interventions that include imperfect vaccination and quarantine.

We developed a mathematical model to compare various emergency responses in the event of an airborne anthrax attack. The system consists of an atmospheric dispersion model, an age-dependent dose–response model, a disease progression model, and a set of spatially distributed two-stage queueing systems consisting of antibiotic distribution and hospital care. Our results underscore the need for the extremely aggressive and timely use of oral antibiotics by all asymptomatics in the exposure region, distributed either preattack or by nonprofessionals postattack, and the creation of surge capacity for supportive hospital care via expanded training of nonemergency care workers at the local level and the use of federal and military resources and nationwide medical volunteers. The use of prioritization (based on disease stage and/or age) at both queues, and the development and deployment of modestly rapid and sensitive biosensors, while helpful, produce only second-order improvements.

We model an isolated portion of a competitive supply chain as a M/M/1 make-to-stock queue. The retailer carries finished goods inventory to service a Poisson demand process, and specifies a policy for replenishing his inventory from an upstream supplier. The supplier chooses the service rate, i.e., the capacity of his manufacturing facility, which behaves as a single-server queue with exponential service times. Demand is backlogged and both agents share the backorder cost. In addition, a linear inventory holding cost is charged to the retailer, and a linear cost for building production capacity is incurred by the supplier. The inventory level, demand rate, and cost parameters are common knowledge to both agents. Under the continuous-state approximation where the M/M/1 queue has an exponential rather than geometric steady-state distribution, we characterize the optimal centralized and Nash solutions, and show that a contract with linear transfer payments replicates a cost-sharing agreement and coordinates the system. We also compare the total system costs, the agents' decision variables, and the customer service levels of the centralized versus Nash versus Stackelberg solutions.

Motivated by the recent development of highly specific agents for brain tumours, we develop a mathematical model of the spatio-temporal dynamics of a brain tumour that receives an infusion of a highly specific cytotoxic agent (e.g. IL-4-PE, a cytotoxin comprised of IL-4 and a mutated form of Pseudomonas exotoxin). We derive an approximate but accurate mathematical formula for the tumour cure probability in terms of the tumour characteristics (size at time of detection, proliferation rate, diffusion coefficient), drug design (killing rate, loss rate and convection constants for tumour and tissue), and drug delivery (infusion rate, infusion duration). Our results suggest that high specificity is necessary but not sufficient to cure malignant gliomas; a nondispersed spatial profile of pretreatment tumour cells and/or good drug penetration are also required. The most important levers to improve tumour cure appear to be earlier detection, higher infusion rate, lower drug clearance rate and better convection into tumour, but not tissue. In contrast, the tumour cure probability is less sensitive to a longer infusion duration and enhancements in drug potency and drug specificity.

In the event of a smallpox bioterrorist attack in a large U.S. city, the interim response policy is to isolate symptomatic cases, trace and vaccinate their contacts, quarantine febrile contacts, but vaccinate more broadly if the outbreak cannot be contained by these measures. We embed this traced vaccination policy in a smallpox disease transmission model to estimate the number of cases and deaths that would result from an attack in a large urban area. Comparing the results to mass vaccination from the moment an attack is recognized, we find that mass vaccination results in both far fewer deaths and much faster epidemic eradication over a wide range of disease and intervention policy parameters, including those believed most likely, and that mass vaccination similarly outperforms the existing policy of starting with traced vaccination and switching to mass vaccination only if required.

Fecal microbiota transplantation is being assessed as a treatment for chronic microbiota-related diseases such as ulcerative colitis. Results from an initial randomized trial suggest that remission rates depend on unobservable features of the fecal donors and observable features of the patients. We use mathematical modeling to assess the efficacy of pooling stools from different donors during multiple rounds of treatment. In the model, there are two types of patients and two types of donors, where the patient type is observable and the donor type (effective or not effective) is not observable. In the model, clinical outcomes from earlier rounds of treatment are used to estimate the current likelihood that each donor is effective, and then each patient in each round is treated by a pool of donors that are currently deemed to be the most effective. Relative to the no-pooling case, pools of size two or three significantly increase the proportion of patients in remission during the first several rounds of treatment. Although based on data from a single randomized trial, our modeling suggests that pooling of stools - via daily cycling of encapsulated stool from several different donors - may be beneficial in fecal microbiota transplantation for chronic microbiota-related diseases.

We analyse the sequence in which the three most commonly prescribed cancer treatments--surgery (S), chemotherapy (C) and radiotherapy (R)--should be administered. A system of ordinary differential equations is formulated that captures the various local and systemic effects of the three modes of treatment, as well as the first-order effects of the inter-relationship between the primary tumour and the distant metastatic tumours, including primary tumour shedding and the primary tumour's effect on the rate of angiogenesis in the metastatic tumours. Under a set of stated assumptions on the parameter values, we find the exact cancer cure probability (subject to toxicity constraints) for the six permutation schedules (i.e. SCR, CSR, CRS, SRC, RSC, RCS) and for two novel schedules, SRCR and RSCR, that apply radiotherapy in disjoint, optimally timed portions. We show analytically that SRCR and RSCR are the two best-performing (i.e. highest cure probability) schedules among the eight considered. Further, SRCR is shown to be optimal among all possible schedules, provided a modest condition is satisfied on the delay of initial angiogenesis experienced by the patient's dormant tumours.

This paper examines how setups, due dates, and the mix of standardized and customized products affect the scheduling of a single machine operating in a dynamic and stochastic environment. We restrict ourselves to the class of dynamic cyclic policies, where the machine busy/idle policy and lot-sizing decisions are controlled in a dynamic fashion, but different products must be produced in a fixed sequence. As in earlier work, we conjecture that an averaging principle holds for this queueing system in the heavy traffic limit, and optimize over the class of dynamic cyclic policies. The results allow for a detailed discussion of the interactions between the due-date, setup, and product mix facets of the problem.

We examine the component procurement problem in a single-item, make-to-stock assembly system. The suppliers are uncapacitated and have independent but non-identically distributed stochastic delivery lead times. Assembly is instantaneous, product demand follows a Poisson process and unsatisfied demand is backordered. The objective is to minimize the sum of steady-state holding and backorder costs over a pre-specified class of replenishment policies. To keep the analysis tractable, we impose a synchronization assumption that no mixing occurs between sets of component orders. Combining existing results from queueing theory with original results concerning distributions that are closed under maximization and translation, we derive a simple approximate solution to the problem when lead time variances are identical. In simulations, our derived policy is within 2% of optimal and significantly outperforms policies that ignore either component dependence or lead time stochasticity. It is also quite robust with respect to various model assumptions, except the synchronization one.

We consider a production stage that produces a single item in a make-to-stock manner. Demand for finished goods is stationary. In each time period, an updated vector of demand forecasts over the forecast horizon becomes available for use in production decisions. We model the sequence of forecast update vectors using the Martingale model of forecast evolution developed by Graves et al. (1986, 1998) and Heath and Jackson (1994). The production stage is modeled as a single-server, discrete-time, continuous-state queue. We focus on a modified base-stock policy incorporating forecast information and use an approximate analysis rooted in heavy traffic theory and random walk theory to obtain a closed-form expression for the (forecast-corrected) base-stock level that minimizes the expected steady-state inventory holding and backorder costs. This expression, which is shown to be accurate under certain conditions in a simulation study, sheds some light on the interrelationships among safety stock, stochastic correlated demand, inaccurate forecasts, and random and capacitated production in forecasting-production-inventory systems.

The crux of the kidney allocation problem is the trade-off between clinical efficiency and equity. We consider a dynamic resource allocation problem with the tri-criteria objective of maximizing the quality-adjusted life expectancy of transplant candidates (clinical efficiency) and minimizing two measures of inequity: a linear function of the likelihood of transplantation of the various types of patients, and a quadratic function that quantifies the differences in mean waiting times across patient types. The dynamic status of patients is modeled by a set of linear differential equations, and an approximate analysis of the optimal control problem yields a dynamic index policy. We construct a large-scale simulation model using data from over 30,000 transplants, and the simulation results demonstrate that, relative to the organ allocation policy currently employed in the United States, the dynamic index policy increases the quality-adjusted life expectancy and reduces the mean waiting time until transplantation for all six demographic groups (two sexes, races, and age groups) under consideration.

Purpose: The linear-quadratic model typically assumes that tumor sensitivity and repopulation are constant over the time course of radiotherapy. However, evidence suggests that the growth fraction increases and the cell-loss factor decreases as the tumor shrinks. We investigate whether this evolution in tumor geometry, as well as the irregular time intervals between fractions in conventional hyperfractionation schemes, can be exploited by fractionation schedules that employ time-varying fraction sizes. Methods: We construct a mathematical model of a spherical tumor with a hypoxic core and a viable rim, which is most appropriate for a prevascular tumor, and is only a caricature of a vascularized tumor. This model is embedded into the traditional linear-quadratic model by assuming instantaneous reoxygenation. Dynamic programming is used to numerically compute the fractionation regimen that maximizes the tumor-control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissues. Results: In several numerical examples that employ five or 10 fractions per week on a 1-cm or 5-cm diameter tumor, optimally varying the fraction sizes increases the TCP significantly. The optimal regimen incorporates large Friday (afternoon, if 10 fractions per week) fractions that are escalated throughout the course of treatment, and larger afternoon fractions than morning fractions. Conclusion: Numerical results suggest that a significant increase in tumor cure can be achieved by allowing the fraction sizes to vary throughout the course of treatment. Several strategies deserve further investigation: using larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment.

We consider two queueing control problems that are stochastic versions of the economic lot scheduling problem: A single server processes N customer classes, and completed units enter a finished goods inventory that services exogenous customer demand. Unsatisfied demand is backordered, and each class has its own general service time distribution, renewal demand process, and holding and backordering cost rates. In the first problem, a setup cost is incurred when the server switches class, and the setup cost is replaced by a setup time in the second problem. In both problems we employ a long-run average cost criterion and restrict ourselves to a class of dynamic cyclic policies, where idle periods and lot sizes are state-dependent, but the N classes must be served in a fixed sequence. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these scheduling problems by diffusion control problems. Our analysis of the approximating setup cost problem yields a closed-form dynamic lot-sizing policy and a computational procedure for an idling threshold. We derive structural results and an algorithmic procedure for the setup time problem. A computational study compares the proposed policy and several alternative policies to the numerically computed optimal policy.

We analyze two joint decisions in the management of HIV-infected patients on antiretroviral therapy: how frequently to measure a patient's virus level, and when to switch therapy. The underlying stochastic model captures the initial suppression and eventual rebound of the virus level in the blood of a typical HIV-infected patient undergoing treatment. We consider two classes of policies: a viral load policy, which triggers a change in therapy when the current virus level divided by the smallest level achieved thus far exceeds a prespecified threshold, and a proactive policy, which is similar to the viral load policy but also switches drugs at a prespecified time if no evidence of viral rebound has been seen. We find approximate analytical expressions for the probability of switching before the virus reaches its nadir (minimum value) and the mean delay in detection of viral rebound (i.e., the time interval from when the viral nadir occurs until the switch in therapy). Numerical results show that the proactive policy outperforms (i.e., a smaller detection delay for a given probability of prenadir switching) the viral load policy and recent recommendations by an expert AIDS panel, and may delay the onset of multidrug resistance in a significant proportion of patients who experience drug failure.

We address the procurement of new components for recyclable products in the context of Kodak's single-use camera. The objective is to find an ordering policy that minimizes the total expected procurement, inventory holding, and lost sales cost. Distinguishing characteristics of the system are the uncertainty and unobservability associated with return flows of used cameras. We model the system as a closed queueing network, develop a heuristic procedure for adaptive estimation and control, and illustrate our methods with disguised data from Kodak. Using this framework, we investigate the effects of various system characteristics such as informational structure, procurement delay, demand rate, and length of the product's life cycle.

We analyze three queueing control problems that model a dynamic stochastic distribution system, where a single capacitated vehicle serves a finite number of retailers in a make-to-stock fashion. The objective in each of these vehicle routing and inventory problems is to minimize the long run average inventory (holding and backordering) and transportation cost. In all three problems, the controller dynamically specifies whether a vehicle at the warehouse should idle or embark with a full load. In the first problem, the vehicle must travel along a prespecified (TSP) tour of all retailers, and the controller dynamically decides how many units to deliver to each retailer. In the second problem, the vehicle delivers an entire load to one retailer (direct shipping) and the controller decides which retailer to visit next. The third problem allows the additional dynamic choice between the TSP and direct shipping options. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these queueing control problems by diffusion control problems, which are explicitly solved in the fixed route problems, and numerically solved in the dynamic routing case. Simulation experiments confirm that the heavy traffic approximations are quite accurate over a broad range of problem parameters. Our results lead to some new observations about the behavior of this complex system.

We analyze the performance of a tandem queueing network populated by two customer types. The interarrival times of each type and the service times of each type at each station are independent random variables with general distributions, but the load on each station is assumed to be identical. A setup time is incurred when a server switches from one customer type to the other, and each server employs an exhaustive polling scheme. We conjecture that a time scale decomposition, which is known to occur at the first station under heavy traffic conditions, holds for the entire tandem system, and we employ heavy traffic approximations to compute the sojourn time distribution for a customer that arrives to find the network in a particular state. When setup times are zero (except perhaps at the first station) and additional “product-form” type assumptions are imposed, we find the steady-state sojourn time distribution for each customer type.

BACKGROUND: There are not enough cadaveric kidneys to meet the demands of transplant candidates. The equity and efficiency of alternative organ allocation strategies have not been rigorously compared. METHODS: We developed a five-compartment Monte Carlo simulation model to compare alternative organ allocation strategies, accommodating dynamic changes in recipient and donor characteristics, patient and graft survival rates, and quality of life. The model simulated the operations of a single organ procurement organization and attempted to predict the evolution of the transplant waiting list for 10 years. Four allocation strategies were compared: a first-come first-transplanted system; a point system currently utilized by the United Network of Organ Sharing; an efficiency-based algorithm that incorporated correlates of patient and graft survival; and a distributive efficiency algorithm, which had an additional goal of promoting equitable allocation among African-American and other candidates. RESULTS: A 10-year computer simulation was performed. The distributive efficiency policy was associated with a 3.5% ± 0.8% (mean ± SD) increase in quality-adjusted life expectancy (33.9 months vs 32.7 months), a decrease in the median waiting time to transplantation among those who were transplanted (6.6 months vs 16.3 months), and an increase in the overall likelihood of transplantation (61% vs 45%), compared with the United Network of Organ Sharing algorithm. Improved equity and efficiency were also seen by race (African-American vs other), sex, and age ( CONCLUSION: Evidence-based organ allocation strategies in cadaveric kidney transplantation would yield improved equity and efficiency measures compared with existing algorithms.

We consider a serial inventory system with N stages. The material flows from an outside supplier to stage N, then to stage N − 1, etc., and finally to stage 1 where random customer demand arises. Each stage replenishes a stage-specific inventory position according to a stage-specific reorder point/order quantity policy. Two variations of this policy are considered. One is based on echelon stock, and the other installation stock. The former requires centralized demand information, while the latter does not. The relative cost difference between the two policies is called the value of centralized demand information. For fixed order quantities, we develop efficient algorithms for computing both the optimal echelon reorder points and the optimal installation reorder points. These algorithms enable us to conduct an extensive computational study to assess the value of centralized demand information and to understand how this value depends on several key system parameters, i.e., the number of stages, leadtimes, batch sizes, demand variability, and the desired level of customer service.

We consider a stock cutting problem for a paper plant that produces sheets of various sizes for a finished goods inventory that services random customer demand. The controller decides when to shut down and restart the paper machine and how to cut completed paper rolls into sheets of paper. The objective is to minimize long-run expected average costs related to paper waste (from inefficient cutting), shutdowns, backordering, and holding finished goods inventory. A two-step procedure (linear programming in the first step and Brownian control in the second step) is developed that leads to an effective, but suboptimal, solution. The linear program greatly restricts the number of cutting configurations that can be employed in the Brownian analysis, and hence the proposed policy is easy to implement, and the resulting production process is considerably simplified. In an illustrative numerical example using representative data from an industrial facility, the proposed policy outperforms several policies that use a larger number of cutting configurations. Finally, we discuss some alternative production settings where this two-step procedure may be applicable.

We analyze two scheduling problems for a queueing system with a single server and two customer classes. Each class has its own renewal arrival process, general service time distribution, and holding cost rate. In the first problem, a setup cost is incurred when the server switches from one class to the other, and the objective is to minimize the long-run expected average cost of holding customers and incurring setups. The setup cost is replaced by a setup time in the second problem, where the objective is to minimize the average holding cost. By assuming that a recently derived heavy traffic principle holds not only for the exhaustive policy but for nonexhaustive policies, we approximate (under standard heavy traffic conditions) the dynamic scheduling problems by diffusion control problems. The diffusion control problem for the setup cost problem is solved exactly, and asymptotics are used to analyze the corresponding setup time problem. Computational results show that the proposed scheduling policies are within several percent of optimal over a broad range of problem parameters.