Cover Image

PAPERBACK
$39.75



View/Hide Left Panel

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.



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 103
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 requires strong assumptions about the availability of mutual critical purposes related to counterterrorism, resources must information and the rationality of both opponents. Empiri- be allocated in the most effective ways possible. Statistical cal research by many people (e.g., Kahneman and Tversky, risk analysis can help by accounting for uncertainties in the 1972) shows that these assumptions fail in practice, leading costs and benefits of particular efforts, and game theory can to the development of modified theories with weaker as- help by accounting for the fact that terrorists adapt their sumptions or the use of prior probabilities in the spirit of attacks in response to homeland defense initiatives. This Bayesian decision theory. paper describes a procedure that uses risk analysis to gener- This paper considers both traditional game theory (mini- ate random payoff matrices for game theory solution, and max solution for a two-person zero-sum game in normal then pools the solutions from multiple realizations of the form) and also a minimum expected loss criterion appropri- payoff matrix to estimate the probability that a given play is ate for extensive-form games with prior probabilities. How- optimal with respect to one of several criteria. The strategy is ever, we emphasize that for terrorism, the zero-sum model is illustrated for risk management in the context of a simplified at best an approximation; the valuation of the wins and the model of the threat of smallpox attack. losses is likely to differ between the opponents. Game theory requires numerical measures of payoffs (or losses) that correspond to particular sets of decisions. 1. INTRODUCTION In practice, those payoffs are rarely known. Statistical risk The U.S. government wishes to invest its resources as analysis allows experts to determine reasonable probability wisely as possible in defense. Each wasted dollar diverts distributions for the random payoffs. This paper shows money that could be used to harden crucial vulnerabilities, how risk analysis can support game theory solutions, and prevents investment in future economic growth, and in- how Monte Carlo methods provide insight into the optimal creases taxpayer burden. This is a classic conflict situation; game theory solutions in the presence of uncertainty about a good strategy for the player with fewer resources is to payoffs. leverage disproportionate resource investment by its wealthy Our methodology is demonstrated in the context of opponent. That strategy rarely wins, but it makes the conflict risk management for a potential terrorist attack using the sufficiently debilitating that the wealthy opponent may be smallpox virus. The analysis we present here is a simpli- forced to consider significant compromises. fied version that aims at methodological explanation rather Game theory is a traditional method for choosing resource than analysis or justification of specific healthcare policies. investments in conflict situations. The standard approach 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 NOTE: Reprinted, with permission, from Statistical Methods in Coun- preliminary and unvalidated data and do not represent the terterrorism: Game Theory, Modeling, Syndromic Sureillance, and Bio- opinion of the FDA, the Department of Health and Human metric Authenticationon. G. Wilson, and D. Olwell (eds.), Springer, 2006. Services or any branch of the U.S. government. pp. 9-22. 0

OCR for page 103
0 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT 2. GAME THEORY FOR SMALLPOX already entailed) may be known, but the total cost in each cell is a random variable. These random variables are not The smallpox debate in the United States has focused independent, since components of the total cost are common upon three kinds of attack and four kinds of defense. The to multiple cells. Thus it is appropriate to regard the entire three attack scenarios suppose that there might be: game theory table as a multivariate random variable whose joint distribution is required for a satisfactory analysis that • no smallpox attack propagates uncertainty in the costs through to uncertainty • a lone terrorist attack on a small area (similar to the about best play. likely scenario for the anthrax letters) Classical game theory (cf. Myerson 1991, Chapter 3) • a coordinated terrorist attack upon multiple population determines the optimal strategies for the antagonists via the centers. minimax theorem. This theorem asserts that for any two- person cost matrix in a strictly competitive game (which The four defense scenarios that have been publicly con- is the situation for our example), there is an equilibrium sidered by United States agency officials are: strategy such that neither player can improve their expected payoff by adopting a different attack or defense. This equilib- • stockpile smallpox vaccine rium strategy may be a pure strategy, in which case optimal • stockpile vaccine and develop biosurveillance capabilities play is a specific attack-defense pair. This happens when the • stockpile vaccine, develop biosurveillance, and inocu- attack that maximizes the minimum damage and the defense late key personnel that minimizes the maximum damage coincide in the same • provide mass vaccination to non-immunocompromised cell. Otherwise, the solution is a mixed strategy, in which citizens in advance. case the antagonists pick attacks and defenses according to a probability distribution that must be calculated from the Although there are many refinements that can be considered cost matrix. There may be multiple equilibria that achieve for both the attack and the defense scenarios, these represent the same expected payoff, and for large matrices it can be the possibilities discussed in the public meetings held in May difficult to solve the game. and June 2002 (McKenna, 2002). Alternatively, one can use Bayesian decision theory to Suppose that analysts used game theory as one tool to solve the game. Here a player puts a probability distribution evaluate potential defense strategies. Then the three kinds of over the actions of the opponent, and then chooses their own attack and four kinds of defense determine a classic normal- action so as to minimize the expected cost (cf. Myerson 1991, form payoff matrix for the game [see Table 1]. Chapter 2). Essentially, one just multiplies the cost in each The Cij entries are the costs (or payoffs) associated with row by the corresponding probability, sums these by row, and each combination of attack and defense, and we have used picks the defense with the smallest sum. This formulation is abbreviated row and column labels to identify the defenses easier to solve, but it requires one to know or approximate and attacks, respectively, as described before. the opponent’s probability distribution and it does not take For each of the 12 attack-defense combinations, there is full account of the mutual strategic aspects of adversarial an associated cost. These costs may include dollars, human games (i.e., the assigned probabilities need not correspond lives, time, and other resources. For our calculation, all of to any kind of “if I do this then he’ll do that” reasoning). these costs are monetized, according to principles detailed Bayesian methods are often used in extensive-form games, in Section 3. And the monetized value of a human life is set where players make their choices over time, conditional on to $750,000, following the Department of Transportation’s the actions of their opponent. human capital model that estimates value from average lost In developing our analysis of the smallpox example we productivity (non-market approaches tend to give larger make two assumptions about time. First, we use only the values). information available by June 1, 2002; subsequent informa- Note that there is very large uncertainty in the Cij values. tion on the emerging program costs is not included. This Portions of the cost (e.g., those associated with expenses 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, TABLE 1 Attack-Defense Cost Matrix but ultimately they chose the third scenario with stockpiling, biosurveillance, and very limited vaccination of some first No Attack Single Attack Multiple Attack responders). Second, all of the estimated cost forecasts run Stockpile Vaccine C11 C12 C13 to October 1, 2007. The likelihood of changing geopolitical Biosurveillance C21 C22 C23 Key Personnel C31 C32 C33 circumstances makes it unrealistic to attempt cost estimates Everyone C41 C42 C43 beyond that fiscal year.

OCR for page 103
0 APPENDIX F 3. RISK ANALYSIS FOR SMALLPOX arise in multiple cells, introducing statistical dependency among the entries. (That is, if a given random payoff matrix Statistical risk analysis is used to estimate the probability assumes an unusually large cost for stockpiling in one cell of undesirable situations and their associated costs. In the of the random table, then the same high value should appear same way that it is used in engineering (e.g., for assessing in all other cells in which stockpiling occurs.) nuclear reactor safety; cf. Speed, 1985) or the insurance industry (e.g., for estimating the financial costs associated 3.1 Cell (1,1): Stockpile Vaccine/No Attack Scenario with earthquakes in a specific area; cf. Brillinger, 1993), this paper uses risk analysis to estimate the costs associated with Consider the problem of trying to estimate the costs different kinds of smallpox attack/defense combinations. associated with the (1,1) cell of the payoff matrix, which Risk analysis involves careful discussions with domain corresponds to no smallpox attack and the stockpiling of experts and structured elicitation of their judgments about vaccine. This estimate involves combining costs with very probabilities and costs. For smallpox planning, this requires different levels of uncertainty. input from physicians, public health experts, mathematical At the conceptual level, the cost C11 is the sum of four epidemiologists, economists, emergency response adminis- terms: trators, government accountants, and other kinds of experts. C11 = ETdry + ETAvent + ETAcamb + VIG + PHIS, We have not conducted the in-depth elicitation from multiple experts in each area that is needed for a fully rigorous risk where ETdry and ETAvent are the costs of efficacy and safety analysis; however, we have discussed the cost issues with testing for the Dryvax and Aventis vaccines, respectively; representatives from each area, and we believe that the esti- ETAcamb is the cost of new vaccine production and testing mates in this section are sufficiently reasonable to illustrate, from Acambis; VIG is the cost of producing sufficient doses qualitatively, the case for combining statistical risk analysis of vaccinia immune globulin to treat adverse reactions and with game theory for threat management in the context of possible exposures; and PHIS is the cost of establishing terrorism. the public healthcare infrastructure needed to manage this Expert opinion was typically elicited in the following stockpiling effort. way. Each expert was given a written document with back- There is no uncertainty about ETAcamb; the contract fixes ground on smallpox epidemiology and a short description of this cost at $512 million. But there is substantial uncertainty the attacks and defenses considered in this paper. The expert about ETdry and ETAvent since these entail clinical trials and often had questions; these were discussed orally with one of may require follow-on studies; based on discussions with the authors and, to the extent possible, resolved on the basis experts, we believe these costs may be realistically modeled of the best available information. Then the expert was asked as independent uniform random variables, each ranging to provide a point estimate of the relevant cost or outcome between $2 and $5 million. There is also large uncertainty and the range in which that value would be expected to fall about the cost for producing and testing sufficient doses of in 95% of similar realizations of the future. If these values VIG to be prepared for a smallpox attack; our discussions disagreed with those from other experts, then the expert was suggest this is qualitatively described by a normal random told of the discrepancy and invited to alter their opinion. variable with mean $100 million and a standard deviation Based on point estimate and the range, the authors and the of $20 million. And there is great uncertainty about PHIS expert chose a distribution function with those parameters (which includes production of bifurcated inoculation nee- which also respected real-world requirements for positiv- dles, training, storage costs, shipment readiness costs, etc.); ity, integer values, known skew, or other properties. As the based on the five-year operating budget of other government last step in the interview, the expert was given access to all offices with analogous missions, we assume this cost is the other expert opinions obtained to that point and asked normally distributed with mean $940 million and standard if there were any that seemed questionable; this led to in deviation $100 million. one case to an expert being recontacted and a subsequent revision of the elicitation. But it should be emphasized that 3.2 Cell (2,1): Biosurveillance/No Attack Scenario these interviews were intended to be short, and did not use the full range of probes, challenges, and checks that are part Biosurveillance programs are being piloted in several of serious elicitation work. major metropolitan areas. These programs track data, on The next three subsections describe the risk analysis as- a daily basis, from emergency room admission records in sumptions used to develop the random costs for the first three order to quickly discover clusters of disease symptoms that cells (C11, C21, C31) in the game theory payoff matrix. Details suggest bioterrorist attack. Our cost estimates are based for developing the costs in the other cells are available from upon discussions with the scientists working in the Boston the authors. These assumptions are intended to be representa- area (cf. Ross et al., 2002) and with the Pittsburgh team that tive, realistic, and plausible, but additional input by experts developed monitoring procedures for the Salt Lake City could surely improve upon them. Many of the same costs Olympic games.

OCR for page 103
0 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT distributed between 400,000 and 600,000 (this reflects un- The cost C21 includes the cost C11 since this defense certainty about how many personnel would be designated strategy uses both stockpiling of vaccine and increased bio- as “key”). The IM is tied to units of 25,000 people, since surveillance. Thus this is a one-time cost and represents the number of people C21 = C11 + PHIB + PHM + NFA · FA 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 where PHIB is the cost of the public health infrastructure standard deviation $10,000. needed for biosurveillance, including the data input require- The probability of an adverse event is taken from An- ments and software; PHM is the cost of a public health derson (2002), which is based upon Lane et al. (1970); the monitoring center, presumably at the Centers for Disease point estimate for all adverse events is .293, but since there Control, that reviews the biosurveillance information on is considerable variation and new vaccines are coming into a daily basis; NFA is the number of false alarms from the production, we have been conservative about our uncertainty biosurveillance system over five years of operation; and FA and assumed that the probability of an adverse event is uni- is the cost of a false alarm. formly distributed between .15 and .45. Of course, most of For this exercise, we assume that PHIB is normally dis- these events will be quite minor (such as local soreness) and tributed with mean $900 million and standard deviation $100 would not entail any real economic costs. million (for a five-year funding horizon); this is exclusive of The AEC is extremely difficult to estimate. For purposes the storage, training, and other infrastructure costs in PHIS, of calculation, we have taken the value of a human life to and it includes the cost of hospital nursing-staff time to en- be $2.86 million (the amount used by the National Highway ter daily reports on emergency room patients with a range Transportation Safety Administration in cost-benefit analy- of disease symptoms (not just those related to smallpox). ses of safety equipment). But most of the events involve no PHM is modeled as a normal random variable with mean cost, or perhaps a missed day of work that has little mea- $20 million and standard deviation $4 million (this standard surable impact on productivity. After several calculations deviation was proposed by a federal administrator, and may and consultations, this analysis assumes that AEC can be understate the real uncertainty). approximated as a gamma random variable with mean $40 False alarms are a major problem for monitoring systems; and standard deviation $100 (this distribution has a long it is difficult to distinguish natural contagious processes from right tail). 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 4. ANALYSIS mean 60. The cost for a single false alarm is modeled as a The statistical risk analysis used in Section 3, albeit normal random variable with mean $500,000 and standard crude, shows how expert judgment can generate the random deviation $100,000. payoff matrices. The values in the cells of such tables are not independent, since many of the cost components are 3.3 Cell (3,1): Key Personnel/No Attack Scenario shared between cells. In fact, it is appropriate to view the table as a matrix-valued random variable with a complex One option, among several possible policies that have joint distribution. been discussed, is for the United States to inoculate about Random tables from this joint distribution can be gener- 500,000 key personnel, most of whom would be first- ated by simulation. For each table, one can apply either the responders in major cities (i.e., emergency room staff, police, minimax criterion to determine an optimal strategy in the and public health investigators who would be used to trace sense of von Neumann and Morgenstern (1944), or a mini- people who have come in contact with carriers). If chosen, mum expected loss criterion to determine an optimal solution this number is sufficiently large that severe adverse reactions in the sense of Bayesian decision theory (cf. Myerson, 1991, become a statistical certainty. Chapter 2). By doing this repeatedly, for many different The cost of this scenario subsumes the cost C21 of the random tables, one can estimate the proportion of time that previous scenario, and thus each defense strategy is superior. C31 = C21 + (NKP × IM/25000) + (PAE × NKP × AEC) 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 where NKP is the number of key personnel; IM is the cost the costs of the game under competing defenses. For exam- of the time and resources needed to inoculate 25,000 key ple, if two defenses yield game payoffs that differ only by an personnel and monitor them for adverse events; PAE is the insignificant amount, it seems unrealistic to give no credit to probability of an adverse event; and AEC is the average cost the second-best strategy. For this reason we also use a scor- of one adverse event. We assume that NKP is uniformly

OCR for page 103
0 APPENDIX F ing algorithm in which the score a strategy receives depends formation that does the minimax criterion. The analyst needs upon how well-separated it is from the optimal strategy. to know the probabilities of a successful smallpox attack Specifically, suppose that defense strategy i has value Vi conditional on the U.S. selecting each of the four possible on a given table. Then the score Si that strategy i receives defenses. This is difficult to determine, but we illustrate how is one can do a small sensitivity analysis that explores a range of probabilities for smallpox attack. Si = 1 - Vi /{max Vj} Table 2 shows a set of probabilities that we treat as the baseline case. We believe it accords with a prudently cau- and this ensures that strategies are weighted to reflect the tious estimate of the threat of a smallpox attack. To interpret magnitude of the monetized savings that accrue from using Table 2, it says that if the United States were to only stockpile them. The final rating of the strategies is obtained by averag- vaccine, then the probability of no smallpox attack is .95, ing their scores from many random tables. 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 4.1 Minimax Criterion must sum to one. We performed the simulation experiment described above The minimum expected loss criterion multiplies the prob- 100 times and compared the four defense strategies in terms abilities in each row of Table 2 by the corresponding costs in of the minimax criterion. Although one could certainly do the same row of Table 1, and then sums across the columns. more runs, we believe that the approximations in the cost The criterion selects the defense that has the smallest sum. modeling are so uncertain that additional simulation would As with the minimax criterion, one can simulate many only generate spurious accuracy. payoff tables and then apply the minimum expected loss Among the 100 runs, we found that the Stockpile strategy criterion to each. In 100 repetitions, Stockpile won 96 times, won 9 times, the Biosurveillance strategy won 24 times, the Biosurveillance won 2 times, and Vaccinate Everyone won Key Personnel strategy won 26 times, and the Vaccinate twice. The scores showed roughly the same pattern, strongly Everyone strategy won 41 times. This lack of a clear winner favoring the Stockpile defense. may be, at some intuitive level, the cause of the widely dif- We now consider two alternative sets of probabilities, ferent views that have been expressed in the public debate shown in Table 3 and Table 4. Table 3 is more pessimis- on preparing for a smallpox attack. tic, and has larger attack probabilities. Table 4 is more If one uses scores, the results are even more ambiguous. optimistic, and has smaller attack probabilities. A serious The average score for the four defense strategies ranged sensitivity analysis would investigate many more tables, between .191 and .326, indicating that the expected perfor- but our purpose is illustration and we doubt that the quality mances were, on average, quite similar. of the assessments that underlie the cost matrix can warrant From a public policy standpoint, this may be a fortunate further detail. result. It indicates that in terms of the minimax criterion, any For Table 3, 100 simulation runs found that Stockpile decision is about equally defensible. This gives managers won 15 times, Biosurveillance won 29 times, Key Person- flexibility to incorporate their own judgment and to respond nel won 40 times, and Vaccinate Everyone won 16 times. In to extrascientific considerations. 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 4.2 Minimum Expected Loss Criterion are quite similar. In contrast, for Table 4 nearly all the weight The minimax criterion may not be realistic for the game of the score was on the Stockpile defense. These results theory situation presented by the threat of smallpox. In par- show that the optimal strategy is sensitive to the choice of ticular, the normal-form game assumes that both players are probabilities used in the analysis. Determining those prob- ignorant of the decision made by their opponent until com- mitted 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 TABLE 2 Baseline Probabilities of Attack Given States has decided to do, and presumably this will affect Different Defenses their choice of attack. Therefore the extensive-form version No Attack Single Attack Multiple Attack of game theory seems preferable. This form can be thought of as a decision tree, in which players alternate their moves. Stockpile Vaccine 0.95 0.04 0.01 Biosurveillance 0.96 0.035 0.005 At each stage, the player can use probabilistic assessments Key Personnel 0.96 0.039 0.001 about the likely future play of the opponent. Everyone 0.99 0.005 0.005 The minimum expected loss criterion requires more in-

OCR for page 103
0 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE 3 Pessimistic Probabilities of Attack Given viously, is not adequate to support public policy Different Defenses formulation. No Attack Single Attack Multiple Attack Nonetheless, despite these limitations, the methodology Stockpile Vaccine 0.70 0.20 0.10 has attractive features. First, it is easy to improve the quality Biosurveillance 0.80 0.15 0.05 of the result through better risk analysis. Second, it auto- Key Personnel 0.85 0.10 0.05 matically raises issues that have regularly emerged in policy Everyone 0.90 0.05 0.05 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 ad- TABLE 4 Optimistic Probabilities of Attack Given versarial situations and because classical game theory does Different Defenses not use random costs. No Attack Single Attack Multiple Attack Stockpile Vaccine 0.98 0.01 0.01 NOTES: BACKGROUND ON SMALLPOX Biosurveillance 0.99 0.005 0.005 Key Personnel 0.99 0.005 0.005 Although the probability that the smallpox virus (Variola Everyone 0.999 0.0005 0.0005 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 abilities requires input from the intelligence community and probably force mass vaccination programs, causing addi- the judgment of senior policy-makers. tional loss of life due to adverse reactions. Other economic consequences could easily be comparable to those of the attacks of September 11, 2001. 5. CONCLUSIONS A smallpox attack could potentially be initiated through This paper has outlined an approach combining statistical infected humans or through an aerosol (Henderson et al., risk analysis with game theory in order to evaluate defense 1999). In 12-14 days after natural exposure patients experi- strategies that have been considered for the threat of small- ence fever, malaise, body aches, and a body rash (Fenner et pox. We believe that this approach may offer a useful way al., 1988). During the symptomatic stages of the disease the of structuring generic problems in resource investment for patient can have vesicles in the mouth, throat, and nose that counterterrorism. rupture to spread the virus during a cough or sneeze. The analysis in this paper is incomplete: Person-to-person spread usually occurs through inhala- tion of virus-containing droplets or from close contact with 1. We have focused upon smallpox, because the problem an infected person. As the disease progresses the rash spreads has been framed rather narrowly and quite definitively to the head and extremities and evolves into painful, scarring by public discussion. But a proper game theory analy- vesicles and pustules. Smallpox has a mortality rate of ap- sis would not artificially restrict the options of the ter- proximately 30%, based on data from the 1960s and 1970s rorists, and should consider other attacks, such as truck (Henderson, 1999). bombs, chemical weapons, other diseases, and so forth Various mathematical models of smallpox spread ex- (which would get difficult, but there may be ways to ist and have been used to forecast the number of people approximate). It can be completely misleading to seek infected under different exposure conditions and different a local solution, as we have done. public health responses (cf. Kaplan, Craft, and Wein, 2002; 2. Similarly, we have not fully treated the options of Meltzer et al., 2001). There is considerable variation in the the defenders. For example, heavy investment in predictions from these models, partly because of differing intelligence sources is a strategy that protects against assumptions about the success of the “ring vaccination” strat- many different kinds of attacks, and might well be egy that has been planned by the Centers for Disease Control the superior solution in a less local formulation of the (2002), and this is reflected in the public debate on the value problem. of preemptive inoculation versus wait-and-see preparation. 3. We have not considered constraints on the resources However, the models are in essential agreement that a major of the terrorists. The terrorists have limited resources determinant of the size of the epidemic is the number of and can invest in a portfolio of different kinds of at- people who are exposed in the first attack or attacks. tacks. Symmetrically, the U.S. can invest in a portfolio The current vaccine consists of live vaccinia or cowpox of defenses. This aspect of the problem is not ad- virus and is effective at preventing the disease. Also, vac- dressed—we assume that both parties can fund any of cination can be performed within the first 2 to 4 days post the choices without sacrificing other goals. exposure to reduce the severity or prevent the occurrence of 4. The risk analysis presented here, as discussed pre- the disease (Henderson, 1999).

OCR for page 103
0 APPENDIX F But vaccination is not without risk; the major complica- further test the safety and efficacy of the new vaccine (cf. tions are serious infections and skin disease such as progres- Rosenthal et al., 2001). sive vaccinia, eczema vaccinatum, generalized vaccinia, and encephalitis. Approximately 12 people per million have se- REFERENCES vere adverse reactions that require extensive hospitalization, Anderson, S. (2002). “A risk-benefit assessment of smallpox and smallpox and about one-third of these die—vaccinia immune globulin vaccination,” Technical Report, Office of Biostatistics and Epidemiol- (VIG) is the recommended therapy for all of these reactions ogy, Center for Biologics Evaluation and Research, U.S. Food and Drug except encephalitis. Using data from Lane et al. (1970), we Administration, Rockville, MD, 2002. estimate that 1 in 71,429 people suffer postvaccinial enceph- Brillinger, D.R. (1993). “Earthquake risk and insurance,” EniroMetrics, alitis, 1 in 588,235 suffer progressive vaccinia, 1 in 22,727 4, 1-21. Centers for Disease Control, (2002). Smallpox response plan and guidelines suffer eczema vaccinatum, and 1 in 3,623 suffer generalized (Version .0), www.bt.cdc.gov/agent/smallpox/response-plan/index. vaccinia. Additionally, 1 in 1,656 people suffer accidental asp. infection (usually to the eye) and 1 in 3,289 suffer some other Cohen, J. (2001). “Smallpox vaccinations: How much protection remains?” kind of mild adverse event, typically requiring a person to Science, 294, 985. miss a few days of work. (Other studies give somewhat dif- Enserink, M. (2002). “New cache eases shortage worries,” Science, 296 (April 5) 25-26. ferent numbers; cf. Neff et al., 1967a, 1967b). People who Fenner, F., Henderson, D.A., Arita, I., Jezek, Z., Ladnyi, I.D. (1988). Small- have previously been successfully vaccinated for smallpox pox and Its Eradication, World Health Organization, Geneva. are less likely to have adverse reactions, and people who are Frey, S.E., Couch, R.B, Tacket, C.O., Treanor, J.J., Wolff, M., Newman, immunocompromised (e.g., transplant patients, those with F.K. Atmar, R.L., Edelman, R., Nolan, C.M., Belshe, R.B. (2002). AIDS) are at greater risk for adverse reactions (cf. Centers “Clinical responses to undiluted and diluted smallpox vaccine,” New England Journal of Medicine, 346:17, 1265-1274. for Disease Control, 2002, Guide B, parts 3, 5, and 6). Halloran, M.E., Haber, M., Longini, I.M, Jr., and Struchiner, C.J. (1991). Because the risk of smallpox waned in the 1960s, vac- “Direct and indirect effects in vaccine efficacy and effectiveness,” cination of the U.S. population was discontinued in 1972. It American Journal of Epidemiology, 133, 323-331. is believed that the effectiveness of a smallpox vaccination Henderson, D.A. (1999). “Smallpox: Clinical and epidemiological fea- diminishes after about 7 years, but residual resistance persists tures,” Emerging Infectious Diseases, 5, 537-539. Henderson, D.A., Ingelsby, T.V., Bartlett, J.G., Ascher, M.S., Eitzen, even decades later. It has been suggested that people who E., Jahrling, P.B., Hauer, J., Layton, M., McDade, J., Osterholm, were vaccinated before 1972 may be substantially protected M.T., O’Toole, T., Parker, G., Perl, T., Russel, P.K., and Tonat, K. against death, if not strongly protected against contracting (1999). “Smallpox as a biological weapon—Medical and public health the disease (cf. Cohen, 2001). management,” Journal of the American Medical Association, 281:22, The U.S. currently has about 15 million doses of the 2127-2137. Kahnemann, D., and Tversky, A. (1972). “Subjective probability: A judg- Wyeth Dryvax smallpox vaccine available. The vaccine ment of representativeness,” Cognitie Psychology, 3, 430-454. was made by scarification of calves with the New York City Kaplan, E., Craft, D.L., and Wein, W.M. (2002). “Emergency response to Board of Health strain and fluid containing the vaccinia virus a smallpox attack: The case for mass vaccination,” Proceedings of the was harvested by scraping (Rosenthal et al., 2001). Recent National Academy of Sciences, 99, 5237-5240. clinical trials on the efficacy of diluted vaccine indicate that Lane, M.J., Ruben, F.L., Neff, J.M., and Millar, J.D. (1970). “Complica- tions of smallpox vaccination, 1968: Results of ten statewide surveys,” both the five-fold and ten-fold dilutions of Dryvax achieve Journal of Infectious Diseases, 122, 303-309. a take rate (i.e., a blister forms at the inoculation site, which McKenna, M.A.J. (2002). “No mass smallpox vaccinations, panel recom- is believed to be a reliable indicator of immunization) of at mends,” Atlanta Journal-Constitution, June 21, p. 1. least 95%, so the available vaccine could be administered Meltzer, M.I., Damon, I., LeDuc, J.W., and Millar, J.D. (2001). “Modeling to as many as 150 million people should the need arise (cf. potential responses to smallpox as a bioterrorist weapon,” Emerging Infectious Diseases, 7, 201-208. Frey et al., 2002; NIAID, 2002). Myerson, R.B. (1991). Game Theory: Analysis of Conflict, Harvard Univer- The disclosure by the pharmaceutical company Aventis sity Press, Cambridge, MA, 1991. (Enserink, 2002) of the existence in storage of 80 to 90 mil- NIAID. (2002). “NIAID study results support diluting smallpox vaccine lion doses of smallpox vaccine that were produced more stockpile to stretch supply,” NIAID News, March 28. National Institute than 30 years ago has added to the current stockpile. Testing of Allergy and Infectious Diseases, www.niaid.nih.gov/newsroom/ releases/smallpox.htm. is being done on the efficacy of the Aventis vaccine stock, Neff, J.M., Lane, J., Pert, J.P., et al. (1967). “Complications of smallpox vac- including whether it, too, could be diluted if needed. cination, I: National survey in the United States, 1963,” New England Contracts to make new batches of smallpox vaccine us- Journal of Medicine, 276, 1-8. ing cell culture techniques have been awarded to Acambis. Neff, J.M., Levine, R.H., Lane, J.M., et al. (1967). “Complications of The CDC amended a previous contract with Acambis in smallpox vaccination, United States, 1963, II: Results obtained from four statewide surveys,” Pediatrics, 39, 16-923. September 2001 to ensure production of 54 million doses Rosenthal, S.R., Merchilinsky, M., Kleppinger, C., and Goldenthal, K.L. by late 2002. Another contract for the production of an ad- (2001). “Developing new smallpox vaccines,” Emerging Infectious ditional 155 million doses was awarded to Acambis in late Diseases, 7, 920-926. November 2001, and the total cost of these contracts is $512 Ross, L., Kleinman, K., Dashevsky, I., Adams, C., Kludt, P., DeMaria, A., million. After production, additional time may be needed to Jr., and Platt, R. (2002). “Use of automated ambulatory-care encounter

OCR for page 103
0 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT records for detection of acute illness clusters, including potential bioter- Treaster, J.B., (2002). “The race to predict terror’s costs,” New York Times, rorism events,” Emerging Infectious Diseases, 8, 753-760. Sept. 1, section 3, p. 1. Speed, T.P. (1985). “Probabilistic risk assessment in the nuclear industry: von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Eco- WASH-1400 and Beyond,” in Proceedings of the Berkeley Conference nomic Behaior, Princeton University Press, Princeton NJ. in Honor of Jerzy Neyman and Jack Kiefer, Vol. 2, L. LeCam and R. Olshen, eds., Wadsworth, Pacific Grove, CA, pp.~173-200.