A mathematical model suggests that a HEPA/vaccine approach is viable for most buildings after a large-scale anthrax attack.

We developed a mathematical model to compare 2 indoor remediation strategies in the aftermath of an outdoor release of 1.5 kg of anthrax spores in lower Manhattan. The 2 strategies are the fumigation approach used after the 2001 postal anthrax attack and a HEPA/vaccine plan, which relies on HEPA vacuuming, HEPA air cleaners, and vaccination of reoccupants. The HEPA/vaccine approach leads to few anthrax cases among reoccupants if applied to all but the most heavily contaminated buildings, and recovery is much faster than under the decades-long fumigation plan. Only modest environmental sampling is needed. A surge capacity of 10,000 to 20,000 Hazmat workers is required to perform remediation within 6 to 12 months and to avoid permanent mass relocation. Because of the possibility of a campaign of terrorist attacks, serious consideration should be given to allowing or encouraging voluntary self-service cleaning of lightly contaminated rooms by age-appropriate, vaccinated, partially protected (through masks or hoods) reoccupants or owners.

In addition to killing 5 of its 11 victims, the 2001 anthrax attack on the U.S. Postal Service and federal facilities also contaminated a number of buildings. The U.S. government spent several hundred million dollars recovering buildings with large-area contamination by using chlorine dioxide fumigation. The last of these federal facilities, the Hamilton New Jersey Mail Sorting Facility, is not expected to reopen until early 2005, >3 years after the attack (

A mathematical model (

Model. This document describes the mathematical model that generated the reported results. The calculation of the postattack indoor contamination levels is described in Section 1, the chlorine dioxide parameters are given in Section 2, and the various aspects of the HEPA/vaccine proposal are formulated in Section 3. Values of the model parameters are given in Tables 1 and 2.

Graphic overview of the mathematical model. Mathematical submodels are in boxes. NYC, New York City.

Since chlorine dioxide fumigation eliminated all detectable spores from the Hart Senate Office Building and several mail-sorting facilities, we assumed that it successfully eliminates all spores in the buildings of our model. In the 2001 attack, chlorine dioxide was used to decontaminate the 700-km^{2} Brentwood postal facility, which took 1 year at a cost of $130 million (

^{n}

_{5 }postcleaning samples suggest that the floor spore concentration in the room is below the target level

^{c̅}

_{ƒ}; this approach is reminiscent of that taken during the asbestos remediation after the World Trade Center collapse (

^{n}

_{5}samples, we estimate the number of 2-h vacuumings of the room's surfaces and contents that are required to achieve the target concentration

^{c̅}

_{ƒ}. After these vacuumings, a new set of

^{n}

_{5}samples are taken. If the estimated concentration from these new samples is below

^{c̅}

_{ƒ}then remediation ceases; otherwise, another round of vacuuming and testing is performed. Consecutive vacuumings are 48 h apart, and testing (if needed) occurs midway between these 2 vacuumings, both to allow reaerosolized spores to resettle before testing and to permit the testing results to be received before the next scheduled vacuuming. We varied the 2 decision variables

^{n}

_{5}and

^{c̅}

_{ƒ }to explore the tradeoffs among our performance measures.

Graphic depiction of the compartments in the differential equation model and the spore movement among compartments.

^{c̅}

_{ƒ }each generic room is reoccupied by 1 person for 12 h per day. After reoccupation, a portable HEPA air cleaner (at 3 air exchanges per h [

_{50}) of 8,000 spores (

_{15}from the probit model in (

_{50}denotes the dose that infects half of the population; because inhalational anthrax is nearly always lethal (in the absence of treatment), the ID

_{50}coincides with the 50% lethal dose (LD

_{50}). The differential equation model is used to measure the cumulative number of spores inhaled by each reoccupant in a 10-year period. Combining these cumulative doses, the dose-response model, the atmospheric dispersion model, and the population density of reoccupants allows us to compute the total number of inhalation anthrax cases.

^{c̅}

_{ƒ}) and the number of samples per round (

^{n}

_{5}).

^{7}m

^{2}, which is >4 million 12x12x8-ft rooms. For this base-case scenario, the fumigation plan costs $2.7 billion and takes 42 years.

^{c̅}

_{ƒ}) and the number of floor samples per round (

^{n}

_{5}). Because of the random sample measurements, 50 simulations were performed to estimate each of the points in

^{2}to 28 cases when the floor concentration threshold is 0.1 spores/m

^{2}. To put these numbers in perspective, we also found that 15,760 cases would occur if no cleaning was performed (i.e.,

^{c̅}

_{ƒ }

^{=∞}). The total cost in

^{n}

_{5}= 1 and

^{c̅}

_{ƒ}= 0.1 or 1, and sampling is the bottleneck for the other values of

^{n}

_{5}tested. Because using

^{n}

_{5}> 1 increases the cost and time without decreasing anthrax cases, we focus in

^{n}

_{5}=1.

^{2}) and uses the HEPA/vaccine approach for lightly contaminated rooms (<100 spores/m^{2}). This hybrid approach results (on average) in only 2 anthrax cases, and the mean remediation time for the lightly contaminated rooms is 5.9 years. It takes 8.4 years to fumigate the highly contaminated rooms. Hence, the total remediation time ranges from 8.4 to 14.3 years, depending upon whether different workers are involved in the 2 decontamination modalities. For the 3 other threshold levels pictured in ^{2} to decide between fumigation and vacuuming in

The horizontal axes in these 4 plots give the original room deposition level before remediation begins. These plots show how the total number of anthrax cases (the stars and the left vertical axes) are distributed across room deposition levels, e.g., in plot A, most of the anthrax cases occur in rooms with original deposition levels >100 spores/m^{2}. Similarly, the 2 curves and the right vertical axis of each plot show how much time is spent cleaning and sampling in rooms of various deposition levels. These 4 plots are identical except that the spore concentration threshold in spores/m^{2} (^{c̅}
_{ƒ}), which dictates when remediation is stopped, is A) ^{c̅}
_{ƒ }= 0.1; B) ^{c̅}
_{ƒ } = 1; C) ^{c̅}
_{ƒ } = 10; D) ^{c̅}
_{ƒ } = 100. These plots motivate the hybrid policy, which fumigates heavily contaminated rooms and uses the HEPA/vaccine approach in lightly contaminated rooms.

The amount of indoor floor area in lower Manhattan (vertical axes) that is contaminated at various anthrax concentration levels (horizontal axes) as a result of an outdoor release of 1.5 kg of anthrax spores. Plot A, an average of 92 scenarios (9 release locations in Manhattan times 8 wind directions, plus 20 release locations on the outskirts of Manhattan). Plot B, provides similar information for the scenario that generated the largest total area of contamination.

A number of aspects of the model contain considerable uncertainty: the cost and time of the fumigation plan, the indoor spatial deposition after an attack, the reaerosolization and deposition rates inside a room, spore dynamics in a duct, air-cleaning efficacy, vacuum efficacy, Hazmat logistics, the spatial heterogeneity in sampling, vaccine coverage, and the low end of the dose-response curve. Before discussing each of these 10 variables in turn, we note that our general approach to these uncertainties is to be conservative with respect to assessing the HEPA/vaccine option; i.e., we err on the side of overstating the mean number of anthrax cases that would result under this approach or understating the cost and time of the fumigation plan.

Although fumigation was successful during the cleanup after the 2001 postal attack, the fumigation of a skyscraper is a challenge that has yet to be tackled. Given the 42 years it would take to fumigate the exposed area, an alternative technology could be developed.

The estimated indoor spatial deposition contains orders-of-magnitude of uncertainty, depending upon the size of the release, the spore characteristics (e.g., dry versus wet, size, purity, viability, surface electrostatic properties), the weather conditions, building and canopy terrain in lower Manhattan, building HVAC infrastructure, and whether or not windows and vents were open. The goal of the atmospheric modeling is neither to accurately predict the probability distribution of indoor spatial concentrations for a possible future attack (such an attempt would be greatly limited by the irreducible uncertainty in the release size) nor to provide postattack situational awareness (which would require a much more detailed spatial model), but rather to generate a comparative set of plausible scenarios to evaluate remediation strategies before an attack. Hence, we focused on the average of 92 plausible scenarios. To give some sense of the upper range, we present in

Performance of the HEPA/vaccine plan under the base-case scenario. The horizontal axes in plots A-C are the number of floor samples per round (). Each of plots A–C have 4 curves, 1 for each value of the floor concentration threshold (). Cleaning stops after the estimated floor concentration from samples per room is below the threshold . The vertical axes in plots A-C are A) the mean number of inhalation anthrax cases, B) the mean cost, and C) the mean recovery time. In plots A-C, the concentration threshold () has a much bigger impact than the number of samples per room () on these 3 performance measures. Plot D shows the tradeoff of anthrax cases versus recovery time in the base case. The number of samples per room is assumed to be = 1 in this plot, which is derived from plots A–C. Plot D also contains tradeoff curves for 3 sensitivity analyses: a lower air-cleaning rate, increased sampling variability of spore concentration, and the most severe of the 92 cases depicted in

Because air and surfaces are concomitantly remediated, the number of anthrax cases is rather insensitive to the reaerolization and deposition rates in the room.

The large uncertainty with respect to duct modeling led us to adopt a worst-case approach and use the spore disengagement rate that maximizes the number of anthrax cases. Many new buildings and some retrofitted older buildings have HEPA filters built into the HVAC system (

We have focused on portable air cleaners, whereas dilution ventilation, in which 15%–25% of the total airflow rate consists of outside airflow (

To the extent that reaerosolized spores resettled before or during postvacuum testing in the referenced study (

Our assumption that each Hazmat worker has 4 productive hours of work per day underestimates the rate that could be achieved over a several-week time frame but is prudent over a longer period of time and would help avoid worker fatigue and burnout.

Because the amount of spatial heterogeneity of spores in a room is difficult to assess, we considered the case where 95% of samples within a room fall within 2 orders of magnitude rather than 1.

As noted in section 3.8 of the mathematical model, our 85% vaccine coverage of reoccupants may be a considerable underestimate. No age groups are being left behind in the plans for the next-generation anthrax vaccine, and persons with weak immune systems may achieve partial protection.

^{c̅}

_{ƒ }= 10 spores/m

^{2}and

^{n}

_{5}= 1 sample per round would be reduced from 341 to 72. Even within the class of probit models, others have used a probit slope twice as steep, which results in many fewer cases (

^{c̅}

_{ƒ }= 10 spores/m

^{2}and

^{n}

_{5}=1 sample per round decreases from 341 to 3 x 10

^{-5}, which highlights the value of further research into the low end of the dose-response relationship. However, in the mathematical model we note that the slope of 0.7 is more consistent with data from the 2001 anthrax attack. Dahlgren et al. (

The base-case release, which is an average of 92 different scenarios under various weather conditions and locations in lower Manhattan, contaminates the equivalent of 4 million 12x12x8-ft rooms. Our analysis suggests that an outdoor release would generate a more diffuse depositional distribution of spores than an indoor attack: we estimate that ≈10,000 spores/m^{2} were deposited in parts of the Hart Senate Office Building (section 3.2 of the mathematical model), which is considerably higher than the concentrations in

A key finding of our study is that only a moderate amount of sampling appears to be required. In theory, additional sampling reduces type I and type II errors, thereby avoiding anthrax cases in rooms that were inadvertently thought to be sufficiently safe, and reducing unnecessary remediation of rooms that were mistakenly perceived as overly contaminated. However, the number of anthrax cases was essentially independent of the number of room samples per round, as long as at least 1 sample was taken. Indeed, with current vacuuming and sampling capacity, the only impact from taking >1 sample per 12x12x8-ft room is prolonged remediation and increased cost. However, in the absence of exhaustive environmental testing, on-site coordinators need to validate that work is performed according to the required standards (i.e., vacuuming is actually being done for the specified number of minutes/m^{2}).

^{c̅}

_{ƒ}, the number of samples per round

^{n}

_{5}, and the level of concentration that requires fumigation versus vacuuming should be determined with greater precision. These threshold values should be chosen so that the reoccupant risk level (in terms of quality-adjusted life years) is consistent with those for other hazards (e.g., asbestos, radiation).

Large-area urban remediation strategies must confront a number of difficult issues, the most important of which is surge Hazmat capacity. We have assumed that remediation and vaccination are initiated simultaneously 1 week after the attack. The initial vaccination of reoccupants would require ≈1 week; protective immunity is believed to develop at 35 days after initial vaccination (

There are other aspects to optimizing surge remediation and recovery capacity. Just as the worried well caused a surge in ciproflaxin sales in 2001, many people outside of the exposed region will attempt to buy HEPA air cleaners and vacuums. Hence, demand will come not only from the exposed area but also from surrounding regions. In the same way that the U.S. government is working with pharmaceutical companies to provide surge capacity of medical countermeasures (including anthrax vaccine) in the event of a biologic attack, it needs to develop cooperative agreements with building protection service companies so that equipment shortages do not block the critical path to recovering the exposed area.

Another key aspect of a detailed plan is exception management: the HEPA/vaccine plan will not work for 100% of the buildings in the exposed area. More aggressive remediation of critical assets (hospitals; nursing homes; daycare centers; emergency response facilities; electrical, water and sanitation facilities; transportation facilities) will be desirable. Some nonresidential buildings (such as the buildings contaminated in the 2001 attack) have extremely high ceilings, and achieving a high air-exchange rate in these spaces may be not be feasible with portable air cleaners. Another confounding issue is visitors to the impacted region. In the aftermath of a catastrophic anthrax attack, the public would expect nationwide voluntary mass vaccination. Visitors to the exposed areas should be offered an anthrax vaccine, and guidelines for unvaccinated visitors should be developed. Also, because the spore concentration continues to decrease exponentially during reoccupation (but not during semiquiescent periods), more vulnerable residents might delay their reoccupation until several months after the other residents. A significant logistical issue is the disposal of contaminated carpets, furniture, and other household goods. Some reoccupants will insist on discarding these items, even after they have been heavily cleaned. Reoccupant education and outreach measures, including perhaps temporal or financial disincentives for disposal, need to be taken to avoid overwhelming solid waste disposal capacity. Emergency plans (e.g., medical incinerator capacity) should be developed for the HEPA vacuum bags and other items that need to be discarded during remediation. Another difficult issue is postevent building maintenance, particularly of HVAC systems, which must minimize spore reaerosolization during maintenance and disposal of old ducts. Safe procedures to rid ducts of asbestos (asbestos fibers are roughly the same size as anthrax spores, but the U.S. Environmental Protection Agency limit for asbestos is 900 fibers/m^{3} [

In summary, this study suggests that a HEPA/vaccine approach is viable for most buildings after a large-scale anthrax attack. This outcome is dependent on a qualitative increase in surge Hazmat remediation capacity to reduce the recovery delay to a level that would not invite permanent mass relocation. Detailed mass remediation plans need to be developed now; as noted by Danzig (

Figure 6 in PDF format

L.M.W. thanks Richard Danzig for suggesting the anthrax remediation problem, David Miller for performing the calculations in section 1 of the mathematical model, and Scott Fredericks, John Mason, Chris Weis, and Burt Fried for helpful discussions.

This research was partially supported by the Center for Social Innovation, Graduate School of Business, Stanford University.

Dr. Wein is the Paul E. Holden Professor of Management Science at the Graduate School of Business, Stanford University. His primary research interests are mathematical modeling and analysis in manufacturing, medicine, and biology. In the last several years, he has worked on problems in homeland security, including bioterrorism and border issues.