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informs Vol. 39, No. 5, September–October 2009, pp. 476–490 issn 0092-2102 eissn 1526-551X 09 3905 0476 ® doi 10.1287/inte.1090.0463 © 2009 INFORMS Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Eva K. Lee, Chien-Hung Chen Center for Operations Research in Medicine and HealthCare, School of Industrial and Systems Engineering, Georgia Institute of Technology, and NSF I/UCRC Center for Health Organization Transformation, Georgia Institute of Technology, Atlanta, Georgia 30332 {[email protected], [email protected]} Ferdinand Pietz, Bernard Benecke Strategic National Stockpile, Coordinating Office for Terrorism Preparedness and Emergency Response, Centers for Disease Control and Prevention, Atlanta, Georgia 30333 Public-health emergencies, such as bioterrorist attacks or pandemics, demand fast, efficient, large-scale dispensing of critical medical countermeasures. By combining mathematical modeling, large-scale simulation, and powerful optimization engines, and coupling them with automatic graph-drawing tools and a user-friendly interface, we designed and implemented RealOpt© , a fast and practical emergency-response decision-support tool. RealOpt allows public-health emergency coordinators to (1) determine locations for point-of-dispensing (POD) facility setup; (2) design customized and efficient floor plans for PODs via an automatic graph-drawing tool; (3) determine required labor resources and provide efficient placement of staff at individual stations within a POD; (4) perform disease-propagation analysis, understand and monitor the intra-POD disease dilemma, and help to derive dynamic response strategies to mitigate casualties; (5) assess resources and determine minimum needs to prepare for treating their regional populations in emergency situations; (6) carry out large-scale virtual drills and performance analyses, and investigate alternative strategies; and (7) design a variety of dispensing scenarios that include emergency-event exercises to train personnel. These advanced and powerful computational strategies allow emergency coordinators to quickly analyze design decisions, generate feasible regional dispensing plans based on best estimates and analyses available, and reconfigure PODs as an event unfolds. The ability to analyze planning strategies, compare the various options, and determine the most cost-effective combination of dispensing strategies is critical to the ultimate success of any mass dispensing effort. Key words: public health; emergency response; mass dispensing; resource allocation; facility location; disease propagation; medical countermeasures; bioterrorism; pandemic; infectious disease; anthrax; disaster medicine; all-hazard emergency response; public-health informatics; integer programming; simulation; decision-support system. P ublic-health emergencies, such as bioterrorist attacks or pandemics, demand fast, efficient, large-scale dispensing of critical medical countermeasures (i.e., vaccines, drugs, and therapeutics). Such dispensing is complex and requires careful planning and coordination from multiple federal, state, and local agencies, as well as the potential involvement of the private sector. Dispensing medications quickly (within 48 hours for anthrax prophylactic) to large population centers (with tens of thousands or even millions of people) is urgent; moreover, the multifaceted nature of dispensing (e.g., sending federal stockpiles to local points of dispensing (PODs), coordination at the local level to manage the transportation of citizens to PODs, and the POD operations) makes the process highly unpredictable. Thus, emergency managers and public-health administrators must be able to quickly investigate alternative response strategies as an emergency unfolds. The focus of this paper is on mass dispensing of medical countermeasures for protection of the general population; however, large-scale public-health emergencies may involve thousands of sick or injured people who will require various levels of medical care, ranging from patient evacuation, hospital care, and sustainable and potentially long-term health-recovery procedures. Thus, such emergencies present a daunting set of challenges, including the surge capability 476 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Interfaces 39(5), pp. 476–490, © 2009 INFORMS and flexibility of our existing medical systems, federal and state emergency capacity for rapid medical dispatching, and the resolve and resilience of health-care workers and emergency responders to perform under critical timelines and exceedingly stressful conditions. In the wake of the 2001 anthrax attacks, the Department of Health and Human Services (HHS) increased its order for smallpox vaccine, accelerated production, and began working to develop a detailed plan for the public-health response to an outbreak of smallpox. By January 2003, the United States had sufficient quantities of the vaccine for every person in the country in an emergency situation (Gerberding 2003). HHS subsequently required each state to submit a mass-vaccination plan for administering smallpox vaccine. Furthermore, states are charged with developing city-readiness programs that deal with establishing regional treatment and dispensing centers, and developing procedures, policies, and a planning framework for efficient allocation of staff and resources in response to these events. The importance of such population protection has been carefully studied for human, social, and economic benefits. Kaplan et al. (2002) argued that immediate mass vaccination after a smallpox bioterrorist attack would result in fewer deaths and faster eradication of the potential epidemic; Wein et al. (2003) concluded that immediate and aggressive dispersion of oral antibiotics and the full use of available resources (local nonemergency care workers, federal and military resources, and nationwide medical volunteers) are extremely important. Mass Dispensing: Challenges Mass dispensing requires the rapid establishment of a network of dispensing sites and health facilities that are flexible, scalable, and sustainable for medical prophylaxis and treatment of the general population. Moreover, each POD must be capable of serving the affected local population within a specified short time frame. Clearly, for very large-scale dispensing, the sophisticated logistical expertise needed to deal with the complexities of selecting an adequate number of strategically well-placed POD locations, and of designing and staffing each POD, is beyond the capability of any human planner or public-health 477 administrator. The limited availability of trained critical staff, such as public-health professionals, further compounds the inherent complexities. Rapid distribution of medical countermeasures to a large population requires significant resources within individual communities. Few, if any, cities are presently able to meet the objective of dispensing countermeasures to their entire population within 48 hours (the mortality rate is very steep after 48 hours for anthrax exposure). The Strategic National Stockpile (SNS) is available to help agencies respond to public-health threats that can be mitigated or eliminated by treating the affected population with the antibiotics or vaccines that it contains. Although distribution of countermeasures by the federal government can leverage the distribution infrastructure of couriers, such as FedEx, UPS, and USPS, the last mile of dispensing to the broad regional population requires strategic and operational planning of a network of POD sites, including the determination of appropriate staffing at each POD, to ensure that a practical plan is in place to accomplish the task within the given time constraint. These last-mile issues form the crux of the discussion and advances we present in this paper. Since 2003, exercise drills have been conducted regularly throughout the nation to better prepare public-health personnel to realistically plan for mass dispensing. Beaton et al. (2003) reported an exit survey of a drill held in the state of Washington. Giovachino et al. (2005) presented postdrill analysis of an exercise conducted in the District of Columbia. Aaby et al. (2006) explained how Montgomery County (Maryland) Public Health Services used operations research techniques to improve its clinic planning. Lee et al. (2005; 2006a, b; Lee 2008) described RealOpt’s development for POD design, resource allocation, and real-time dynamic response. Gebbie et al. (2006) defined criteria for the performance evaluation of drills. Some researchers, such as Lien et al. (2006), examined the role that the private sector (e.g., retail chains, such as grocery stores and wholesale clubs) can play as community centers in partnership with public-health authorities for the mass distribution of vaccines or antibiotics. Retail grocery and wholesale stores already have experience with dispensing annual influenza vaccinations and could be an excellent resource in a public-health emergency. Nelson et al. 478 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response (2007) addressed the definition, measurement, and determination of sufficiency of public-health emergency preparedness, and they reviewed the current approaches. Lee et al. (2009c) offered a multimodality strategy for planning a realistic mass-dispensing event in a region with over five million people. The work includes discussion of cost-effective operations analyses and of public and private sector involvement. SNS stockpiles sufficient anthrax antibiotics and smallpox vaccine for the entire population; however, a key mass-dispensing challenge is the need to dispense to the entire regional population under scarce staffing resources and within a very tight time line. Mason and Washington (2003) used operations research techniques to assist in staffing a smallpox POD site when limited staffing is available. The discrete-event simulation model they developed, Maxi-Vac, offers insight on the practicality of a simulation system in emergencies. However, its implementation revealed the severe bottlenecks of commercial simulation and optimization software. Each scenario involved about 30 staff and had an objective of maximizing throughput; however, each required more than 10 hours to generate a usable, feasible solution. Furthermore, the resulting cycle time tended to be too long for practical purposes. A subsequent field exercise and study highlighted the importance of a real-time system in which users can develop operational plans based on their regional needs, analyze trade-offs, and perform dynamic changes on staffing assignment and (or) floor-plan reconfiguration in response to patient flow as an event unfolds. Since 2003, we have worked with the Centers for Disease Control and Prevention (CDC) and state public-health agencies to design and implement RealOpt to aid users in determining large-scale, realtime resource allocation. The system enables publichealth administrators to enter values of their choosing and obtain results that best reflect their emergencyresponse operating environment. Through the design and integration of efficient optimization technology and large-scale simulation, RealOpt allows users to investigate (1) locations for dispensing-facility setup, (2) clinic and POD layout design, (3) staff allocation, and (4) disease-propagation analysis. Optimal or nearoptimal solutions of instances of staff allocation models for regions with populations in the hundreds of Interfaces 39(5), pp. 476–490, © 2009 INFORMS thousands or millions can be achieved within two to four CPU minutes, with queue lengths, wait time, cycle time, and staff utilization rates that are acceptable and practical for actual operations. Medical Countermeasures Dispensing: Modeling and Computation Depending on the type of medical countermeasures that are to be dispensed, PODs can have various layouts. Lee et al. (2006a, b; 2009c) give detailed descriptions and contrast trade-offs for various POD layout designs of drive-through and walk-through models for prophylaxis medical dispensing in response to anthrax, smallpox, and flu pandemic scenarios. Briefly, within a POD facility, the tasks include (1) assessing client health status; (2) assessing client eligibility to receive service; (3) assessing implications of each case and referring case for further investigation, if necessary; (4) counseling clients regarding services and associated risks; (5) administering services; (6) educating clients regarding adverse events; (7) documenting services; (8) monitoring vaccine and medical prophylaxis take rates; (9) monitoring adverse reactions; and (10) monitoring disease development. Figure 1 shows a POD for anthrax antibiotics dispensing that was set up in one of the national drill exercises. Citizens must either drive to the POD site or Arrival Triage decision Triage Normal Registration Special Special medical care Medical screening Drug dispensing Exit Figure 1: The flowchart shows a POD that was set up in a national drill exercise to dispense anthrax antibiotics. Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Interfaces 39(5), pp. 476–490, © 2009 INFORMS to a central location from which a bus transports them to the POD. This POD consists of five main blocks: triage, registration, medical screening, drug dispensing, and special medical care. At triage, staff members greet clients, ask if they have questions, and assess how to direct clients. They direct most clients to registration. However, they might direct some (e.g., clients with preexisting conditions who are taking medications that require extra medical advice or assistance) to special medical care. At the special medical care stations, staff members assist clients with registration, attend to their special medical concerns, and dispense drugs (when appropriate). Staff members might also direct families with children to special medical care because the drug dispensed will depend on a child’s age and weight. At registration, clients fill out forms about their health. At medical screening, the staff members review the forms and determine which antibiotics should be given. At the drug dispensing station, clients receive the drug and final consultation from the staff, and then they exit the system. Mathematical and Computational Advances Maxi-Vac and drill exercises have demonstrated that a simulation system that captures the stochasticity of the emergency operations within a POD—and seeks to optimize the resource allocation and throughput— is essential. However, the CDC benchmark using commercial systems proved that these systems are computationally not feasible for solving even a small scenario (e.g., one that involves 30 staff members and service to approximately 1,000 households). A closedform solution of a system assumed to be in steady state is not helpful because in the short time window in which dispensing must be completed, the system will not achieve a steady state. Moreover, a deterministic, analytic resource calculation often underestimates resource needs and is inefficient and impractical for drill planning and actual usage. The challenge herein involves the integration of simulation and optimization into a seamless decisionsupport system that can perform rapid optimization over each simulation iteration. The coupling of the simulation and optimization processes is useful because a simulation is typically a much more realistic evaluator of system performance (objective function), and a good optimization strategy helps in speedy convergence toward the 479 desired set of parameters that produce the best system performance. However, this approach has considerable difficulties. First, a realistic simulation is typically computationally intensive, thereby limiting the number of times it can be used to evaluate the objective function for optimization. Second, most optimization algorithms work best with convex objective functions. A realistic simulation will seldom meet this condition. Therefore, the coupling of simulation and optimization remains a big challenge for the research community. Resource-Allocation Model and Solution Strategies Given a staff assignment (obtained from an initial optimization step) and input of service distributions at each station, we model and simulate the movement of individuals inside a POD. The simulation output is a set of parameters (including statistics of average flow time, queue length, wait time, utilization rate, etc.) that enables evaluation of the objective function being optimized (e.g., the resulting throughput). The optimization of labor resources involves placement of staff at various stations in the POD to maximize throughput or minimize the staffing needs to satisfy a preset throughput. The cost at each station depends on the type and number of workers who are assigned to that station and have the required skills, and on the average wait time, queue length, and utilization rate of the station. The total system cost depends on the cost at each station and on system parameters, such as cycle time and throughput. These cost functions are not necessarily expressible in closed form. Constraints in the model include maximum limits on average wait time and queue length, range of utilization desired at each station, and upper and lower bounds on the number of workers with the required skills who are needed to perform various tasks at the POD. Constraining the average cycle time to be less than a prespecified upper bound is critical for emergency response because individuals must move through the system as quickly as possible to facilitate crowd control, reduce sources of human frustration and potential disorderly outbursts, and reduce the potential spread of disease or contamination. Our CDC collaborator indicated that an acceptable upper 480 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response bound is 90 minutes. The resulting nonlinear mixedinteger program (NLMIP) poses a challenge for existing optimization engines (see the appendix). Mason and Washington (2003) illustrated the difficulty of a simplified version of this resource-allocation problem for a smallpox POD scenario running on a popular commercial simulator and optimization solver. Their input included the distribution of service times at various stations in the clinic and the availability of 30 public-health staff members to work during a 12-hour shift. The objective was to determine the staffing assignments that result in the best throughput. We confirmed CDC’s benchmarking results by testing several commercial simulation and optimization systems; many exhibited similar computational bottlenecks: After running for roughly 10 hours, the staff assignment returned did not satisfy the desired requirements related to flow and queue and wait time; in addition, the throughput achieved was fewer than 800 households. Working closely with public-health emergency directors, our team developed a simulation and optimization system from scratch using the Java programming language. Our crude implementation in winter 2003 solved the instance described in the previous paragraph in 1.09 seconds and returned a feasible assignment with a throughput of 1,117 households. For the 2007 benchmark of 100 workers running the simulation over a 36-hour period, our system returned a feasible assignment within 30 seconds with a throughput of 12,013 households. At each simulation, the NLMIP is solved using a fast, adaptive local search heuristic. The algorithm uses a fluid model to approximate system dynamics and estimate performance; it estimates the average delay and service time associated with the current staff assignment. Then it performs a greedy adaptive step to increase staffing for those stations with a total wait time exceeding the constraint limit until it reaches the desired cycle time. Similarly, it increases staffing at those stations with the longest average wait time until it reaches a desirable level. It repeats this greedy adaptive process until it achieves feasibility. When the objective is to minimize resources, the heuristic terminates. When the objective is to maximize throughput, it repeatedly performs the greedy adaptive step until it assigns all available workers. It Interfaces 39(5), pp. 476–490, © 2009 INFORMS then matches up the resulting staff assignment and optimizes over the types of available (prioritized) workers via a minimum-cost network flow algorithm. When optimization completes, it performs the next simulation iteration, which feeds the service information resulting from the simulation run into the heuristic algorithm. The process of simulation and optimization repeats and, upon termination, returns a feasible, near-optimal staffing assignment. Emergency planners can maximize the throughput under limited staffing availability, or minimize the staffing needs to satisfy a prespecified throughput. They can also manually input their staff assignment and allow our simulator to return the service statistics for evaluation of performance. We have fine-tuned the search step such that the heuristic seeks to simultaneously optimize the staffing assignment, equalize worker utilization rate, and minimize the average cycle time. It achieves this by varying the different greedy criteria within the adaptive heuristic procedure to achieve the desirable service statistics upon termination. The adaptive heuristic returns a good staffing assignment within two CPU minutes when the desired throughput ranges from tens of thousands to millions. Moreover, the resulting utilization rates, wait times, and cycle times appear to be superior to other solutions. Because individuals should move through the system as quickly as possible, short cycle and wait times are critical; a balanced utilization rate is desirable for both the morale and physical wellbeing of workers. To measure performance, we have also benchmarked the quality of this heuristic against the solution from our in-house nonlinear mixed-integer program (MIP) solver. Although the solver can solve to optimality instances of approximately hundreds of thousands of households, its computational time is over 10,000 CPU seconds. The heuristic algorithm consistently returns solutions that are within 5 percent of optimality in less than two minutes. POD-Location Model and Computational Challenges To best use public-health resources to maximize region-wide throughput of emergency treatment, our team works with local emergency-preparedness directors to investigate POD locations that best serve the Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response 481 Interfaces 39(5), pp. 476–490, © 2009 INFORMS 10,000,000 1,300,000 1,200,000 1,100,000 1,000,000 900,000 800,000 700,000 600,000 500,000 400,000 300,000 200,000 100,000 10 – 0 05 – 2 04 – 0 03 – 5 03 – 4 03 – 3 03 – 2 03 – 1 02 – 0 01 – 2 0 01 – 1 regional population. For a given regional population, we determine the number of POD locations needed for cost-effective operations and determine the assignments of households to the various PODs. We solve this via a two-stage approach by first solving for the minimum number (and operating cost) of PODs and then minimizing the travel distance and time for each household to reach its assigned POD. To model the POD-location problem and population needs, we first discretize the targeted region into subregions called grids. To each grid, we associate its population based on known US Census population densities. Each grid has a selection of potential POD locations (these locations could include large warehouses, community centers, or churches) depending on the mode of dispensing. Given the total number of jurisdictions in the region, we formulate the capacitated POD-location problem COVER-CAP (see the appendix) to ensure that at least two PODs are opened for each jurisdiction. If a catastrophic event at one site necessitates shutting down a POD, the remaining location can continue to carry out the emergency dispensing. The model also ensures that each household will travel at most dmax miles (the maximum allowed travel distance), that every household is served, and that the capacity of the facility is not violated (e.g., POD parking capacity is limited and fire codes limit the number of individuals who can be inside a facility simultaneously). COVER-CAP returns the minimum total number of PODs needed for each jurisdiction. Next, the MINAVG-CAP problem (see the appendix) is formulated. This formulation seeks to minimize the travel distance and time over all households while keeping the number of PODs in each jurisdiction fixed to the optimal value that the COVER-CAP problem determined. When there is no sharing of staff resources, these two problems can be solved independently for each jurisdiction. The metro Atlanta area, with a population of 5.2 million people, has 11 districts (jurisdictions), each of which has multiple counties. Using a grid size of one mile by one mile, the number of grid squares per district ranges between 36 and 3,275, and the number of households within grid squares ranges Figure 2: The graph shows the number of 0/1 variables in each PODlocation MIP instance for the 11 districts. Only two instances have fewer than 100,000 0/1 variables. between 140 and 3,074. Figure 2 graphically compares the number of 0/1 variables for COVER-CAP for each of the 11 districts; problem sizes range from 20880 × 21025 constraints and variables in the smallest instance to 9452550 × 9455625 in the largest instance. Although there have been many computational advances in facility location, the general problem remains NP-hard. We performed benchmark tests on these instances using CPLEX V11. CPLEX returned an optimal number of facilities for the smallest instance within 30 CPU minutes when the POD capacity was set to 2,000 people per hour; however, for this same instance, it cannot solve the problem within a week of CPU time when the facility capacity is 500 per hour. CPLEX was not able to solve any of the other instances after running for several months of CPU time. Our challenge was to find optimal or near-optimal solutions rapidly so as to be practical for emergency directors. Using recent computational advances for solving intractable facility-like instances (Lee and Zaider 2008), we were able to solve all except one instance to optimality within 300,000 CPU seconds. Specifically, we ran COVER-CAP and MINAVG-CAP for dmax ranges from 2 to 34 for the 11 districts. Of the total 2,904 MIP instances, we were able to solve all except 5 to proven optimality within 40,000 CPU seconds. Uniformly, the capacity-500 instances were 482 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response the most difficult to solve. Four of the remaining instances required approximately 300,000 CPU seconds to solve to proven optimality, and one remains unsolved. More importantly, in all solved instances, we obtained a feasible solution that is within 5 percent of optimality within 5,000 CPU seconds. To reduce the computational time further for scenario-based analyses, we designed a specialized heuristic approach that couples features of a genetic algorithm and an adaptive greedy search. When using the permutation representation of a chromosome, the challenge is to find feasible opened facilities that satisfy both capacity and distance constraints. In addition, maintaining solution feasibility during the evolutionary process can require extra computational effort and reduce solution quality. To overcome this, for each location we introduced the concept of potential served set (i.e., a subset of the population) that the location can serve feasibly. Instead of using the indexes on a chromosome as the sequence of open facilities and attempting to assign population grids to their closest facility in a feasible manner, we generate a potential served set for each candidate facility. This is analogous to Aickelin’s set-covering approach (Aickelin 2002). By doing so, we can consider indexes on a chromosome as the sequence of potential served sets. Using potential served sets in the decoding procedure ensures the solution feasibility is independent of the evolutionary process of the genetic algorithm and also provides better opportunities to eliminate redundant open facilities. Our heuristic routine also implements and embeds some features of adaptive greedy search procedures to further improve the solution quality. The initial population is partly randomly generated and partly generated via a k-mean clustering algorithm to ensure better initial and finalsolution quality. Furthermore, we apply kick-move and local search to maintain the balance between search diversification and intensification. Specifically, kick-move search is implemented by replacing a fraction of chromosomes that carry the current best objective values with randomly generated chromosomes. The rapid solution engine and quality of returned solutions facilitate the study of efficient frontiers to analyze the trade-offs in determining the most suitable number of strategically placed POD sites to best serve the regional population. For the metro Atlanta Interfaces 39(5), pp. 476–490, © 2009 INFORMS area, our POD-location solver allows us to obtain good feasible solutions that are within 8 percent of optimality in less than 15 minutes. (The hardest two instances with 500-capacity constraints ran for about 15 minutes; most other instances required less than 3 minutes.) Figure 3 shows the efficient frontier (tradeoff) between the number of facilities needed to serve the regional population and the maximum distance traveled by each household. This required solving 2,904 very challenging MIPs. Figure 4 shows the distribution of distances traveled by households for a solution with a capacity limit of 1,500 per hour and a maximum travel distance set to 10 miles. The Decision-Support System: RealOpt To provide a framework for modeling and optimizing the public-health infrastructure for all hazard emergency responses, we have designed and implemented a software suite of decision-support systems, RealOpt (RealOpt 2003–2009). The enterprise system consists of stand-alone software and decision-support systems, including RealOpt-Regional© , RealOpt-POD© , RealOpt-RSS© , and RealOpt-CRC© . It has been used in the areas of biological or radiological terrorism preparedness, infectious-disease outbreak planning, and natural-disaster response planning. RealOpt-Regional is an interactive online software enterprise system for large-scale regional medical dispensing and emergency preparedness. It features interactive visualization tools to assist users with spatial understanding of important landmarks in the region, assess the population densities and demographic makeup of the region, and identify potential facility locations. It features the backend mathematical models for large-scale facility-location problems and the novel and rapid solution engine, as described here in the POD-location section, for strategic and operational planning and real-time dynamic optimization. The rapid computational engine for solving large-scale instances in the POD-location module is critical for regional planning, in which emergency managers must map out facility locations that offer the most effective dispensing strategies, and analyze the economic and potential benefits of resource sharing across counties. Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response 483 Interfaces 39(5), pp. 476–490, © 2009 INFORMS 300 Cap = 2,000 Cap = 1,000 Number of facilities required 250 Cap = 1,500 Cap = 500 200 150 100 50 0 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Maximum allowed travel distance (mile) Figure 3: The graph shows the efficient frontier (PODs versus distance traveled) under various hourly capacity (Cap) restrictions. The solid lines represent the proven optimal solutions, and the dotted lines represent the heuristic solutions. RealOpt-POD is a stand-alone computerized decision-support system for facility layout and resource allocation. It consists of three core components: the Simulation Manager, the Optimization Manager, and the User Interface and Linker Manager. The Simulation Manager is responsible for running the simulations and extracting the various statistics (e.g., average wait time, average queue length, average utilization rate, etc.) from the POD (service facility). The Optimization Manager contains the various exact algorithms and Capacity = 1,500, dmax = 10 0.16 Fractional frequency of occurence 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 Distance traveled (miles) Figure 4: The graph shows the actual distance that each household travels to its assigned POD when the facility capacity is 1,500. 10 484 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response fast heuristics for resource allocation. In the resourceallocation module, the Simulation Manager is called repeatedly to resolve and update resource-allocation statistics. The User Interface and Linker Manager is responsible for all functions related to input of data and displaying of results. These three components and a graph-drawing tool with drag-and-drop features are integrated seamlessly into RealOpt-POD. This latter tool allows users to design specific floor plans to enable lay users to easily build simulation models. The system allows input of raw data and goodness-of-fit parameters to capture the service time into the model in real time. Moreover, we developed the entire system using Java for easy portability across different computer platforms and PDAs, eliminating the need for administrators of cash-strapped publichealth agencies to purchase proprietary licenses for compilers. RealOpt-POD takes only a few minutes to determine the staffing allocation for instances with required throughput in the order of hundreds of thousands or millions. This allows regions to develop operational plans for mass dispensing, analyze facility design and study labor trade-offs, and estimate resources needed for protection of the general population. It also allows state emergency managers to assess their current regional labor resources and to sufficiently and objectively allocate resources to various jurisdictions to ensure the most cost-effective POD operations. RealOpt-RSS is a tool for the efficient management of the logistics of receipt, stage, and storage (RSS) facilities and regional distribution nodes for medical countermeasures (RealOpt 2003–2009), and RealOptCRC relates to population health monitoring for radiological emergency planning and response. It shares some key components of RealOpt-POD, with added components to deal with radiological screening and decontamination logistics (Lee et al. 2009a). The real-time capability of RealOpt means that users can enter different parameters into the system and obtain results very quickly. It facilitates analysis of “what-if” scenarios; thus, it serves as an invaluable tool for planning and reconfigurations. Interfaces 39(5), pp. 476–490, © 2009 INFORMS Disease-Propagation Analysis: Mitigation Strategies and Choice of Dispensing Modalities Large-scale dispensing clinics could facilitate the spread of disease because of their high-volume population flow. The field of dynamical systems (mostly differential equation systems) provides the principle methods of modeling in classical mathematical epidemiology (Anderson et al. 1992, Diekmann and Heesterbeek 2000). Despite their simplicity when compared to recent complex simulation studies (Ferguson et al. 2005, 2006; Longini et al. 2005; Germann et al. 2006), these methods have helped generate functional insights, such as the transmission threshold for the start of an epidemic and the vaccination threshold for containment of an outbreak. As modelers attempt to incorporate more realistic dynamics into their models (such as stochasticity, nonexponential waiting times, sample-path dependent events, demographical and geographical data, etc.), more flexible tools, such as individual-based stochastic simulations, are preferable. Although simulation is a powerful approach, it is less mathematically tractable (i.e., it requires intensive computing time) than the classical methods. The rapid simulation and optimization modeling and computational ability of RealOpt opens up an opportunity to explore disease-propagation studies in which stochasticity of systems can be incorporated readily. RealOpt includes a disease-propagation module that aids users in understanding facility design and flow strategies that mitigate the spread of disease. The module incorporates the standard four-stage SEIR (susceptible, exposed, infectious, and recovered) model (Kermack and McKendrick 1991) and a novel six-stage SEPAIR model (Figure 5) to capture the disease development (i.e., asymptomatic or symptomatic). By distinguishing the symptomatic stage from the asymptomatic stage, this model allows one to examine the effect of triage accuracy in facility design. We can show that improving triage accuracy is critical when a disease has a higher probability of explicitly showing symptoms in patients. Lee et al. (2009b) give a detailed theoretical and computational analysis of disease propagation and strategies for mitigation during biological or pandemic outbreaks and mass dispensing. In addition to Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response 485 Interfaces 39(5), pp. 476–490, © 2009 INFORMS S E R P I D Figure 5: The flowchart shows an epidemiology model in which the infectious disease develops in six stages: susceptible (S), exposed (E), infectious (P), asymptomatic (A), symptomatic (I), and recovered (R). A small percentage of patients might not recover from the asymptomatic or symptomatic stage and therefore die (D). the stochasticity of client arrival and service distribution that RealOpt can accept, it also accommodates the following factors. • The clinic model can be represented as an n-server system with queuing; transmission can occur between clients or between clients and staff. In a real emergency, staff members will be given medical countermeasures to protect them from the disease prior to their assignment to POD services. However, a medical countermeasure does not provide 100 percent protection; each staff member still has a small probability of being infected by clients. • The intraclinic infectivity between clients and staff can vary. • If symptomatic individuals are not triaged out properly during the initial screening, they could infect other people inside the POD. The system allows users to observe the effect of triage and screening errors, determine improved strategies for triage and screening, and establish guidelines for mitigating the spread of disease because of such errors. • Inhomogeneous mixing within the community is possible. • The infectious, asymptomatic, and symptomatic individuals can infect at various rates. RealOpt offers very flexible and realistic clinicdesign and modeling features that facilitate diseasepropagation analysis. For example, it allows one to investigate and contrast the effect of standard incidence versus simple mass-action incidence infection. Likewise, we can investigate the effect of centralized versus decentralized POD design. We can also model batch processing in areas such as orientation or bus transportation, situations in which relatively large numbers of patients are in contact with one another 30 No. of intra-POD infection A 25 Triangular Exponetial ODE 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Time (hr.) Figure 6: The graph contrasts the number of intra-POD infections over a 36-hour period (with a total throughput of 36,000) when various stochasticity is incorporated into the RealOpt simulation run using the sixstage epidemiology model. Triangular and exponential correspond to the service-distributions input to RealOpt. The symptomatic proportion is 67 percent, contact number is 193 (for outer-POD disease propagation), and −5 transmission coefficient = 018E /min for intra-POD disease propagation. The incoming percentage for susceptible is 95 percent, and infectious is 5 percent. The mean dwell time is one day for both exposed and infectious and three days for asymptomatic and symptomatic. for extended periods, and thus can be areas of high infectivity. Figure 6 highlights that the ordinary differential equations (ODEs) model underestimates the number of intra-POD infections. Our disease-propagation module considers the stochasticity of patient arrival and service-probability distributions, thereby generating more accurate estimates. Figure 7 contrasts the triage accuracy with respect to the symptomatic proportion, when simple mass-action incidence infection, as Lee et al. (2009b) describe in detail, is considered. This analysis assesses errors in triage and their infection consequences. It provides estimates for POD planners and epidemiologists to use to help them determine the level and expertise of triage that should be in place with respect to the transmission coefficient. Such analyses may influence the selection of dispensing modalities. Specifically, over the past few years, we have observed more use of drive-through PODs for infectious-disease prophylaxic dispensing (e.g., seasonal flu vaccination for communities). In January 2008, a hepatitis A confirmation of a grocery worker triggered the prophylaxic vaccination of 10,000 residents in Erie County, New York, who were potentially exposed to the disease, costing the 486 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Interfaces 39(5), pp. 476–490, © 2009 INFORMS No. of intra-POD infections and collected data from actual clinical visits. We also performed role-playing with public-health officials. In addition, since 2005, the team has been collecting time-motion study data at various anthrax and smallpox drill exercises as well as actual flu and hepatitis vaccination events. The User Interface and Linker Manager includes a goodness-of-fit panel. The goodness-of-fit test, based on a chi-square test, allows users to determine the probability distribution from data collected in real time; the module includes different probability distributions, significance levels, and numbers of intervals. Symptomatic proportion (%) Triaging accuracy Figure 7: The graph shows the triage accuracy versus symptomatic proportion and the importance of using the SEPAIR six-stage propagation model, because it allows us to examine the effect of implementing triage accuracy. The graph shows the number of intra-POD infections under different triage accuracy and symptomatic proportions. The throughput is 36,000 over a period of 36 hours. The contact number is 193 (for outer-POD disease propagation), and the transmission coefficient is −5 018E /min. The incoming percentage for susceptible is 95 percent and infectious is 5 percent. The mean dwell time is one day for both exposed and infectious and three days for asymptomatic and symptomatic. county’s public-health agency at least $500,000. The health department dispensed the first vaccination in February when it set up a stationary clinic (walkthrough POD). Because of the medical logistics and infectious nature of the disease, some people had to wait for hours in frigid temperatures. In September 2008, the health department used a hepatitis A followup drive-through POD to provide a second round of vaccinations to individuals potentially exposed to the disease. The POD was also the first test of the county’s drive-through plan. The drive-through process is quick, efficient, and convenient, and it minimizes the potential of intra-infectivity (Tan 2008). Input Data: Challenge, Time-Motion Study, and Real-Time Data Processing In addition to the computational challenges, an early difficulty that we encountered was the lack of historical data for mass dispensing services. To perform optimization and resource-allocation analysis, it is imperative that we enter realistic distributions for client-service and arrival times. To obtain data for service distribution, we performed a time-motion study Validation and Successful Usage of RealOpt Since 2003, we have distributed RealOpt to more than 1,000 public-health and emergency coordinators from tribal, local, state, and federal public-health departments, including the CDC. In 2005, the planning and resource-estimation capability of RealOpt was tested and validated in an eight-county anthrax emergency drill in Georgia. This exercise involved between 600 and 700 public-health workers, hundreds of law-enforcement officers, and thousands of volunteers. Each county was responsible for its own planning and execution of the drill; only one county, DeKalb, used RealOpt to determine its POD layout and staffing needs. DeKalb achieved the highest throughput among all counties that simultaneously conducted the same scale of anthrax drill at various locations. Moreover, it was the only county that achieved and exceeded the targeted throughput; it processed 50 percent more individuals than the second-place county. Its labor usage was at or below that of the other counties. The independent state external evaluators commented that DeKalb produced the most efficient floor plan (with no path crossing), the most cost-effective dispensing (lowest labor-to-throughput ratio), and the smoothest operations (shortest average wait time, average queue length, equalized utilization rate) (Moriarty 2006). This anthrax exercise suggests that even without historical data, using our system, one can plan wisely and obtain good estimates of required labor resources. Currently, RealOpt includes time-motion study data from anthrax and Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Interfaces 39(5), pp. 476–490, © 2009 INFORMS smallpox exercise drills, seasonal flu-vaccine events (walk-through and drive-through), and drive-through hepatitis A booster shoots (Tan 2008). RAND Corporation employed RealOpt to analyze the most effective POD layout design for drill exercises. New Orleans emergency-response planners used it for their October 2007 mass-vaccination drill in which the community received free flu shots when the emergency team tested its capabilities of running a POD. The local team used RealOpt for the clinic design and for estimating staffing needs. It correctly predicted bottlenecks for the planners; the throughput numbers the system returned were fairly close to the actual numbers (these individuals actually received flu shots), thus validating the importance of such a decision-support system. Strategic and Regional Planning for Effective Multimodality Mass Dispensing To illustrate RealOpt’s strategic capability and importance, we have been working with a team of emergency-response directors in the Georgia City Readiness Initiative to develop a mass-dispensing plan for an anthrax event for the metropolitan Atlanta area (Lee et al. 2009c). Briefly, working together with public-health directors, and with the aid of the systems approach and decision tools we present in this paper, we designed a cost-effective massdispensing network for anthrax prophylaxic dispensing (Figure 8). This heterogeneous mix of PODs, capable of serving the entire regional population in a 48-hour period, is selected based on both operational efficiency and optimal staff utilization. The study reveals that (1) sharing labor resources across counties and districts within the same jurisdiction is important; (2) the most cost-effective dispensing plan across a region consists of a combination of drive-through, walk-through, and closed PODs, each operating at a throughput rate that depends on the surrounding population density, facility type, and labor availability; (3) the optimal combination of POD modalities changes according to various facility capacity restrictions, and the availability of critical public-health personnel; (4) an increase in the number of PODs in operation does not necessarily increase the total number of core public-health personnel needed; 487 (5) optimal staffing is nonlinear with respect to throughput; thus we cannot estimate the optimal staffing and throughput by simply using an average estimate; and (6) depending on the population, an “optimal” capacity that provides the most effective staffing exists for each POD location. If a POD is operating above its optimal capacity, reduction in capacity (and thus hourly throughput) eases the crowd-control tasks of law enforcement personnel and helps to minimize potential operational problems inside the POD. RealOpt has been used successfully in planning for biodefense drills (e.g., anthrax, smallpox) and pandemic response events in various locations in the United States since 2005. Users have tested various POD layouts, both drive-through and walk-through. Because of the system’s rapid speed, it facilitates analysis of what-if scenarios and serves not only as a decision tool for operational planning, actual drill preparation, and personnel training, but also allows dynamic reconfigurations as an emergency event unfolds. In addition, it supports performing “virtual field exercises,” offering insight into operation flows and bottlenecks when mass dispensing is required. Appendix. Resource-Allocation Model The optimization of labor resources involves placement of staff at various stations in a POD to maximize throughput or minimize staffing needs to satisfy a predetermined throughput. Constraints in the model include maximum limits on wait time and queue length; range of utilization desired at each station, type, and number of skilled workers, respectively, who can perform various tasks in the POD; and a maximum limit on the cycle time of the individual. The model parameters are as follows: S: the set of stations in the POD; T : the set of available types of workers; kij : the cost of assigning a worker of type i to station j, i ∈ T , j ∈ S; ij and mij : the maximum and minimum numm ber of workers, respectively, of type i that may be assigned to station j, for i ∈ T , j ∈ S; T ⊆ T : for K ∈ T, nK is the number of available workers who can assume the role of the worker types represented by K; Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response 488 Interfaces 39(5), pp. 476–490, © 2009 INFORMS Drive-through PODs throughput rate: 0 < 500 500 < 1,000 1,000 < 1,500 Walk-through PODs 0 < 500 500 < 1,000 1,000 < 1,500 Closed PODs Airport Prisons/jails Assisted living Colleges /universities Large corporate offices Homeless shelters Mobile PODs Figure 8: The map shows an effective dispensing strategy that involves a network of public and private dispensing sites of various throughput sizes and dispensing modalities. Blue (red) represents walk-through (drive-through) dispensing sites, with various hourly throughput rates. In addition to these public PODs, closed PODs are set up in large corporate offices, university and college campuses, assisted living facilities, prisons and jails, homeless shelters, and at the airport. The map does not show the mobile PODs. wj qj , and uj : the average wait time, average queue length, and average utilization rate, respectively, at station j, j ∈ S; C: the average cycle time (i.e., length of time a patient spends in the system); and : the average throughput (number of patients serviced in a specified period). Let the decision variable xij ∈ Z+ = the number of workers of type i ∈ T assigned to station j ∈ S. We can represent the cost at each station j as gj i∈T kij xij wj qj uj , j ∈ S. The total system cost depends on the cost at each station, and on system parameters, such as cycle time and throughput. Thus, we may represent the total cost as f j∈S gj c . Here, gj and f are functions that are not necessarily expressible in closed form. We can formulate a general representation of the resource-allocation problem as min z = f gj c j∈S ij st mij ≤ xij ≤ m xij ≤ nK ∀ i ∈ T j ∈ S (1) ∀ K ∈ T (2) i∈K j∈S wxj ≤ wmax qxj ≤ qmax umin ≤ uxj ≤ umax (3) Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response Interfaces 39(5), pp. 476–490, © 2009 INFORMS x ≥ max xij ∈ Z+ cx ≤ cmax ∀ i ∈ T j ∈ S (4) (5) POD-Location Model To model the POD-location problem and population needs, we first discretize the targeted region into grids (e.g., one mile by one mile), where each grid has a specified population according to census information. A selection of potential POD locations lies within these grids. Let k be the total number of jurisdictions in the region. The parameters in the model include Gi = dr l = dmax = cl = pr = set of grids in jurisdiction i; distance between grids r and l; maximum allowed travel distance; the capacity of the facility at grid l; and population of grid r. Let the decision variable yl = 1 if facility site at grid l is selected for setting up a dispensing facility, 0 otherwise; xrl = 1 if the population in grid r is served by the facility at grid l, 0 otherwise. We can formulate the capacitated POD-location problem as follows: (COVER-CAP) min k yl i=1 l∈Gi s.t. yl ≥ 2 ∀ i = 1 k (6) l∈Gi dr lxrl ≤ dmax yl ∀ r l ∈ Gi i = 1 k (7) xrl = 1 ∀ r ∈ Gi i = 1 k (8) xrl pr ≤ cl (9) l∈Gi ∀l ∈ Gi i = 1k r∈Gi xrl yl ∈ 0 1 ∀ r l ∈ Gi i = 1 k (10) Constraint (6) ensures that at least two PODs are opened. This is required if a catastrophic event at one site necessitates shutting down a POD; in such a case, emergency dispensing can still be carried out in the remaining location. Constraint (7) ensures that 489 each household will travel at most dmax miles, constraint (8) ensures that each household is served, and constraint (9) represents the capacity of the facility. Let ni = number of facilities in jurisdiction i as determined by COVER-CAP. MINAVG-CAP minimizes the distance traveled by all households, ki=1 r∈Gi l∈Gi xrl dr lpr , while satisfying the constraint sets (7)–(10). Constraint (6) for MINAVG-CAP is given by l∈Gi yl = ni ∀i = 1 k, where ni is the number of PODs required for jurisdiction i as returned by COVER-CAP. When there is no sharing of staff resources, these two problems can be solved independently for each jurisdiction. For each jurisdiction, COVER-CAP and MINAVG-CAP have identical problem sizes, with k + k k 2 2 i=1 Gi + 2Gi constraints and i=1 Gi + Gi 0/1 variables. Although there have been many computational advances in facility location, the general problem remains NP-hard. Acknowledgments The material and results reported herein are based on our work in this area and interaction with public-health agencies. We are grateful to have had the opportunity to participate in exercise drills and time-motion studies, and to have had discussions with many state and federal public-health and emergency-response experts. Although we would like to thank many people who took part in this multiagency and multidisciplinary collaboration, we would like to particularly thank Jacquelyn Mason of the CDC and Tom Tubesing, formerly of the CDC; Dr. Duane Caneva, Dr. James Lawler, and Dr. Carter Mecher at the White House Homeland Security Council; William Glisson at ESi; Bernard Hicks at the DeKalb Emergency Preparedness Department; and the many public-health and emergency managers throughout the nation who have used RealOpt. We thank the Wagner judges for their comments that helped to improve this paper. We acknowledge funding from CDC to conduct the time-motion study and postevent operations analysis, and from the National Institutes of Health for translational biomedical informatics advances. This research was funded by the National Institutes of Health, and the author will add the Web-published pdf file of the article to the National Library of Medicine’s PubMed Central database no later than 12 months after publication. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the CDC. 490 Lee et al.: Modeling and Optimizing the Public-Health Infrastructure for Emergency Response References Aaby, K., J. W. Herrmann, C. S. Jordan, M. Treadwell, K. Wein. 2006. Montgomery County’s public health service uses operations research to plan emergency mass dispensing and vaccination clinics. Interfaces 36(6) 569–579. Aickelin, U. 2002. An indirect genetic algorithm for set covering problems. J. Oper. Res. 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