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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change Appendix F Combining Game Theory and Risk Analysis in Counterterrorism: A Smallpox Example David L. Banks Professor, Institute of Statistics and Decision Sciences Duke University, Durham, North Carolina Steven Anderson Director, Office of Biostatistics and Epidemiology Center for Biologics Evaluation and Research U.S. Food and Drug Administration, Rockville, Maryland Abstract: Federal agencies have finite resources. Even for critical purposes related to counterterrorism, resources must be allocated in the most effective ways possible. Statistical risk analysis can help by accounting for uncertainties in the costs and benefits of particular efforts, and game theory can help by accounting for the fact that terrorists adapt their attacks in response to homeland defense initiatives. This paper describes a procedure that uses risk analysis to generate random payoff matrices for game theory solution, and then pools the solutions from multiple realizations of the payoff matrix to estimate the probability that a given play is optimal with respect to one of several criteria. The strategy is illustrated for risk management in the context of a simplified model of the threat of smallpox attack. 1. INTRODUCTION The U.S. government wishes to invest its resources as wisely as possible in defense. Each wasted dollar diverts money that could be used to harden crucial vulnerabilities, prevents investment in future economic growth, and increases taxpayer burden. This is a classic conflict situation; a good strategy for the player with fewer resources is to leverage disproportionate resource investment by its wealthy opponent. That strategy rarely wins, but it makes the conflict sufficiently debilitating that the wealthy opponent may be forced to consider significant compromises. Game theory is a traditional method for choosing resource investments in conflict situations. The standard approach requires strong assumptions about the availability of mutual information and the rationality of both opponents. Empirical research by many people (e.g., Kahneman and Tversky, 1972) shows that these assumptions fail in practice, leading to the development of modified theories with weaker assumptions or the use of prior probabilities in the spirit of Bayesian decision theory. This paper considers both traditional game theory (minimax solution for a two-person zero-sum game in normal form) and also a minimum expected loss criterion appropriate for extensive-form games with prior probabilities. However, we emphasize that for terrorism, the zero-sum model is at best an approximation; the valuation of the wins and the losses is likely to differ between the opponents. Game theory requires numerical measures of payoffs (or losses) that correspond to particular sets of decisions. In practice, those payoffs are rarely known. Statistical risk analysis allows experts to determine reasonable probability distributions for the random payoffs. This paper shows how risk analysis can support game theory solutions, and how Monte Carlo methods provide insight into the optimal game theory solutions in the presence of uncertainty about payoffs. Our methodology is demonstrated in the context of risk management for a potential terrorist attack using the smallpox virus. The analysis we present here is a simplified version that aims at methodological explanation rather than analysis or justification of specific healthcare policies. As a tabletop exercise, the primary aim is only to provide a blueprint for a more rigorous statistical risk analysis. The underlying assumptions, modeling methods used here, and any results or discussion of the modeling are based on preliminary and unvalidated data and do not represent the opinion of the FDA, the Department of Health and Human Services or any branch of the U.S. government. NOTE: Reprinted, with permission, from Statistical Methods in Counterterrorism: Game Theory, Modeling, Syndromic Surveillance, and Biometric Authenticationon. G. Wilson, and D. Olwell (eds.), Springer, 2006. pp. 9-22.
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change 2. GAME THEORY FOR SMALLPOX The smallpox debate in the United States has focused upon three kinds of attack and four kinds of defense. The three attack scenarios suppose that there might be: no smallpox attack a lone terrorist attack on a small area (similar to the likely scenario for the anthrax letters) a coordinated terrorist attack upon multiple population centers. The four defense scenarios that have been publicly considered by United States agency officials are: stockpile smallpox vaccine stockpile vaccine and develop biosurveillance capabilities stockpile vaccine, develop biosurveillance, and inoculate key personnel provide mass vaccination to non-immunocompromised citizens in advance. Although there are many refinements that can be considered for both the attack and the defense scenarios, these represent the possibilities discussed in the public meetings held in May and June 2002 (McKenna, 2002). Suppose that analysts used game theory as one tool to evaluate potential defense strategies. Then the three kinds of attack and four kinds of defense determine a classic normal-form payoff matrix for the game [see Table 1]. The Cij entries are the costs (or payoffs) associated with each combination of attack and defense, and we have used abbreviated row and column labels to identify the defenses and attacks, respectively, as described before. For each of the 12 attack-defense combinations, there is an associated cost. These costs may include dollars, human lives, time, and other resources. For our calculation, all of these costs are monetized, according to principles detailed in Section 3. And the monetized value of a human life is set to $750,000, following the Department of Transportation’s human capital model that estimates value from average lost productivity (non-market approaches tend to give larger values). Note that there is very large uncertainty in the Cij values. Portions of the cost (e.g., those associated with expenses TABLE 1 Attack-Defense Cost Matrix No Attack Single Attack Multiple Attack Stockpile Vaccine C11 C12 C13 Biosurveillance C21 C22 C23 Key Personnel C31 C32 C33 Everyone C41 C42 C43 already entailed) may be known, but the total cost in each cell is a random variable. These random variables are not independent, since components of the total cost are common to multiple cells. Thus it is appropriate to regard the entire game theory table as a multivariate random variable whose joint distribution is required for a satisfactory analysis that propagates uncertainty in the costs through to uncertainty about best play. Classical game theory (cf. Myerson 1991, Chapter 3) determines the optimal strategies for the antagonists via the minimax theorem. This theorem asserts that for any two-person cost matrix in a strictly competitive game (which is the situation for our example), there is an equilibrium strategy such that neither player can improve their expected payoff by adopting a different attack or defense. This equilibrium strategy may be a pure strategy, in which case optimal play is a specific attack-defense pair. This happens when the attack that maximizes the minimum damage and the defense that minimizes the maximum damage coincide in the same cell. Otherwise, the solution is a mixed strategy, in which case the antagonists pick attacks and defenses according to a probability distribution that must be calculated from the cost matrix. There may be multiple equilibria that achieve the same expected payoff, and for large matrices it can be difficult to solve the game. Alternatively, one can use Bayesian decision theory to solve the game. Here a player puts a probability distribution over the actions of the opponent, and then chooses their own action so as to minimize the expected cost (cf. Myerson 1991, Chapter 2). Essentially, one just multiplies the cost in each row by the corresponding probability, sums these by row, and picks the defense with the smallest sum. This formulation is easier to solve, but it requires one to know or approximate the opponent’s probability distribution and it does not take full account of the mutual strategic aspects of adversarial games (i.e., the assigned probabilities need not correspond to any kind of “if I do this then he’ll do that” reasoning). Bayesian methods are often used in extensive-form games, where players make their choices over time, conditional on the actions of their opponent. In developing our analysis of the smallpox example we make two assumptions about time. First, we use only the information available by June 1, 2002; subsequent information on the emerging program costs is not included. This keeps the analysis faithful in spirit to the decision problem actually faced by U.S. government policy makers in the spring of 2002 (their initial plan was universal vaccination, but ultimately they chose the third scenario with stockpiling, biosurveillance, and very limited vaccination of some first responders). Second, all of the estimated cost forecasts run to October 1, 2007. The likelihood of changing geopolitical circumstances makes it unrealistic to attempt cost estimates beyond that fiscal year.
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change 3. RISK ANALYSIS FOR SMALLPOX Statistical risk analysis is used to estimate the probability of undesirable situations and their associated costs. In the same way that it is used in engineering (e.g., for assessing nuclear reactor safety; cf. Speed, 1985) or the insurance industry (e.g., for estimating the financial costs associated with earthquakes in a specific area; cf. Brillinger, 1993), this paper uses risk analysis to estimate the costs associated with different kinds of smallpox attack/defense combinations. Risk analysis involves careful discussions with domain experts and structured elicitation of their judgments about probabilities and costs. For smallpox planning, this requires input from physicians, public health experts, mathematical epidemiologists, economists, emergency response administrators, government accountants, and other kinds of experts. We have not conducted the in-depth elicitation from multiple experts in each area that is needed for a fully rigorous risk analysis; however, we have discussed the cost issues with representatives from each area, and we believe that the estimates in this section are sufficiently reasonable to illustrate, qualitatively, the case for combining statistical risk analysis with game theory for threat management in the context of terrorism. Expert opinion was typically elicited in the following way. Each expert was given a written document with background on smallpox epidemiology and a short description of the attacks and defenses considered in this paper. The expert often had questions; these were discussed orally with one of the authors and, to the extent possible, resolved on the basis of the best available information. Then the expert was asked to provide a point estimate of the relevant cost or outcome and the range in which that value would be expected to fall in 95% of similar realizations of the future. If these values disagreed with those from other experts, then the expert was told of the discrepancy and invited to alter their opinion. Based on point estimate and the range, the authors and the expert chose a distribution function with those parameters which also respected real-world requirements for positivity, integer values, known skew, or other properties. As the last step in the interview, the expert was given access to all the other expert opinions obtained to that point and asked if there were any that seemed questionable; this led to in one case to an expert being recontacted and a subsequent revision of the elicitation. But it should be emphasized that these interviews were intended to be short, and did not use the full range of probes, challenges, and checks that are part of serious elicitation work. The next three subsections describe the risk analysis assumptions used to develop the random costs for the first three cells (C11, C21, C31) in the game theory payoff matrix. Details for developing the costs in the other cells are available from the authors. These assumptions are intended to be representative, realistic, and plausible, but additional input by experts could surely improve upon them. Many of the same costs arise in multiple cells, introducing statistical dependency among the entries. (That is, if a given random payoff matrix assumes an unusually large cost for stockpiling in one cell of the random table, then the same high value should appear in all other cells in which stockpiling occurs.) 3.1 Cell (1,1): Stockpile Vaccine/No Attack Scenario Consider the problem of trying to estimate the costs associated with the (1,1) cell of the payoff matrix, which corresponds to no smallpox attack and the stockpiling of vaccine. This estimate involves combining costs with very different levels of uncertainty. At the conceptual level, the cost C11 is the sum of four terms: where ETdry and ETAvent are the costs of efficacy and safety testing for the Dryvax and Aventis vaccines, respectively; ETAcamb is the cost of new vaccine production and testing from Acambis; VIG is the cost of producing sufficient doses of vaccinia immune globulin to treat adverse reactions and possible exposures; and PHIS is the cost of establishing the public healthcare infrastructure needed to manage this stockpiling effort. There is no uncertainty about ETAcamb; the contract fixes this cost at $512 million. But there is substantial uncertainty about ETdry and ETAvent since these entail clinical trials and may require follow-on studies; based on discussions with experts, we believe these costs may be realistically modeled as independent uniform random variables, each ranging between $2 and $5 million. There is also large uncertainty about the cost for producing and testing sufficient doses of VIG to be prepared for a smallpox attack; our discussions suggest this is qualitatively described by a normal random variable with mean $100 million and a standard deviation of $20 million. And there is great uncertainty about PHIS (which includes production of bifurcated inoculation needles, training, storage costs, shipment readiness costs, etc.); based on the five-year operating budget of other government offices with analogous missions, we assume this cost is normally distributed with mean $940 million and standard deviation $100 million. 3.2 Cell (2,1): Biosurveillance/No Attack Scenario Biosurveillance programs are being piloted in several major metropolitan areas. These programs track data, on a daily basis, from emergency room admission records in order to quickly discover clusters of disease symptoms that suggest bioterrorist attack. Our cost estimates are based upon discussions with the scientists working in the Boston area (cf. Ross et al., 2002) and with the Pittsburgh team that developed monitoring procedures for the Salt Lake City Olympic games.
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change The cost C21 includes the cost C11 since this defense strategy uses both stockpiling of vaccine and increased biosurveillance. Thus where PHIB is the cost of the public health infrastructure needed for biosurveillance, including the data input requirements and software; PHM is the cost of a public health monitoring center, presumably at the Centers for Disease Control, that reviews the biosurveillance information on a daily basis; NFA is the number of false alarms from the biosurveillance system over five years of operation; and FA is the cost of a false alarm. For this exercise, we assume that PHIB is normally distributed with mean $900 million and standard deviation $100 million (for a five-year funding horizon); this is exclusive of the storage, training, and other infrastructure costs in PHIS, and it includes the cost of hospital nursing-staff time to enter daily reports on emergency room patients with a range of disease symptoms (not just those related to smallpox). PHM is modeled as a normal random variable with mean $20 million and standard deviation $4 million (this standard deviation was proposed by a federal administrator, and may understate the real uncertainty). False alarms are a major problem for monitoring systems; it is difficult to distinguish natural contagious processes from terrorist attacks. We expect about one false alarm per month over five years in a national system of adequate sensitivity, and thus FA is taken to be a Poisson random variable with mean 60. The cost for a single false alarm is modeled as a normal random variable with mean $500,000 and standard deviation $100,000. 3.3 Cell (3,1): Key Personnel/No Attack Scenario One option, among several possible policies that have been discussed, is for the United States to inoculate about 500,000 key personnel, most of whom would be first-responders in major cities (i.e., emergency room staff, police, and public health investigators who would be used to trace people who have come in contact with carriers). If chosen, this number is sufficiently large that severe adverse reactions become a statistical certainty. The cost of this scenario subsumes the cost C21 of the previous scenario, and thus where NKP is the number of key personnel; IM is the cost of the time and resources needed to inoculate 25,000 key personnel and monitor them for adverse events; PAE is the probability of an adverse event; and AEC is the average cost of one adverse event. We assume that NKP is uniformly distributed between 400,000 and 600,000 (this reflects uncertainty about how many personnel would be designated as “key”). The IM is tied to units of 25,000 people, since this is a one-time cost and represents the number of people that a single nurse might reasonably inoculate and maintain records upon in a year. Using salary tables, we approximate this cost as a normal random variable with mean $60,000 and standard deviation $10,000. The probability of an adverse event is taken from Anderson (2002), which is based upon Lane et al. (1970); the point estimate for all adverse events is .293, but since there is considerable variation and new vaccines are coming into production, we have been conservative about our uncertainty and assumed that the probability of an adverse event is uniformly distributed between .15 and .45. Of course, most of these events will be quite minor (such as local soreness) and would not entail any real economic costs. The AEC is extremely difficult to estimate. For purposes of calculation, we have taken the value of a human life to be $2.86 million (the amount used by the National Highway Transportation Safety Administration in cost-benefit analyses of safety equipment). But most of the events involve no cost, or perhaps a missed day of work that has little measurable impact on productivity. After several calculations and consultations, this analysis assumes that AEC can be approximated as a gamma random variable with mean $40 and standard deviation $100 (this distribution has a long right tail). 4. ANALYSIS The statistical risk analysis used in Section 3, albeit crude, shows how expert judgment can generate the random payoff matrices. The values in the cells of such tables are not independent, since many of the cost components are shared between cells. In fact, it is appropriate to view the table as a matrix-valued random variable with a complex joint distribution. Random tables from this joint distribution can be generated by simulation. For each table, one can apply either the minimax criterion to determine an optimal strategy in the sense of von Neumann and Morgenstern (1944), or a minimum expected loss criterion to determine an optimal solution in the sense of Bayesian decision theory (cf. Myerson, 1991, Chapter 2). By doing this repeatedly, for many different random tables, one can estimate the proportion of time that each defense strategy is superior. Additionally, it seems appropriate to track not just the number of times a defense strategy is optimal, but also weight this count by some measure of the difference between the costs of the game under competing defenses. For example, if two defenses yield game payoffs that differ only by an insignificant amount, it seems unrealistic to give no credit to the second-best strategy. For this reason we also use a scor-
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change ing algorithm in which the score a strategy receives depends upon how well-separated it is from the optimal strategy. Specifically, suppose that defense strategy i has value Vi on a given table. Then the score Si that strategy i receives is and this ensures that strategies are weighted to reflect the magnitude of the monetized savings that accrue from using them. The final rating of the strategies is obtained by averaging their scores from many random tables. 4.1 Minimax Criterion We performed the simulation experiment described above 100 times and compared the four defense strategies in terms of the minimax criterion. Although one could certainly do more runs, we believe that the approximations in the cost modeling are so uncertain that additional simulation would only generate spurious accuracy. Among the 100 runs, we found that the Stockpile strategy won 9 times, the Biosurveillance strategy won 24 times, the Key Personnel strategy won 26 times, and the Vaccinate Everyone strategy won 41 times. This lack of a clear winner may be, at some intuitive level, the cause of the widely different views that have been expressed in the public debate on preparing for a smallpox attack. If one uses scores, the results are even more ambiguous. The average score for the four defense strategies ranged between .191 and .326, indicating that the expected performances were, on average, quite similar. From a public policy standpoint, this may be a fortunate result. It indicates that in terms of the minimax criterion, any decision is about equally defensible. This gives managers flexibility to incorporate their own judgment and to respond to extrascientific considerations. 4.2 Minimum Expected Loss Criterion The minimax criterion may not be realistic for the game theory situation presented by the threat of smallpox. In particular, the normal-form game assumes that both players are ignorant of the decision made by their opponent until committed to a course of action. For the smallpox threat, there has been a vigorous public discussion on what preparations the United States should make. Terrorists know what the United States has decided to do, and presumably this will affect their choice of attack. Therefore the extensive-form version of game theory seems preferable. This form can be thought of as a decision tree, in which players alternate their moves. At each stage, the player can use probabilistic assessments about the likely future play of the opponent. The minimum expected loss criterion requires more information that does the minimax criterion. The analyst needs to know the probabilities of a successful smallpox attack conditional on the U.S. selecting each of the four possible defenses. This is difficult to determine, but we illustrate how one can do a small sensitivity analysis that explores a range of probabilities for smallpox attack. Table 2 shows a set of probabilities that we treat as the baseline case. We believe it accords with a prudently cautious estimate of the threat of a smallpox attack. To interpret Table 2, it says that if the United States were to only stockpile vaccine, then the probability of no smallpox attack is .95, the probability of a single attack is .04, and the probability of multiple attacks is .01. Similarly, one reads the attack probabilities for other defenses across the row. All rows must sum to one. The minimum expected loss criterion multiplies the probabilities in each row of Table 2 by the corresponding costs in the same row of Table 1, and then sums across the columns. The criterion selects the defense that has the smallest sum. As with the minimax criterion, one can simulate many payoff tables and then apply the minimum expected loss criterion to each. In 100 repetitions, Stockpile won 96 times, Biosurveillance won 2 times, and Vaccinate Everyone won twice. The scores showed roughly the same pattern, strongly favoring the Stockpile defense. We now consider two alternative sets of probabilities, shown in Table 3 and Table 4. Table 3 is more pessimistic, and has larger attack probabilities. Table 4 is more optimistic, and has smaller attack probabilities. A serious sensitivity analysis would investigate many more tables, but our purpose is illustration and we doubt that the quality of the assessments that underlie the cost matrix can warrant further detail. For Table 3, 100 simulation runs found that Stockpile won 15 times, Biosurveillance won 29 times, Key Personnel won 40 times, and Vaccinate Everyone won 16 times. In contrast, for Table 4, the Stockpile strategy won 100 times in 100 runs. The scores for Table 3 ranged from 18.2 to 38.8, which are quite similar. In contrast, for Table 4 nearly all the weight of the score was on the Stockpile defense. These results show that the optimal strategy is sensitive to the choice of probabilities used in the analysis. Determining those prob- TABLE 2 Baseline Probabilities of Attack Given Different Defenses No Attack Single Attack Multiple Attack Stockpile Vaccine 0.95 0.04 0.01 Biosurveillance 0.96 0.035 0.005 Key Personnel 0.96 0.039 0.001 Everyone 0.99 0.005 0.005
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change TABLE 3 Pessimistic Probabilities of Attack Given Different Defenses No Attack Single Attack Multiple Attack Stockpile Vaccine 0.70 0.20 0.10 Biosurveillance 0.80 0.15 0.05 Key Personnel 0.85 0.10 0.05 Everyone 0.90 0.05 0.05 TABLE 4 Optimistic Probabilities of Attack Given Different Defenses No Attack Single Attack Multiple Attack Stockpile Vaccine 0.98 0.01 0.01 Biosurveillance 0.99 0.005 0.005 Key Personnel 0.99 0.005 0.005 Everyone 0.999 0.0005 0.0005 abilities requires input from the intelligence community and the judgment of senior policy-makers. 5. CONCLUSIONS This paper has outlined an approach combining statistical risk analysis with game theory in order to evaluate defense strategies that have been considered for the threat of smallpox. We believe that this approach may offer a useful way of structuring generic problems in resource investment for counterterrorism. The analysis in this paper is incomplete: We have focused upon smallpox, because the problem has been framed rather narrowly and quite definitively by public discussion. But a proper game theory analysis would not artificially restrict the options of the terrorists, and should consider other attacks, such as truck bombs, chemical weapons, other diseases, and so forth (which would get difficult, but there may be ways to approximate). It can be completely misleading to seek a local solution, as we have done. Similarly, we have not fully treated the options of the defenders. For example, heavy investment in intelligence sources is a strategy that protects against many different kinds of attacks, and might well be the superior solution in a less local formulation of the problem. We have not considered constraints on the resources of the terrorists. The terrorists have limited resources and can invest in a portfolio of different kinds of attacks. Symmetrically, the U.S. can invest in a portfolio of defenses. This aspect of the problem is not addressed—we assume that both parties can fund any of the choices without sacrificing other goals. The risk analysis presented here, as discussed previously, is not adequate to support public policy formulation. Nonetheless, despite these limitations, the methodology has attractive features. First, it is easy to improve the quality of the result through better risk analysis. Second, it automatically raises issues that have regularly emerged in policy discussions. And third, it captures facets of the problem that are not amenable to either game theory or risk analysis on their own, because classical risk analysis is not used in adversarial situations and because classical game theory does not use random costs. NOTES: BACKGROUND ON SMALLPOX Although the probability that the smallpox virus (Variola major) might be used against the U.S. is thought to be small, the public health and economic impact of even a limited release would be tremendous. Any serious attack would probably force mass vaccination programs, causing additional loss of life due to adverse reactions. Other economic consequences could easily be comparable to those of the attacks of September 11, 2001. A smallpox attack could potentially be initiated through infected humans or through an aerosol (Henderson et al., 1999). In 12-14 days after natural exposure patients experience fever, malaise, body aches, and a body rash (Fenner et al., 1988). During the symptomatic stages of the disease the patient can have vesicles in the mouth, throat, and nose that rupture to spread the virus during a cough or sneeze. Person-to-person spread usually occurs through inhalation of virus-containing droplets or from close contact with an infected person. As the disease progresses the rash spreads to the head and extremities and evolves into painful, scarring vesicles and pustules. Smallpox has a mortality rate of approximately 30%, based on data from the 1960s and 1970s (Henderson, 1999). Various mathematical models of smallpox spread exist and have been used to forecast the number of people infected under different exposure conditions and different public health responses (cf. Kaplan, Craft, and Wein, 2002; Meltzer et al., 2001). There is considerable variation in the predictions from these models, partly because of differing assumptions about the success of the “ring vaccination” strategy that has been planned by the Centers for Disease Control (2002), and this is reflected in the public debate on the value of preemptive inoculation versus wait-and-see preparation. However, the models are in essential agreement that a major determinant of the size of the epidemic is the number of people who are exposed in the first attack or attacks. The current vaccine consists of live vaccinia or cowpox virus and is effective at preventing the disease. Also, vaccination can be performed within the first 2 to 4 days post exposure to reduce the severity or prevent the occurrence of the disease (Henderson, 1999).
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Department of Homeland Security Bioterrorism Risk Assessment: A Call for Change But vaccination is not without risk; the major complications are serious infections and skin disease such as progressive vaccinia, eczema vaccinatum, generalized vaccinia, and encephalitis. Approximately 12 people per million have severe adverse reactions that require extensive hospitalization, and about one-third of these die—vaccinia immune globulin (VIG) is the recommended therapy for all of these reactions except encephalitis. Using data from Lane et al. (1970), we estimate that 1 in 71,429 people suffer postvaccinial encephalitis, 1 in 588,235 suffer progressive vaccinia, 1 in 22,727 suffer eczema vaccinatum, and 1 in 3,623 suffer generalized vaccinia. Additionally, 1 in 1,656 people suffer accidental infection (usually to the eye) and 1 in 3,289 suffer some other kind of mild adverse event, typically requiring a person to miss a few days of work. (Other studies give somewhat different numbers; cf. Neff et al., 1967a, 1967b). People who have previously been successfully vaccinated for smallpox are less likely to have adverse reactions, and people who are immunocompromised (e.g., transplant patients, those with AIDS) are at greater risk for adverse reactions (cf. Centers for Disease Control, 2002, Guide B, parts 3, 5, and 6). Because the risk of smallpox waned in the 1960s, vaccination of the U.S. population was discontinued in 1972. It is believed that the effectiveness of a smallpox vaccination diminishes after about 7 years, but residual resistance persists even decades later. It has been suggested that people who were vaccinated before 1972 may be substantially protected against death, if not strongly protected against contracting the disease (cf. Cohen, 2001). The U.S. currently has about 15 million doses of the Wyeth Dryvax smallpox vaccine available. The vaccine was made by scarification of calves with the New York City Board of Health strain and fluid containing the vaccinia virus was harvested by scraping (Rosenthal et al., 2001). Recent clinical trials on the efficacy of diluted vaccine indicate that both the five-fold and ten-fold dilutions of Dryvax achieve a take rate (i.e., a blister forms at the inoculation site, which is believed to be a reliable indicator of immunization) of at least 95%, so the available vaccine could be administered to as many as 150 million people should the need arise (cf. Frey et al., 2002; NIAID, 2002). The disclosure by the pharmaceutical company Aventis (Enserink, 2002) of the existence in storage of 80 to 90 million doses of smallpox vaccine that were produced more than 30 years ago has added to the current stockpile. Testing is being done on the efficacy of the Aventis vaccine stock, including whether it, too, could be diluted if needed. Contracts to make new batches of smallpox vaccine using cell culture techniques have been awarded to Acambis. The CDC amended a previous contract with Acambis in September 2001 to ensure production of 54 million doses by late 2002. Another contract for the production of an additional 155 million doses was awarded to Acambis in late November 2001, and the total cost of these contracts is $512 million. After production, additional time may be needed to further test the safety and efficacy of the new vaccine (cf. Rosenthal et al., 2001). REFERENCES Anderson, S. (2002). “A risk-benefit assessment of smallpox and smallpox vaccination,” Technical Report, Office of Biostatistics and Epidemiology, Center for Biologics Evaluation and Research, U.S. Food and Drug Administration, Rockville, MD, 2002. Brillinger, D.R. (1993). “Earthquake risk and insurance,” EnviroMetrics, 4, 1-21. Centers for Disease Control, (2002). Smallpox response plan and guidelines (Version 3.0), www.bt.cdc.gov/agent/smallpox/response-plan/index.asp. Cohen, J. (2001). “Smallpox vaccinations: How much protection remains?” Science, 294, 985. Enserink, M. (2002). “New cache eases shortage worries,” Science, 296 (April 5) 25-26. Fenner, F., Henderson, D.A., Arita, I., Jezek, Z., Ladnyi, I.D. (1988). Smallpox and Its Eradication, World Health Organization, Geneva. Frey, S.E., Couch, R.B, Tacket, C.O., Treanor, J.J., Wolff, M., Newman, F.K. Atmar, R.L., Edelman, R., Nolan, C.M., Belshe, R.B. (2002). “Clinical responses to undiluted and diluted smallpox vaccine,” New England Journal of Medicine, 346:17, 1265-1274. Halloran, M.E., Haber, M., Longini, I.M, Jr., and Struchiner, C.J. (1991). “Direct and indirect effects in vaccine efficacy and effectiveness,” American Journal of Epidemiology, 133, 323-331. Henderson, D.A. (1999). “Smallpox: Clinical and epidemiological features,” Emerging Infectious Diseases, 5, 537-539. 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Meltzer, M.I., Damon, I., LeDuc, J.W., and Millar, J.D. (2001). “Modeling potential responses to smallpox as a bioterrorist weapon,” Emerging Infectious Diseases, 7, 201-208. Myerson, R.B. (1991). Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA, 1991. NIAID. (2002). “NIAID study results support diluting smallpox vaccine stockpile to stretch supply,” NIAID News, March 28. National Institute of Allergy and Infectious Diseases, www.niaid.nih.gov/newsroom/releases/smallpox.htm. Neff, J.M., Lane, J., Pert, J.P., et al. (1967). “Complications of smallpox vaccination, I: National survey in the United States, 1963,” New England Journal of Medicine, 276, 1-8. Neff, J.M., Levine, R.H., Lane, J.M., et al. (1967). “Complications of smallpox vaccination, United States, 1963, II: Results obtained from four statewide surveys,” Pediatrics, 39, 16-923. Rosenthal, S.R., Merchilinsky, M., Kleppinger, C., and Goldenthal, K.L. 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