Some Limitations of “Risk = Threat × Vulnerability × Consequence” for Risk Analysis of Terrorist Attacks

Several important risk analysis methods now used in setting priorities for protecting U.S. infrastructures against terrorist attacks are based on the formula: Risk=Threat×Vulnerability×Consequence. This article identifies potential limitations in such methods that can undermine their ability to guide resource allocations to effectively optimize risk reductions. After considering specific examples for the Risk Analysis and Management for Critical Asset Protection (RAMCAP?) framework used by the Department of Homeland Security, we address more fundamental limitations of the product formula. These include its failure to adjust for correlations among its components, nonadditivity of risks estimated using the formula, inability to use risk‐scoring results to optimally allocate defensive resources, and intrinsic subjectivity and ambiguity of Threat, Vulnerability, and Consequence numbers. Trying to directly assess probabilities for the actions of intelligent antagonists instead of modeling how they adaptively pursue their goals in light of available information and experience can produce ambiguous or mistaken risk estimates. Recent work demonstrates that two‐level (or few‐level) hierarchical optimization models can provide a useful alternative to Risk=Threat×Vulnerability×Consequence scoring rules, and also to probabilistic risk assessment (PRA) techniques that ignore rational planning and adaptation. In such two‐level optimization models, defender predicts attacker's best response to defender's own actions, and then chooses his or her own actions taking into account these best responses. Such models appear valuable as practical approaches to antiterrorism risk analysis.     

Improving Risk Management: From Lame Excuses to Principled Practice

The three classic pillars of risk analysis are risk assessment (how big is the risk and how sure can we be?), risk management (what shall we do about it?), and risk communication (what shall we say about it, to whom, when, and how?). We propose two complements as important parts of these three bases: risk attribution (who or what addressable conditions actually caused an accident or loss?) and learning from experience about risk reduction (what works, and how well?). Failures in complex systems usually evoke blame, often with insufficient attention to root causes of failure, including some aspects of the situation, design decisions, or social norms and culture. Focusing on blame, however, can inhibit effective learning, instead eliciting excuses to deflect attention and perceived culpability. Productive understanding of what went wrong, and how to do better, thus requires moving past recrimination and excuses. This article identifies common blame‐shifting “lame excuses” for poor risk management. These generally contribute little to effective improvements and may leave real risks and preventable causes unaddressed. We propose principles from risk and decision sciences and organizational design to improve results. These start with organizational leadership. More specifically, they include: deliberate testing and learning—especially from near‐misses and accident precursors; careful causal analysis of accidents; risk quantification; candid expression of uncertainties about costs and benefits of risk‐reduction options; optimization of tradeoffs between gathering additional information and immediate action; promotion of safety culture; and mindful allocation of people, responsibilities, and resources to reduce risks. We propose that these principles provide sound foundations for improving successful risk management.     

Validation of Quantitative Microbial Risk Assessment Using Epidemiological Data from Outbreaks of Waterborne Gastrointestinal Disease

The assumptions underlying quantitative microbial risk assessment (QMRA) are simple and biologically plausible, but QMRA predictions have never been validated for many pathogens. The objective of this study was to validate QMRA predictions against epidemiological measurements from outbreaks of waterborne gastrointestinal disease. I screened 2,000 papers and identified 12 outbreaks with the necessary data: disease rates measured using epidemiological methods and pathogen concentrations measured in the source water. Eight of the 12 outbreaks were caused by Cryptosporidium, three by Giardia, and one by norovirus. Disease rates varied from 5.5 × 10^?6 to 1.1 × 10^?2 cases/person‐day, and reported pathogen concentrations varied from 1.2 × 10^?4 to 8.6 × 10² per liter. I used these concentrations with single‐hit dose–response models for all three pathogens to conduct QMRA, producing both point and interval predictions of disease rates for each outbreak. Comparison of QMRA predictions to epidemiological measurements showed good agreement; interval predictions contained measured disease rates for 9 of 12 outbreaks, with point predictions off by factors of 1.0–120 (median = 4.8). Furthermore, 11 outbreaks occurred at mean doses of less than 1 pathogen per exposure. Measured disease rates for these outbreaks were clearly consistent with a single‐hit model, and not with a “two‐hit” threshold model. These results demonstrate the validity of QMRA for predicting disease rates due to Cryptosporidium and Giardia.     

Stochastic Optimization for Natural Disaster Asset Prepositioning

A key strategic issue in pre‐disaster planning for humanitarian logistics is the pre‐establishment of adequate capacity and resources that enable efficient relief operations. This paper develops a two‐stage stochastic optimization model to guide the allocation of budget to acquire and position relief assets, decisions that typically need to be made well in advance before a disaster strikes. The optimization focuses on minimizing the expected number of casualties, so our model includes first‐stage decisions to represent the expansion of resources such as warehouses, medical facilities with personnel, ramp spaces, and shelters. Second‐stage decisions concern the logistics of the problem, where allocated resources and contracted transportation assets are deployed to rescue critical population (in need of emergency evacuation), deliver required commodities to stay‐back population, and transport the transfer population displaced by the disaster. Because of the uncertainty of the event's location and severity, these and other parameters are represented as scenarios. Computational results on notional test cases provide guidance on budget allocation and prove the potential benefit of using stochastic optimization.     

Consumer Responses to Integrated Risk‐Benefit Information Associated with the Consumption of Food

The risk analysis of the health impact of foods is increasingly focused on integrated risk‐benefit assessment, which will also need to be communicated to consumers. It therefore becomes important to understand how consumers respond to integrated risk‐benefit information. Quality‐adjusted‐life‐years (QALYs) is one measure that can be used to assess the balance between risks and benefits associated with a particular food. The effectiveness of QALYs for communicating both positive and negative health effects associated with food consumption to consumers was examined, using a 3 × 2 experiment varying information about health changes in terms of QALYs associated with the consumption of fish (n = 325). The effect of this information on consumer perceptions of the usefulness of QALYs for describing health effects, on risk and benefit perceptions, attitudes, and intentions to consume fish was examined. Results demonstrated that consumers perceived QALYs as useful for communicating health effects associated with food consumption. QALYs communicated as a net effect were preferred for food products associated with negative net effects on health, while separate communication of both risks and benefits may be preferred for food products associated with positive or zero net health effects. Information about health changes in terms of QALYs facilitated informed decision making by consumers, as indicated by the impact on risk and benefit perceptions as intended by the information. The impact of this information on actual food consumption choices merits further investigation.     

An Approach for Optimal Allocation of Safety Resources: Using the Knapsack Problem to Take Aggregated Cost‐Efficient Preventive Measures

On the basis of the combination of the well‐known knapsack problem and a widely used risk management technique in organizations (that is, the risk matrix), an approach was developed to carry out a cost‐benefits analysis to efficiently take prevention investment decisions. Using the knapsack problem as a model and combining it with a well‐known technique to solve this problem, bundles of prevention measures are prioritized based on their costs and benefits within a predefined prevention budget. Those bundles showing the highest efficiencies, and within a given budget, are identified from a wide variety of possible alternatives. Hence, the approach allows for an optimal allocation of safety resources, does not require any highly specialized information, and can therefore easily be applied by any organization using the risk matrix as a risk ranking tool.     

A Case Study Evaluating the Risk of Infection from Middle Eastern Respiratory Syndrome Coronavirus (MERS‐CoV) in a Hospital Setting Through Bioaerosols

Middle Eastern respiratory syndrome, an emerging viral infection with a global case fatality rate of 35.5%, caused major outbreaks first in 2012 and 2015, though new cases are continuously reported around the world. Transmission is believed to mainly occur in healthcare settings through aerosolized particles. This study uses Quantitative Microbial Risk Assessment to develop a generalizable model that can assist with interpreting reported outbreak data or predict risk of infection with or without the recommended strategies. The exposure scenario includes a single index patient emitting virus‐containing aerosols into the air by coughing, leading to short‐ and long‐range airborne exposures for other patients in the same room, nurses, healthcare workers, and family visitors. Aerosol transport modeling was coupled with Monte Carlo simulation to evaluate the risk of MERS illness for the exposed population. Results from a typical scenario show the daily mean risk of infection to be the highest for the nurses and healthcare workers (8.49 × 10^?4 and 7.91 × 10^?4, respectively), and the lowest for family visitors and patients staying in the same room (3.12 × 10^?4 and 1.29 × 10^?4, respectively). Sensitivity analysis indicates that more than 90% of the uncertainty in the risk characterization is due to the viral concentration in saliva. Assessment of risk interventions showed that respiratory masks were found to have a greater effect in reducing the risks for all the groups evaluated (>90% risk reduction), while increasing the air exchange was effective for the other patients in the same room only (up to 58% risk reduction).     

Cost‐Benefit Analysis for Optimization of Risk Protection Under Budget Constraints

Cost‐benefit analysis (CBA) is commonly applied as a tool for deciding on risk protection. With CBA, one can identify risk mitigation strategies that lead to an optimal tradeoff between the costs of the mitigation measures and the achieved risk reduction. In practical applications of CBA, the strategies are typically evaluated through efficiency indicators such as the benefit‐cost ratio (BCR) and the marginal cost (MC) criterion. In many of these applications, the BCR is not consistently defined, which, as we demonstrate in this article, can lead to the identification of suboptimal solutions. This is of particular relevance when the overall budget for risk reduction measures is limited and an optimal allocation of resources among different subsystems is necessary. We show that this problem can be formulated as a hierarchical decision problem, where the general rules and decisions on the available budget are made at a central level (e.g., central government agency, top management), whereas the decisions on the specific measures are made at the subsystem level (e.g., local communities, company division). It is shown that the MC criterion provides optimal solutions in such hierarchical optimization. Since most practical applications only include a discrete set of possible risk protection measures, the MC criterion is extended to this situation. The findings are illustrated through a hypothetical numerical example. This study was prepared as part of our work on the optimal management of natural hazard risks, but its conclusions also apply to other fields of risk management.     

Development of Dose‐Response Models to Predict the Relationship for Human Toxoplasma gondii Infection Associated with Meat Consumption

Toxoplasma gondii is a protozoan parasite that is responsible for approximately 24% of deaths attributed to foodborne pathogens in the United States. It is thought that a substantial portion of human T. gondii infections is acquired through the consumption of meats. The dose‐response relationship for human exposures to T. gondii‐infected meat is unknown because no human data are available. The goal of this study was to develop and validate dose‐response models based on animal studies, and to compute scaling factors so that animal‐derived models can predict T. gondii infection in humans. Relevant studies in literature were collected and appropriate studies were selected based on animal species, stage, genotype of T. gondii, and route of infection. Data were pooled and fitted to four sigmoidal‐shaped mathematical models, and model parameters were estimated using maximum likelihood estimation. Data from a mouse study were selected to develop the dose‐response relationship. Exponential and beta‐Poisson models, which predicted similar responses, were selected as reasonable dose‐response models based on their simplicity, biological plausibility, and goodness fit. A confidence interval of the parameter was determined by constructing 10,000 bootstrap samples. Scaling factors were computed by matching the predicted infection cases with the epidemiological data. Mouse‐derived models were validated against data for the dose‐infection relationship in rats. A human dose‐response model was developed as P (d) = 1–exp (–0.0015 × 0.005 × d) or P (d) = 1–(1 + d × 0.003 / 582.414)^?1.479. Both models predict the human response after consuming T. gondii‐infected meats, and provide an enhanced risk characterization in a quantitative microbial risk assessment model for this pathogen.     

Game Theory and Risk Analysis

Risk analysts often analyze adversarial risks from terrorists or other intelligent attackers without mentioning game theory. Why? One reason is that many adversarial situations—those that can be represented as attacker‐defender games, in which the defender first chooses an allocation of defensive resources to protect potential targets, and the attacker, knowing what the defender has done, then decides which targets to attack—can be modeled and analyzed successfully without using most of the concepts and terminology of game theory. However, risk analysis and game theory are also deeply complementary. Game‐theoretic analyses of conflicts require modeling the probable consequences of each choice of strategies by the players and assessing the expected utilities of these probable consequences. Decision and risk analysis methods are well suited to accomplish these tasks. Conversely, game‐theoretic formulations of attack‐defense conflicts (and other adversarial risks) can greatly improve upon some current risk analyses that attempt to model attacker decisions as random variables or uncertain attributes of targets (“threats”) and that seek to elicit their values from the defender's own experts. Game theory models that clarify the nature of the interacting decisions made by attackers and defenders and that distinguish clearly between strategic choices (decision nodes in a game tree) and random variables (chance nodes, not controlled by either attacker or defender) can produce more sensible and effective risk management recommendations for allocating defensive resources than current risk scoring models. Thus, risk analysis and game theory are (or should be) mutually reinforcing.     

Secondary Risk Theory: Validation of a Novel Model of Protection Motivation

Protection motivation theory states individuals conduct threat and coping appraisals when deciding how to respond to perceived risks. However, that model does not adequately explain today's risk culture, where engaging in recommended behaviors may create a separate set of real or perceived secondary risks. We argue for and then demonstrate the need for a new model accounting for a secondary threat appraisal, which we call secondary risk theory. In an online experiment, 1,246 participants indicated their intention to take a vaccine after reading about the likelihood and severity of side effects. We manipulated likelihood and severity in a 2 × 2 between‐subjects design and examined how well secondary risk theory predicts vaccination intention compared to protection motivation theory. Protection motivation theory performed better when the likelihood and severity of side effects were both low (R² = 0.30) versus high (R² = 0.15). In contrast, secondary risk theory performed similarly when the likelihood and severity of side effects were both low (R² = 0.42) or high (R² = 0.45). But the latter figure is a large improvement over protection motivation theory, suggesting the usefulness of secondary risk theory when individuals perceive a high secondary threat.     

Potency Factors for Risk Assessment at Libby,Montana

We reanalyzed the Libby vermiculite miners’ cohort assembled by Sullivan to estimate potency factors for lung cancer, mesothelioma, nonmalignant respiratory disease (NMRD), and all‐cause mortality associated with exposure to Libby fibers. Our principal statistical tool for analyses of lung cancer, NMRD, and total mortality in the cohort was the time‐dependent proportional hazards model. For mesothelioma, we used an extension of the Peto formula. For a cumulative exposure to Libby fiber of 100 f/mL‐yr, our estimates of relative risk (RR) are as follows: lung cancer, RR = 1.12, 95% confidence interval (CI) =[1.06, 1.17]; NMRD, RR = 1.14, 95% CI =[1.09, 1.18]; total mortality, RR = 1.06, 95% CI =[1.04, 1.08]. These estimates were virtually identical when analyses were restricted to the subcohort of workers who were employed for at least one year. For mesothelioma, our estimate of potency is K_M = 0.5 × 10^?8, 95% CI =[0.3 × 10^?8, 0.8 × 10^?8]. Finally, we estimated the mortality ratios standardized against the U.S. population for lung cancer, NMRD, and total mortality and obtained estimates that were in good agreement with those reported by Sullivan. The estimated potency factors form the basis for a quantitative risk assessment at Libby.     

Communicating Earthquake Preparedness: The Influence of Induced Mood,Perceived Risk,and Gain or Loss Frames on Homeowners’ Attitudes Toward General Precautionary Measures for Earthquakes

Despite global efforts to reduce seismic risk, actual preparedness levels remain universally low. Although earthquake‐resistant building design is the most efficient way to decrease potential losses, its application is not a legal requirement across all earthquake‐prone countries and even if, often not strictly enforced. Risk communication encouraging homeowners to take precautionary measures is therefore an important means to enhance a country's earthquake resilience. Our study illustrates that specific interactions of mood, perceived risk, and frame type significantly affect homeowners’ attitudes toward general precautionary measures for earthquakes. The interdependencies of the variables mood, risk information, and frame type were tested in an experimental 2 × 2 × 2 design (N = 156). Only in combination and not on their own, these variables effectively influence attitudes toward general precautionary measures for earthquakes. The control variables gender, “trait anxiety” index, and alteration of perceived risk adjust the effect. Overall, the group with the strongest attitudes toward general precautionary actions for earthquakes are homeowners with induced negative mood who process high‐risk information and gain‐framed messages. However, the conditions comprising induced negative mood, low‐risk information and loss‐frame and induced positive mood, low‐risk information and gain‐framed messages both also significantly influence homeowners’ attitudes toward general precautionary measures for earthquakes. These results mostly confirm previous findings in the field of health communication. For practitioners, our study emphasizes that carefully compiled communication measures are a powerful means to encourage precautionary attitudes among homeowners, especially for those with an elevated perceived risk.     

A Quantitative Release Assessment for the Noncommercial Movement of Companion Animals: Risk of Rabies Reintroduction to the United Kingdom

In 2004, the European Union (EU) implemented a pet movement policy (referred to here as the EUPMP) under EU regulation 998/2003. The United Kingdom (UK) was granted a temporary derogation from the policy until December 2011 and instead has in place its own Pet Movement Policy (Pet Travel Scheme (PETS)). A quantitative risk assessment (QRA) was developed to estimate the risk of rabies introduction to the UK under both schemes to quantify any change in the risk of rabies introduction should the UK harmonize with the EU policy. Assuming 100 % compliance with the regulations, moving to the EUPMP was predicted to increase the annual risk of rabies introduction to the UK by approximately 60‐fold, from 7.79 × 10^?5 (5.90 × 10^?5, 1.06 × 10^?4) under the current scheme to 4.79 × 10^?3 (4.05 × 10^?3, 5.65 × 10^?3) under the EUPMP. This corresponds to a decrease from 13,272 (9,408, 16,940) to 211 (177, 247) years between rabies introductions. The risks associated with both the schemes were predicted to increase when less than 100 % compliance was assumed, with the current scheme of PETS and quarantine being shown to be particularly sensitive to noncompliance. The results of this risk assessment, along with other evidence, formed a scientific evidence base to inform policy decision with respect to companion animal movement.     

Quantitative Risk Estimation for a Legionella pneumophila Infection Due to Whirlpool Use

Quantitative microbiological risk assessment was used to quantify the risk associated with the exposure to Legionella pneumophila in a whirlpool. Conceptually, air bubbles ascend to the surface, intercepting Legionella from the traversed water. At the surface the bubble bursts into dominantly noninhalable jet drops and inhalable film drops. Assuming that film drops carry half of the intercepted Legionella, a total of four (95% interval: 1–9) and 4.5×10⁴ (4.4×10⁴ – 4.7×10⁴) cfu/min were estimated to be aerosolized for concentrations of 1 and 1,000 legionellas per liter, respectively. Using a dose‐response model for guinea pigs to represent humans, infection risks for active whirlpool use with 100 cfu/L water for 15 minutes were 0.29 (～0.11–0.48) for susceptible males and 0.22 (～0.06–0.42) for susceptible females. A L. pneumophila concentration of ≥1,000 cfu/L water was estimated to nearly always cause an infection (mean: 0.95; 95% interval: 0.9–～1). Estimated infection risks were time‐dependent, ranging from 0.02 (0–0.11) for 1‐minute exposures to 0.93 (0.86–0.97) for 2‐hour exposures when the L. pneumophila concentration was 100 cfu/L water. Pool water in Dutch bathing establishments should contain <100 cfu Legionella/L water. This study suggests that stricter provisions might be required to assure adequate public health protection.     

Risk Perception and Human Health Risk in Rural Communities Consuming Unregulated Well Water in Saskatchewan,Canada

Rural communities dependent on unregulated drinking water are potentially at increased health risk from exposure to contaminants. Perception of drinking water safety influences water consumption, exposure, and health risk. A community‐based participatory approach and probabilistic Bayesian methods were applied to integrate risk perception in a holistic human health risk assessment. Tap water arsenic concentrations and risk perception data were collected from two Saskatchewan communities. Drinking water health standards were exceeded in 67% (51/76) of households in Rural Municipality #184 (RM184) and 56% (25/45) in Beardy's and Okemasis First Nation (BOFN). There was no association between the presence of a health exceedance and risk perception. Households in RM184 or with an annual income >$50,000 were most likely to have in‐house water treatment. The probability of consuming tap water perceived as safe (92%) or not safe (0%) suggested that households in RM184 were unlikely to drink water perceived as not safe. The probability of drinking tap water perceived as safe (77%) or as not safe (11%) suggested households in BOFN contradicted their perception and consumed water perceived as unsafe. Integration of risk perception lowered the adult incremental lifetime cancer risk by 3% to 1.3 × 10^?5 (95% CI 8.4 × 10^?8 to 9.0 × 10^?5) for RM184 and by 8.9 × 10^?6 (95% CI 2.2 × 10^?7 to 5.9 × 10^?5) for BOFN. Probability of exposure to arsenic concentrations >1:100,000, negligible cancer risk, was 23% for RM184 and 22% for BOFN.     

The Cancer Risk Associated with Residential Exposure to Soil Containing Radioactive Coal Combustion Residuals

Coal combustion residuals (CCRs) are composed of various constituents, including radioactive materials. The objective of this study was to utilize methodology on radionuclide risk assessment from the Environmental Protection Agency (EPA) to estimate the potential cancer risks associated with residential exposure to CCR‐containing soil. We evaluated potential radionuclide exposure via soil ingestion, inhalation of soil particulates, and external exposure to ionizing radiation using published CCR radioactivity values for ²³²Th, ²²⁸Ra, ²³⁸U, and ²²⁶Ra from the Appalachia, Illinois, and Powder River coal basins. Mean and upper‐bound cancer risks were estimated individually for each radionuclide, exposure pathway, and coal basin. For each radionuclide at each coal basin, external exposure to ionizing radiation contributed the greatest to the overall risk estimate, followed by incidental ingestion of soil and inhalation of soil particulates. The mean cancer risks by route of exposure were 2.01 × 10⁻⁶ (ingestion), 6.80 × 10⁻⁹ (inhalation), and 3.66 × 10⁻⁵ (external), while the upper bound cancer risks were 3.70 × 10⁻⁶ (ingestion), 1.18 × 10⁻⁸ (inhalation), and 6.15 × 10⁻⁵ (external), using summed radionuclide‐specific data from all locations. The upper bound cancer risk from all routes of exposure was 6.52 × 10⁻⁵. These estimated cancer risks were within the EPA's acceptable cancer risk range of 1 × 10⁻⁶ to 1 × 10⁻⁴. If the CCR radioactivity values used in this analysis are generally representative of CCR waste streams, then our findings suggest that CCRs would not be expected to pose a significant radiological risk to residents living in areas where contact with CCR‐containing soils might occur.     

Outbreak-Based Giardia Dose–Response Model Using Bayesian Hierarchical Markov Chain Monte Carlo Analysis

Giardia is a zoonotic gastrointestinal parasite responsible for a substantial global public health burden, and quantitative microbial risk assessment (QMRA) is often used to forecast and manage this burden. QMRA requires dose–response models to extrapolate available dose–response data, but the existing model for Giardia ignores valuable dose–response information, particularly data from several well-documented waterborne outbreaks of giardiasis. The current study updates Giardia dose–response modeling by synthesizing all available data from outbreaks and experimental studies using a Bayesian random effects dose–response model. For outbreaks, mean doses (D) and the degree of spatial and temporal aggregation among cysts were estimated using exposure assessment implemented via two-dimensional Monte Carlo simulation, while potential overreporting of outbreak cases was handled using published overreporting factors and censored binomial regression. Parameter estimation was by Markov chain Monte Carlo simulation and indicated that a typical exponential dose–response parameter for Giardia is r = 1.6 × 10⁻² [3.7 × 10⁻³, 6.2 × 10⁻²] (posterior median [95% credible interval]), while a typical morbidity ratio is m = 3.8 × 10⁻¹ [2.3 × 10⁻¹, 5.5 × 10⁻¹]. Corresponding (logistic-scale) variance components were σ_r = 5.2 × 10⁻¹ [1.1 × 10⁻¹, 9.6 × 10⁻¹] and σ_m = 9.3 × 10⁻¹ [7.0 × 10⁻², 2.8 × 10⁰], indicating substantial variation in the Giardia dose–response relationship. Compared to the existing Giardia dose–response model, the current study provides more representative estimation of uncertainty in r and novel quantification of its natural variability. Several options for incorporating variability in r (and m) into QMRA predictions are discussed, including incorporation via Monte Carlo simulation as well as evaluation of the current study's model using the approximate beta-Poisson.     

Implementation of Equity in Resource Allocation for Regional Earthquake Risk Mitigation Using Two‐Stage Stochastic Programming

This article presents a new methodology to implement the concept of equity in regional earthquake risk mitigation programs using an optimization framework. It presents a framework that could be used by decisionmakers (government and authorities) to structure budget allocation strategy toward different seismic risk mitigation measures, i.e., structural retrofitting for different building structural types in different locations and planning horizons. A two‐stage stochastic model is developed here to seek optimal mitigation measures based on minimizing mitigation expenditures, reconstruction expenditures, and especially large losses in highly seismically active countries. To consider fairness in the distribution of financial resources among different groups of people, the equity concept is incorporated using constraints in model formulation. These constraints limit inequity to the user‐defined level to achieve the equity‐efficiency tradeoff in the decision‐making process. To present practical application of the proposed model, it is applied to a pilot area in Tehran, the capital city of Iran. Building stocks, structural vulnerability functions, and regional seismic hazard characteristics are incorporated to compile a probabilistic seismic risk model for the pilot area. Results illustrate the variation of mitigation expenditures by location and structural type for buildings. These expenditures are sensitive to the amount of available budget and equity consideration for the constant risk aversion. Most significantly, equity is more easily achieved if the budget is unlimited. Conversely, increasing equity where the budget is limited decreases the efficiency. The risk‐return tradeoff, equity‐reconstruction expenditures tradeoff, and variation of per‐capita expected earthquake loss in different income classes are also presented.     

Modeling the Cost Effectiveness of Fire Protection Resource Allocation in the United States: Models and a 1980–2014 Case Study

The estimated cost of fire in the United States is about $329 billion a year, yet there are gaps in the literature to measure the effectiveness of investment and to allocate resources optimally in fire protection. This article fills these gaps by creating data‐driven empirical and theoretical models to study the effectiveness of nationwide fire protection investment in reducing economic and human losses. The regression between investment and loss vulnerability shows high R² values (≈0.93). This article also contributes to the literature by modeling strategic (national‐level or state‐level) resource allocation (RA) for fire protection with equity‐efficiency trade‐off considerations, while existing literature focuses on operational‐level RA. This model and its numerical analyses provide techniques and insights to aid the strategic decision‐making process. The results from this model are used to calculate fire risk scores for various geographic regions, which can be used as an indicator of fire risk. A case study of federal fire grant allocation is used to validate and show the utility of the optimal RA model. The results also identify potential underinvestment and overinvestment in fire protection in certain regions. This article presents scenarios in which the model presented outperforms the existing RA scheme, when compared in terms of the correlation of resources allocated with actual number of fire incidents. This article provides some novel insights to policymakers and analysts in fire protection and safety that would help in mitigating economic costs and saving lives.