Resource allocation for demand surge mitigation during disaster response

Decision Support Systems 50 (2010) 304–315

Contents lists available at ScienceDirect

Decision Support Systems j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / d s s

Resource allocation for demand surge mitigation during disaster response Hina Arora a, T.S. Raghu b,⁎, Ajay Vinze b a b

Microsoft Corporation, USA W. P. Carey School of Business, Arizona State University, Temp, AZ 85287, USA

a r t i c l e

i n f o

Article history: Received 21 December 2008 Received in revised form 17 August 2010 Accepted 24 August 2010 Available online 7 September 2010 Keywords: Pandemic flu Decision support Resource allocation Optimization

a b s t r a c t Large-scale public health emergencies can result in an overwhelming demand for healthcare resources. Regional aid in the form of central stockpiles and resource redistribution can help mitigate the resulting demand surge. This paper discusses a resource allocation approach for optimizing regional aid during public health emergencies. We find that, optimal response involves delaying the distribution of resources from the central stockpile as much as possible. Also, smaller counties stand to benefit the most from mutual aid. And finally, policy level decisions that alter the objectives of pandemic relief efforts can significantly impact the allocations to affected regions. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The 20th century has witnessed 4 influenza pandemics. The Spanish flu pandemic in 1918 was the most severe, resulting in about 500,000 deaths in the United States. This was followed by less severe pandemics in 1957 and 1968 that resulted in about 70,000 and 34,000 deaths respectively. Based on this historical data and estimated disease transmission rates, it is predicted that a modern day influenza pandemic in the United States could infect between 75 and 90 million people, and cause between 100,000 and 2 million deaths depending on the severity of the epidemic. The economic effects could be as great as the average postwar recession [19,20]. The swine flu pandemic, unfolding currently, has so far led to over 43,000 infected people and over 300 deaths in the United States. The pandemic is expected to get worse over the winter, where, in conjunction with regular seasonal influenza viruses, it is expected to cause significant illness with associated hospitalizations and deaths. Once the pandemic sets in it is expected to spread rapidly (Table 1). Given the high infection rate of influenza, this can result in a demand surge for scarce healthcare resources such as beds, vaccines, medicines, nursing staff, and doctors to treat those infected and check the infection among the healthy population. In order to combat the epidemic, it becomes imperative to manage these scarce resources and ensure their availability to those that need it the most in a timely manner. Four different strategies have been suggested in the literature to deal with demand fluctuations [16,31]. The first strategy is one of ⁎ Corresponding author. E-mail addresses: [email protected] (H. Arora), [email protected] (T.S. Raghu), [email protected] (A. Vinze). 0167-9236/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2010.08.032

matching capacity to varying demand. This capacity management strategy is a widely used strategy and can be implemented via shared capacity (mutual aid) and/or building excess (redundant) capacity [22]. The second strategy is one of providing constant capacity to meet peak demand. This capacity management strategy results in excess capacity at non-peak times, thereby escalating inventory holding costs. The third strategy is one of controlling demand to be at a constant (average) level, for instance through scheduling. This demand management strategy results in poor service at peak times, thereby making it undesirable from the customer's point of view. The fourth strategy is one of influencing peak demand by shifting it to low demand periods. This demand management strategy reduces demand fluctuations, but may not be feasible for emergency services. The second, third and fourth strategies would be infeasible in a pandemic context due to the unpredictable and real-time nature of such events. While the first strategy does lend itself to the pandemic context, the choice between use of redundant capacity and mutual aid is not an obvious one, and is deemed as one of the biggest challenges in emergency preparedness [7,21]. This research therefore addresses the problem of mutual aid through a cost–benefit-based optimization model. The model explicitly considers resource requirements, and the source of the resources. It captures direct delivery from a central stockpile and lateral transshipment of antivirals for treatment and prophylaxis between regions. The stock of antivirals is considered to be limited, and a supplier capable of manufacturing more antivirals is not considered due to the recommendation by the Centers for Disease Control (CDC) that planners not depend on the assumption that “ongoing production will fill supply gaps” [18]. Three key findings emerge from our modeling approach. First, postponement, or delaying the decision of how much to pre-allocate leads to greater flexibility and more

H. Arora et al. / Decision Support Systems 50 (2010) 304–315 Table 1 Impact of pandemic influenza (reproduced from [16]). Characteristic

Moderate (1958/68-like)

Severe (1918-like)

Illness Outpatient medical care Hospitalization ICU care Mechanical ventilation Deaths

90 million (30%) 45 million (50%) 865,000 128,750 64,875 209,000

90 million (30%) 45 million (50%) 9,900,000 1,485,000 742,500 1,903,000

*Estimates based on extrapolation from past pandemics in the United States. Note that these estimates do not include the potential impact of interventions not available during the 20th century pandemics.

effective response. Second, while mutual aid is most beneficial to the smaller counties, especially when resources are being pre-allocated by population proportion. Third, sub-group allocations based on factors such as age can further optimize resource deployment leading to greater savings. The rest of the paper is organized as follows. In Section 2, we review the current literature on pandemic flu response. In Section 3, we present the model for transshipment for demand surge mitigation. The data and details related to model simulation are discussed in Section 4. In Section 5, we present and discuss the results and in Section 6, we present our concluding remarks and future research directions. 2. Modeling pandemic response Past research has looked at different aspects of pandemic diffusion and containment through mathematical models and simulation studies. However, the idea of mutual aid in the context of pandemic response has been raised only recently. Prominent studies that have addressed epidemic diffusion through populations are discussed briefly below. Brandeau et al. [6] use a quality-adjusted-life-years based cost-effectiveness measure to capture the benefit of health interventions. Additionally, a binding budget constraint is used in the optimization model. Since the federal government will likely spend as much money as might be required in containing the spread of the pandemic, the budget constraint is eliminated in the model developed in this paper. Also, since cost-effectiveness measures are known to be subjective and hard to quantify, this research uses a cost–benefit-

305

based resource allocation approach instead. Craft et al. [15] use a mathematical disease based queuing model to arrive at an optimal distribution mechanism of antibiotics in a hospital setting. Each service zone, or hospital, is considered in isolation. Since the focus of this research is mutual aid between interacting hospitals, distribution mechanisms within hospitals are not explored in this research. The Operations Research literature has dealt with problems similar to mutual aid and redundant capacity in the warehouse–retailer context. As mentioned earlier, a widely used strategy to mitigate demand fluctuations is one of matching capacity to varying demand via shared capacity (mutual aid) and/or building excess (redundant) capacity [22]. This is the idea behind lateral transshipments or inventory pooling (see for instance, [33,35,36]). These systems are usually modeled as multi-echelon systems containing multiple retailers with limited stockpiles, a single warehouse with limited stockpile, and a single supplier with infinite supply. If stock is available at a retailer, the customer is serviced. If stock is unavailable, one of the following restocking methods is used. In direct delivery, the retailers restock by getting items from the central warehouse. This is usually a slower and costlier option. In lateral transshipments (or inventory pooling), retailers restock by getting items from other retailers. Pooling can be complete, where retailers share whatever they have, or partial, where retailers hold back some stock. In general, complete pooling is used, and may even be optimal [33]. Lateral transshipments are usually a faster and cheaper option. While the OR transshipment model takes resource requirements and mutual aid into consideration, the benefits that are taken into account in the OR context are different from the benefits that must be taken into account in the pandemic context. The benefit in the OR context captures the fulfillment of a service agreement with the customer. On the other hand, the benefit in the pandemic context must not only capture the benefit derived from saving an infected individual, but also the future savings derived from reducing the total number of infections. This is often done in healthcare research through models that capture the value of life. Two approaches to valuing life are often taken in the healthcare context: cost-effectiveness analysis (CEA), and cost–benefit analysis (CBA). CEA (see for instance, [30]) uses intermediate outcome indicators (such as number of people fully immunized), final outcome indicators (such as number of years of lives saved), or the years of healthy years lost (in terms of quality-adjusted-life-years). This last method is often

Fig. 1. Pandemic timeline — the six phases of pandemic influenza.

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Table 2 List of variable definitions for resource allocation model. Variable T P k Pk mk GARk nk nd, k w

Definition

Total number of nodes Total population Any node (SNS: k = 1, others: k ≥ 2) T Population of node k. Therefore P = ∑ Pk . k=2 Susceptible population in node k Gross attack rate in node k Infected population in node k = GARk mk Infected population in node k that will die = 0.00202*nk Health outcomes: dead (d), hospitalized (h), outpatient (o), ill but does not seek medical care (n) nh, k Infected population in node k that will be hospitalized = 0.00791*nk no, k Infected population in node k that will receive outpatient care = 0.46276*nk nn, k Infected population in node k that will seek no medical care = 0.52731*nk
 T S Total available stock in SNS Pk Sk . Sk Maximum SNS allocation for node k = P *S. Therefore S = k∑ =2 Proportion of susceptible population in node k that will be provided pPxsk with prophylaxis. p Proportion of Sk that is pre-allocated to node k i Node with an excess of antivirals (stockpile is node 1) Number of antivirals available in node i Ri j Node with a shortage of antivirals rj Number of antivirals required by node j λij Proportion of Ri contributed for treatment to node j cij Cost incurred in transferring a single course of antiviral from node i to node j ep (or et) Overall prophylaxis (or treatment) effectiveness ep, w Effectiveness of prophylaxis (or treatment) in reducing a particular (or et, w) health outcome vw Dollar value saved per person by averting a particular health outcome

referred to as cost utility analysis (CUA). CBA (see for instance, [23]) on the other hand either uses the cost-of-illness (COI) method, or the willingness-to-pay (WTP) method. In COI, benefits are calculated in terms of the production potential of humans, that is, the difference between the evaluated costs of illness with the intervention and the costs of illness without the intervention. In WTP, benefits are calculated in terms of the perceived usefulness of the intervention, that is, the maximum amount of money individuals are willing to pay in order to have access to the intervention. One of the key advantages of CBA over CEA is that both the costs and benefits are measured in monetary terms, making the resulting decisions more explicit and transparent [25]. 2.1. Pandemic characterization As shown in Fig. 1, the World Health Organization defines six phases of pandemic influenza [10]. The first phase is characterized by isolated occurrences of an influenza virus subtype among animals. The risk to human transmission is still considered to be low. WHO recommends that influenza pandemic preparedness be checked and strengthened in this phase. The second phase is characterized by increasing amounts of transmission among animals, and a substantial threat for human transmission. Measures should be taken to minimize the risk of human transmission. Phase three is characterized by isolated occurrences of the virus among humans. Rapid characterization of the new virus should be ensured, and early detection, notification and response to additional cases should be performed. Stage four is characterized by small clusters of human-to-human transmission, and the virus is not yet well adapted to humans. The new virus should be contained as much as possible, and the spread delayed in order to implement preparedness measures. Phase five is characterized by larger clusters of human-to-human transmission, suggesting that the virus is increasingly adapting to humans. Measures to contain and avert a pandemic should be maximized, and pandemic

Fig. 2. Three different mutual aid regions considered in this research. California Region VI, Arizona Regions I and V, with a total of 19 counties.

H. Arora et al. / Decision Support Systems 50 (2010) 304–315

response measures should be implemented. Finally, phase six is the pandemic phase, where there is increased and sustained human-tohuman transmission, and the goal should be to minimize the impact of the pandemic. The avian flu is currently in stage three. Influenza pandemics are characterized by four discrete disease stages [3]: Susceptible, Exposed, Infectious and Recovered. Infectious individuals spread the disease to the (non-immune) susceptible population. Those in the susceptible population to which the disease is transmitted become exposed and after a period of time, the incubation (or latent) period, they become infectious. Individuals remain infectious for a period of time, the infectious period, and then recover (with immunity). Epidemics of contagious diseases such as influenza progress exponentially. As a result, once the pandemic sets in, it is expected to spread rapidly (Table 1). Timely intervention in the form of antiviralbased prophylaxis or vaccine-based immunization (to reduce the number of susceptibles), quarantine (to reduce the number of contacts with the infected) and antiviral-based treatment (to help the infected recover) can help contain the epidemic. Vaccines and antivirals can significantly reduce the incidence of a pandemic. While vaccines provide the best line of defense, they can only be made once the pandemic virus has been identified, and could take 4–6 months to become available for general use. One would therefore have to resort to antivirals, both to provide immunity (prophylaxis), and, for treatment purposes. While the CDC has stockpiled antivirals, there is enough to cover only less than 10% of the US population [17]. In the interim, therefore, it becomes imperative to optimally allocate the limited stock of antivirals to combat the epidemic. Any antiviral allocation policy will therefore need to address the following decision-making stages: antiviral distribution, prophylaxis allocation and treatment allocation. Surge capacity is a health care system's ability to expand quickly beyond normal services to meet an increased demand [1]. The healthcare system can improve surge capacity by maintaining redundant capacity in centrally located stockpiles, and/or reallocating and redirecting existing capacity through mutual aid (transshipment) at the regional level. For instance, the Centers for Disease Control and Prevention (CDC) maintains about 12 Strategic National Stockpiles (SNS) of medicine and medical supplies that can be used in the event of a large-scale disaster [12]. These SNS warehouses are strategically located throughout the United States, and can deliver packages of antivirals and other medical supplies within a 12-hour period to any location in the Country. In addition, various systems have been put in place for mutual aid of medical supplies between various States, Regions and Counties. For instance, the Emergency Management Assistance Compact (EMAC) was established to coordinate interstate mutual aid for sharing resources and personnel among states during emergencies and disasters [13]. Other systems such as the Standardized Emergency Management System (SEMS) in California have been established to coordinate local and mutual aid at the State level [8]. The choice between use of redundant capacity and mutual aid at the regional and local level is not an obvious one. One study [21] lists the choice between centralized versus decentralized stockpiles as one of the major challenges in emergency preparedness. Another report sponsored by the Agency for Healthcare Research and Quality (AHRQ) states that, while “the federal government has spent millions of dollars to acquire and maintain the SNS… local and state organizations are creating their own inventories. No published evidence describes the costs and benefits of establishing these local inventories.” [7]. A cost–benefit-based optimization model is developed in the next section to assess the optimal choice between redundant capacity and mutual aid at the regional and local level. 3. Demand surge mitigation during an influenza pandemic A cost–benefit-based optimal allocation of antivirals in the context of pandemic response should take the following factors into consideration:

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First, locate hospitals with antiviral shortages. Second, locate hospitals with surplus antivirals (including central stockpiles). Third, determine (or estimate) the cost of transferring antivirals from a unit with a surplus to one that is short. This would include transportation costs and time delays. Costs may also vary depending on whether the resource requirement is being fulfilled through mutual aid at the regional level, or the central stockpile. These costs can also be considered to model the level and complexity of coordination required across clusters and with the central node. Fourth, determine the benefit of antiviral-based interventions for prophylaxis and treatment. This entails estimating the value of reducing the number of possible health outcomes such as the number of deaths, hospitalizations, etc. And finally, evaluate different intervention policies such as the distribution of stockpile resources, percentage of antivirals that should be allocated for prophylaxis versus treatment, and transshipment policies. Meltzer et al. [23] performed a cost–benefit analysis to study the economic impact of vaccine-based interventions on four health outcomes: number of deaths, hospitalizations, outpatient visits and those ill and not seeking medical care during the pandemic. Cost of intervention estimates included the cost of the vaccine, the cost of administering the vaccine, and patient-associated travel costs. Benefit of intervention estimates included the savings derived from reducing the amount of direct medical costs and averting the earnings lost due to illness or death. This data was estimated for the four health outcomes across three different age groups and two different risk levels. The cost–benefit analysis was then performed for two levels of vaccine coverage and effectiveness across five different gross attack rates (total number of expected infected individuals). Three different criteria for vaccine prioritization were considered: preventing deaths regardless of age/risk factors, preventing deaths among those at greatest risk, and maximizing net economic returns (or minimizing social disruption). Assessing different criteria is important because prioritization becomes a key determining factor in allocation decisions when resources are in short supply. Groups were prioritized for vaccinations based on the objective of the intervention program. If the objective was to cater to those at highest risk, persons ages 65 and older received top priority. However, when maximizing economic returns was the objective, that group received the lowest priority, while high risk persons ages 20 through 64 received top priority. Since the greatest economic cost is due to death, all other things being equal, the largest economic returns are in fact derived from interventions that avert the largest number of deaths. Hence, even

Table 3 Federal and State antiviral allocations for Counties. Region

Population

Federal SNS allotment by population

State SNS allotment

USA Arizona Maricopa Pinal Gila Pima Santacruz Cochise Graham Greenlee California San Luis Obispo Santa Barbara Ventura Los Angeles Orange Mono Inyo San Bernardino Riverside San Diego Imperial

299,398,484 6,166,318 3,768,123 271,059 52,209 946,362 43,080 127,757 33,660 7738 36,457,549 257,005 400,335 799,720 9,948,081 3,002,048 12,754 17,980 1,999,332 2,026,803 2,941,454 160,301

44,000,000 906,210 553,768 39,835 7673 139,079 6331 18,775 4947 1137 5,357,850 37,770 58,834 117,528 1,461,983 441,185 1874 2642 293,824 297,862 432,280 23,558

67,717 41,381 2977 573 10,393 473 1403 370 85 2,752,151 19,401 30,221 60,370 750,973 226,622 963 1357 150,928 153,002 222,048 12,101

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Table 4 Number of infected people by age and risk factor for the entire US population. Age group

0–19 years 20–64 years 65 + years

Risk

High risk Low risk High risk Low risk High risk Low risk

Number at risk GAR = 15%

GAR = 20%

GAR = 25%

GAR = 30%

GAR = 35%

1,018,598 14,897,002 3,042,426 18,085,533 1,098,176 1,647,265

1,358,131 19,862,669 4,056,568 24,114,044 1,464,235 2,196,353

1,697,664 24,828,336 5,070,710 30,142,555 1,830,294 2,745,441

2,037,197 29,794,003 6,084,852 36,171,066 2,196,353 3,294,529

2,376,730 34,759,670 7,098,994 42,199,577 2,562,412 3,843,617

the objective of maximizing economic returns (or minimizing social disruption) still achieves the objective of minimizing total number of deaths. As mentioned earlier, while vaccines provide the best line of defense, they could take up to 4–6 months to become available for general use. In the interim, therefore, it becomes imperative to optimally allocate the limited stock of antivirals to combat the epidemic. The Meltzer model therefore is extended to derive the optimal allocation of antivirals for prophylaxis and treatment. Also, while Meltzer uses the cost–benefit analysis to look at allocation strategies across risk and age groups, the model described here uses the cost–benefit formulation in an optimization model to come up with the optimal redistribution of resources across different regions. Antivirals can be used in one of three ways: pre-exposure prophylaxis of the general susceptible population; post-exposure prophylaxis of the susceptible population that may have been exposed to the virus; or treatment of the infected population. Each of these forms of interventions typically has different levels of efficacy, and averts health outcomes (death, hospitalization, outpatient, and ill but no medical care) to different degrees. Since there is a limited stock of antivirals, an optimal distribution of antivirals between prophylaxis and treatment should be decided upon. In order to keep the model simple and tractable, only two of these antiviral-based interventions will be considered here: pre-exposure prophylaxis and treatment. It is assumed that treatment would require 1 course (10 pills) of antivirals, and prophylaxis would require 4 courses of antivirals [17]. The following model is proposed to evaluate the different intervention policies and study the optimal choice between maintaining redundant capacity and relying on mutual aid. Consider a multi-region scenario, where each region contains a susceptible population. Further consider a single SNS that contains a stockpile of antivirals. The regions could be at any level of hierarchy: zip codes, cities, counties, states, etc. Some pre-determined proportion of the stockpile is used up for prophylaxis of a certain percentage of the susceptible population in the various regions. This will reduce the number of susceptibles and the number of health outcomes by an amount that is based on the prophylactic effectiveness of the antiviral. At the first incidence of a pandemic, some proportion of the stockpile is allocated to each region for treatment of the expected infected population and the rest is retained in the stockpile for later use. The pandemic might affect one region more than another, thereby leaving some regions with a shortage of antivirals, while leaving others with an excess. Regions with a shortage can therefore receive mutual aid from regions with a surplus. A cost–benefit analysis based optimization model is used to determine the optimal redistribution of resources between the regions through mutual aid. The redistribution of antivirals reduces the number of infected and the number of health outcomes by an amount that is based on the treatment effectiveness of the antiviral. The benefits in this model are derived from the economic savings obtained from reducing the number of health outcomes. The costs in this model are derived from the cost of transferring (transshipping) resources between regions that are involved in mutual aid. The cost–benefit-based optimization model therefore looks to maximize net returns through optimal allocation.

As in Meltzer et al. [23], the following inputs to the model are considered: gross attack rates (GAR) which determines the expected infected population; expected distribution of infected individuals by age; expected number of health outcomes (death, hospitalization, outpatient, ill but no medical care), and expected dollar value savings per health outcome averted. Two additional inputs to the model are considered: antiviral prophylactic and treatment effectiveness; and cost of transshipment of antivirals between regions. The two stage model is detailed below. Table 2 defines the various model parameters.

3.1. Stage 1: Prophylaxis Let us consider (T − 1) regions, and one stockpile (SNS) location. Let k ∈ [1, 2,... T] represent any node in this system, where k = 1 represents the stockpile, and k = 2, 3,... T represent the (T − 1) regions. Let Pk represent the total population in region k ∈ [2, 3,... T], and T

P = ∑ Pk represent the total population across all regions. Ask=2

suming a susceptible population of mk in region k, the expected number of infected individuals is given by, nk = GARk * mk; where GARk is the gross attack rate in region k. Given a total stock of S courses of antivirals, using the proposed CDC equitable allocation approach based on population
proportions, the maximum stockpile allocation  Pk *S. for region k is: Sk = P Prophylaxis is first provided to a percentage, pPxsk of the susceptible population (mk) in each region k. Since prophylaxis requires 4 courses of antivirals, 4 * pPxsk * mk courses of antivirals will be consumed in each region. This requirement is fulfilled by taking the required resources from the SNS. It is assumed that there is enough in the stockpile to cover the prophylaxis requirement. We consider four possible health outcomes (w): death (d), hospitalization (h), outpatient care (o) and those who are ill but seek no medical care (n). Using the number of expected infected individuals, the expected number of health outcomes for each region can be

Table 5 Effectiveness of Tamiflu by age group. Intervention

General effectiveness

Effectiveness by age group 0–19 years

20–64 years

65+ years

Pre-exposure prophylaxis: Death Hospitalization Outpatient visits Ill, no medical care sought

71% 71% 71% 71% 71%

71% 71% 71% 71% 71%

71% 71% 71% 71% 71%

71% 71% 71% 71% 71%

Antiviral therapy: Death Hospitalization Outpatient visits Ill, no medical care sought

65% 56% 44% 44%

70% 60% 44% 44%

70% 60% 44% 44%

30% 30% 44% 44%

Imperial

245 275 333 300 363 398 425 468 414 308 281 213 195 498 366 161 162 113 0 355 385 443 410 473 506 535 578 325 219 192 121 90 445 313 108 98 0 113

San Bernardino

323 385 409 438 501 534 487 530 259 152 125 59 38 351 219 14 0 98 162

Inyo

527 589 613 641 704 736 690 733 324 280 253 227 249 132 0 207 219 313 366

Mono

658 720 745 773 836 868 822 865 324 412 384 358 380 0 132 338 351 445 498

Orange

360 422 446 475 538 570 524 567 234 128 100 33 0 380 249 51 38 90 195 441 503 528 556 619 652 605 648 134 27 0 68 100 384 253 128 125 192 281

Los Angeles

373 435 460 489 552 584 538 581 201 95 68 0 33 358 227 60 59 121 213

k=2

by mk *pPxsk *ep. The revised susceptible population per region after prophylaxis is therefore mk =mk −mk *pPxsk *ep. As before, this would also result in a reduced number of expected infected individuals, nk =GARk *mk, and the expected number of health outcomes, nd, k = 0.00202*nk, nh, k =0.00791*nk, no, k =0.46276*nk, nn, k =0.52731*nk. 3.2. Stage 2: Treatment Let p be the proportion of the maximum available stockpile for each region (Sk) that is pre-allocated to the region at the first incidence of a pandemic. After pre-allocation, each region has p * Sk T

courses of antivirals, and the stockpile retains S = S− ∑ p*Sk . k=2

469 530 555 583 646 679 633 675 107 0 27 95 128 412 280 156 152 219 308

Ventura

k=2

T

Greenlee

208 200 146 194 252 169 47 0 782 675 648 581 567 865 733 528 530 578 468

Graham

575 637 661 690 753 785 739 782 0 107 134 201 234 324 324 262 259 325 414

Santa Barbara

at a cost of: COSTpxs = ∑ 4*pPxsk *mk *c1k .

S = ∑ Sk . It will also reduce the number of susceptibles in each node

165 132 78 127 191 128 0 47 739 633 605 538 524 822 690 486 487 535 425

San Luis Obispo

computed from Meltzer et al. [23] as follows: nd, k = 0.00202* nk, nh,k = 0.00791 * nk, no, k = 0.46276 * nk, nn, k = 0.52731 * nk. The savings from prophylactic intervention can be computed based on the expected effectiveness (ep, w) of the antivirals in averting a particular health outcome. Let vw represent the value saved per person by averting a particular health outcome. Prophylaxis based intervention therefore results in a total savings of SAVGpxs = ∑w nw;k *pPxsk *ep;w *vw ,

Prophylaxis based intervention will reduce the stock availability per region to Sk =Sk −4* pPxsk *mk, resulting in a reduced stockpile of

The objective is to provide treatment to as much of the infected population in each region, nk, as possible. Since treatment requires only 1 course of antivirals, this will require nk courses of antivirals for each region. This requirement in each region is first fulfilled by using up the pre-allocated amount for that region. Let us assume that an amount pak of the requirement in region k was fulfilled through this process. Depending on the GAR numbers for each region, the pre-allocated amount will either be in excess of this requirement (which implies the region has a net excess of resources, Ri), or be short of it (which implies the region has a net shortage of resources, rj). The remaining requirements (rj) are then fulfilled by borrowing resources from nodes (including the stockpile) that have an excess of resources (Ri). This is the process of transshipment, which we formulate as a quadratic programming problem as shown below. We assume that each node with an excess of resource (Ri) shares a proportion λij of its resources with each region j which has a shortage. In other words, each region j that has a shortage, can receive a total of

Cochise

212 168 205 98 156 0 128 169 785 679 652 584 570 868 736 532 534 506 398

represents the cost of transshipping resources from node i to node j. ! ∑ λij *Ri i∈I The net return is therefore: ∑ nw;j * *et;w *vw −∑ cij *λij *Ri . nj

Maricopa

Pima

88 55 0 106 171 205 78 146 661 555 528 460 446 745 613 408 409 443 333 63 0 55 70 135 168 132 200 637 530 503 435 422 720 589 383 385 385 275

node j. Let et, w represent the treatment effectiveness of antivirals in averting a particular health outcome. The above allocation of antivirals for treatment to region j therefore reduces the number of health ! ∑ λij *Ri outcomes in the region by ∑ nw;j * i∈I nj *et;w . This translates to !w ∑ λij *Ri savings of ∑ nw; j * i∈I nj *et;w *vw , at a cost of ∑ cij *λij *Ri , where cij

116 70 106 0 71 98 127 194 690 583 556 489 475 773 641 436 438 410 300

Gila Pinal

Santa Cruz

i∈I

treatment requires one course of antivirals per person, this can provide ! ∑ λij *Ri i∈I of the infected population in treatment for a proportion nj

0 63 88 116 179 212 165 208 575 469 441 373 360 658 527 322 323 355 245

179 135 171 71 0 156 191 252 753 646 619 552 538 836 704 499 501 473 363

∑ λij *Ri courses of antivirals from regions that have a surplus. Since

Maricopa Pinal Gila Pima Santa Cruz Cochise Graham Greenlee San Luis Obispo Santa Barbara Ventura Los Angeles Orange Mono Inyo San Bernardino Riverside San Diego Imperial

Table 6 Distances between county seats in miles.

309

T

322 383 408 436 499 532 486 528 262 156 128 60 51 338 207 0 14 108 161

Riverside

San Diego

H. Arora et al. / Decision Support Systems 50 (2010) 304–315

w

i∈I

w

i∈I

While we track the net returns, in optimizing the transshipment allocations, we note that the value of lives saved dominates the costs of transshipment. Thus costs of transshipment become relevant when benefit values are tied. Further, a linear cost term is considered instead of a square cost term because the cost term is only intended to play a role

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Table 7 FedEx shipping rates. lb/mi

0–150 mi

(a) FedEX one day freight rates 151–499 lb $0.64 500–999 lb $0.63 1000–1999 lb $0.58 2000+ lb $0.56 Min charge $60.00

151–300 mi

301–600 mi

601–1000 mi

1000–1400 mi

1404–1800 mi

1800 mi

$1.12 $1.09 $1.04 $1.02 $105.00

$1.89 $1.84 $1.81 $1.78 $1580.00

$2.97 $2.93 $2.89 $2.83 $205.00

$3.46 $3.43 $3.40 $3.26 $221.00

$3.87 $3.80 $3.73 $3.67 $233.00

$4.21 $4.18 $4.15 $4.01 $257.00

$2.74 $2.72 $2.70 $2.68

$2.91 $2.69 $2.87 $2.85

$3.09 $3.07 $3.05 $3.03

$3.30 $3.28 $3.26 $3.24

$3.42 $3.40 $3.38 $3.36

(b) FedEX Hundredweight/per-pound rates for standard overnight 100–499 lb $1.17 $1.68 500–999 lb $1.15 $1.66 100–1999 lb $1.13 $1.64 2000+ lb $1.11 $1.62

at the margin. A square cost term will make allocation decisions cost averse, which is unrealistic, since the federal government will want to save lives at any cost. This also explains the absence of a budget constraint in the formulation. An intent of the formulation is to ensure equitable allocation across regions. The objective is to minimize the square of the lost benefits due to a non-availability of resources, subject to the following constraints: Constraint 1 ensures that the amount allocated is no more than what is required. Constraint 2 ensures that a shortage is always first fulfilled from the stockpile, and second through mutual aid from other regions. Constraint 3 specifies the lower and upper bounds on the proportion of resources that can be shared by nodes with an excess of resources. Constraint 4 ensures that nodes that have an excess of resources cannot share any more than they have. This is a quadratic optimization problem that can be numerically solved in the standard form.1 Based on the allocations from the optimization program, we

Objective: 0 0 0 112 1 ∑ λij *Ri AA + ∑cij *λt;ij *Ri A Min ∑ @∑ @nw;j *et;w *vw −nw;j *et;w *vw *@ i∈I nj w j i∈I

Subject to: rj ≥ ∑ λ *R for each j ∈ J ij i

…ð1Þ

∑ λ1j ¼ 1 j∈J

…ð2Þ

i∈I

0≤λij ≤1 for all i ∈ I and j ∈ J …ð3Þ ∑ λij ≤1 for each i ∈ I j∈J

…ð4Þ

compute the total savings. If an amount ak of the requirement in each region is fulfilled through this transshipment process, treatment results

 pak + ak in a total savings of: SAVGtrt = ∑k ∑w nw;k *et;w *vw* , nk   at a cost of: COSTtrt = ∑j ∑ cij *λ
t;ij *Ri .

4. Data and model simulation The data required to implement the model include, regions (along with their susceptible populations), gross attack rates for each region, expected number of infected individuals by age, prophylactic and treatment effectiveness numbers, estimate of the stockpile availability, estimate of cost of transshipment of antivirals between regions, and a dollar value estimate of averting a health outcome. Each of these data elements is detailed below. Based on current clinical evidence, treatment will require 1 course (10 pills) of antivirals, and prophylaxis will require 4 courses of antivirals [17]. The US Department of Health and Human Services has made antiviral preparedness recommendations for the scenario where 30% of the US population will need some kind of medical attention. However, the federal stockpiles have enough to cover only about 25% of these cases, i.e., 44 million courses [4]. The Department of Health and Human Services is therefore allocating the available federal stock of antivirals to States based on population. In addition to the federal stockpile of 44 million courses of antivirals, Sates are also preparing to carry their own stockpiles of antivirals. While the model is generic enough to use at any level of hierarchy, we examine mutual aid at the county level for a number of reasons. First, memoranda of understanding for mutual aid are currently being set across the country at the county level due to pre-existing relationships and proximity. Second, while the federal stock is being allocated proportionally by population, states are at different levels of preparedness. For instance, while California has already purchased 102% of the planned State-level stockpile, Arizona has only purchased 11%. Studying mutual aid at the county level would therefore reveal the dependency patterns across the counties based on their level of preparedness. Third, considering all counties/states would lead to increased computational complexity, and would make it more difficult to understand the implications of mutual aid.

Table 8 The expected number of health outcomes across age groups and risk factors. Outcomes: Dead (nd), Hospitalization (nh), Outpatient (no), Ill but no medical care (nn). The following values can be derived from this table for the total infected population (n): nd = 0.00202*n, nh = 0.00791*n, no = 0.46276*n, nn = 0.52731*n. Age/risk

Low risk

High risk

0–19 years

nd = 0.00003*n nh = 0.00140*n no = 0.19157*n nn = 0.18300*n nd = 0.00014*n nh = 0.00308*n no = 0.16021*n nn = 0.29237*n nd = 0.00015*n nh = 0.00059*n no = 0.02521*n nn = 0.01604*n

nd = 0.00001*n nh = 0.00033*n no = 0.02140*n nn = 0.00226*n nd = 0.00095*n nh = 0.00108*n no = 0.04563*n nn = 0.02653*n nd = 0.00074*n nh = 0.00142*n no = 0.01873*n nn = 0.00711*n

i∈I

Net savings are computed as a sum of treatment and prophylaxis savings: (SAVGpxs + SAVGtrt) at a cost of (COSTpxs + COSTtrt).

20–64 years

65 and over

1

The optimization program was implemented in Matlab.

H. Arora et al. / Decision Support Systems 50 (2010) 304–315 Table 9 Dollar savings per averted health outcome. Outcome

Outpatient Hospitalizations Deaths Ill, no medical care sought

Savings ($) per averted health outcome 0–19 years

20–64 years

65+ years

Weighted Average

300 3366 1,019,536 197

330 6842 1,045,278 202

458 7653 74,146 327

338 5945 913,903 217

Eight different counties in Arizona and 11 different counties in California are therefore considered here, resulting in a total of 19 counties (Fig. 2). The “regions” here are based on current fire emergency mutual aid regions [9,32]. The population in each county is obtained from the 2005 census data estimates. It is assumed that the entire population will be susceptible to a new strain of influenza. The federal and state allocations, scaled by population to the county level, are shown in Table 3. Given a GAR, the expected infected population is simply the product of the GAR and the susceptible population. The expected number of infected individuals by age can then be computed using the data in Meltzer et al. [23]. As an example, Table 4 shows the number of cases by age for the entire US population for five different GARs. Tamiflu, one of the recommended antivirals, is considered for prophylaxis and treatment with the effectiveness numbers as shown in Table 5. The SNS stores and ships the antiviral packages in pallets. In the case of Tamiflu, one pallet will contain 3872 courses (number of antiviral doses required to treat one person) of antivirals and weigh up to 437 pounds [11]. Using this information, and the estimated distances between the 19 counties (Table 6), the cost structure is created based on FedEx Hundredweight/per-pound rates when shipping 100–499 pounds via standard overnight as shown in Table 7b. Freight charges apply to packages over 150 pounds, and are a little less than the Hundredweight amounts (Table 7a). However, given that mutual aid would require repackaging and reshipping the SNS pallets, the higher cost may be a better approximation of the costs incurred. The cost of transshipment from the SNS to any of the counties is set to 0. This is done in order to ensure that a county that is short of resources, always first goes to the SNS before it looks to other counties for mutual aid. The motivation behind this setup is twofold. First, the distribution of resources from a central SNS location will involve a lower coordination cost than redistribution of resources among Counties. Second, in a dynamic setup, Counties would want to save the resources they own in order to respond to the possibility of the spread of the pandemic to their regions in a future time period. Also note that, while the model does not directly account for the delay in transferring resources between counties, the delay is indirectly accounted for through the distance-based cost structure. The expected number of health outcomes across age groups as a percentage of the number infected is shown in Table 8. These values are computed based on the approach described in Meltzer et al. [23]. Each of these outcomes will cause a potential economic loss due to direct medical costs and lost earnings. This economic loss is used to indicate the value of each health outcome. In other words, the value or benefit derived from averting each health outcome. Table 9 reproduces these numbers from Meltzer et al. [23]. The following input parameters are varied in the model to simulate different attack scenarios and response strategies. Prophylaxis Percentage is used to capture the percentage of susceptible population in each county that is provided with antiviral-based pre-exposure prophylaxis. Stockpile Pre-allocation Percentage is used to capture the percentage of stockpile that is pre-allocated to the various counties (the rest is retained in the stockpile). Proportional Allocation is used to capture whether the stockpile is allocated to counties by population proportion, or by gross attack rate. Gross Attack Rate is used to capture different severities of the pandemic

311

across the regions. Transshipment is used to capture the presence (transshipment case) or absence (no transshipment case) of mutual aid. Age Group Analysis is used to capture prioritization of antivirals based on age-group effectiveness. Enough to go around refers to whether or not there are enough resources to cover the pandemic scenario under consideration, and depends on the parameters set for the scenario. Table 10 details the different scenarios that were simulated. For each scenario, a pandemic is simulated in the 19 counties by setting different GARs in the three emergency regions. A small percentage of the stockpile is first used up for prophylaxis. A percentage of the remaining antivirals are then pre-allocated to the various counties, while the rest are retained in the stockpile for use on an as-needed basis. Nodes with an excess or shortage of antivirals are then identified. The treatment requirements for each county are determined by the GAR. Pre-allocated resources are first used to meet these requirements. If pre-allocated resources fulfill the requirements, then what is left of the pre-allocated resources is defined as the resource excess at that node. If the pre-allocated resources do not fulfill requirements, then the shortage is defined as the requirement at that node. The optimization problem is then solved to arrive at an optimal redistribution of the antivirals. Model results are presented through three different sets of graphs. The first graph plots the “Required versus Allocated” numbers for each county — i.e., how many antivirals did each county need, and how much was it able to receive from the SNS and/or other counties. The second graph plots the “Proportion Shared” for each county — i.e., what proportion of the excess resources that the county had was given to other counties that were short on resources. The third graph plots the “Proportion of Max Savings” and compares it to the “Proportion of Total Infected”. “Max Savings” is the maximum amount of savings a county could have achieved if it had all the resources it required. “Proportion of Max Savings” is what percentage of “Max Savings” the county was able to achieve with the resources it was actually assigned. “Proportion of Total Infected” depicts the proportion of the total infected population across all counties that reside in the county. As mentioned before, the SNS is node 1, and the counties are nodes 2 through 20. The next section discusses the scenarios and implications of the results. 5. Results and discussion Six different CDC policies were tested, resulting in important policy implications. The policies tested were as follows: (i) How much of the CDC stockpile should be pre-allocated. In other words, is the current CDC policy of pre-allocating 100% of the stockpile at the first incidence of a pandemic optimal? (ii) Does the pre-allocation strategy have any bearing on the severity and extent of the pandemic? (iii) Is mutual aid beneficial? (iv) Is proportional allocation of limited stockpile optimal? (v) What proportion of limited antivirals should be expended on prophylaxis? (vi) Can different allocation objectives lead to more savings? Each of the six scenarios is discussed in more detail below. Table 11 summarizes the savings, costs and policy implications of the different scenarios. 5.1. Scenario 1: Impact of pre-allocation on cost and savings Scenario 1 simulates the impact of different levels of pre-allocation. Counties in AZRI, AZRV and CARVI are assumed to have been affected in such a manner that there are enough antivirals in the stockpile to treat the total infected population across the 19 counties. It is assumed here that 1% of the susceptible population in each county is provided with antiviral-based prophylaxis. The rest of the antivirals are used up for

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H. Arora et al. / Decision Support Systems 50 (2010) 304–315

Table 10 Different scenarios that were simulated in this research. Scenario

Enough to go around?

Prophylaxis percentage

Stockpile pre-allocation Percentage

Proportional allocation by

Gross attack rate

Transshipment

Share percentage

Age group analysis

1: 2: 3: 4: 5: 6:

Yes No Yes Yes Yes No

0.01 0.01 0.01 0.01 0.01, 0.02 0.0

1.0, 0.8, 0.6 1.0, 0.8, 0.6 1.0 1.0 1.0 1.0

Population Population Population Population, GAR Population Population

0.3, 0.4, 0.3, 0.3, 0.3, 0.4,

Yes Yes Yes, No Yes Yes Yes

1.0 1.0 1.0 1.0 1.0 1.0

No No No No No No, Yes

Impact of pre-allocation Impact of higher GAR Impact of no transshipment Impact of allocation by GAR Impact of higher prophylaxis percentage Impact of age group analysis (done with only 5 counties)

treatment. Three different pre-allocation cases are considered. In the first case, 100% (p= 1.0) of the stockpile is pre-allocated to the 19 counties based on population proportions. In the second case, 80% (p= 0.8) of the stockpile is pre-allocated to the 19 counties, and 20% is retained in the stockpile to be used on an as-needed basis. In the third case, 60% (p= 0.6) of the stockpile is pre-allocated to the 19 counties, and 40% is retained in the stockpile to be used on an as-needed basis. Fig. 3a shows the requirement versus allocated bar graphs across the 19 counties for the three pre-allocation cases. In the first case (p= 1.0), counties in AZR1 are short, while counties in CARVI have an excess of antivirals. The total requirements of the AZR1 counties are therefore met by the CARVI counties (Fig. 3b). In the second (p= 0.8) case, the SNS provides most of the resources to the Arizona counties that are short, and the rest of the shortage is met by the California counties. In the third (p= 0.6) case, all counties have a shortage, while the SNS has enough of an excess to cover all of their needs (Fig. 3a and b). Fig. 3c reflects the fact that the stockpile has enough to go around, and therefore, all three cases perform equally well. Note that even when there is enough to go around, the “Proportion of Max Savings” is still not 1.0. This is because the effectiveness of the antivirals is not 100% (as shown in Table 5). The savings in all three cases is $4886 million. The cost is $2.26 million for the first case, and $0.99 million for the second case, and $0 for the third. In other words, when there are enough resources to go around, preallocating fewer resources results in lower transshipment costs. This is because in the first case, counties rely entirely on mutual aid thereby incurring the maximum transshipment cost, in the second case, counties rely partly on transshipment and partly on the SNS (where they incur 0 cost), and in the third case, counties rely entirely on the SNS thereby incurring a zero cost.

0.2, 0.3, 0.2, 0.2, 0.2, 0.3,

0.1 0.2 0.1 0.1 0.1 0.2

are short of antivirals, and there is just not enough to go around. It is assumed that 1% of the susceptible population in each county is provided with antiviral-based prophylaxis. The rest of the antivirals are used up for treatment. As before, three different pre-allocation cases are considered: p = 1.0, p = 0.8 and p = 0.6. In the first case (p= 1.0), all the stockpile antivirals have been pre-allocated and used up for treatment. Neither the SNS, nor any of the counties have an excess of antivirals that can be allocated to counties that require them. In the second (p= 0.8) and third (p= 0.6) case, all counties have a shortage; the SNS has a percentage of the stockpile that it retained, that can now be distributed on an as-needed basis. Since in each case, resources are only borrowed from the SNS, the total cost is $0. Notice that the savings here are more than in the first scenario. This is because more lives are saved when the pandemic is more severe. Also, the total savings is the same across all three cases, viz., $6207 million. However, what does change across the three cases is the allocation of limited quantities of antivirals across counties. As can be seen in Fig. 4, as the SNS retains more and pre-allocates less, resources are allocated in proportion to the infected population rather than the susceptible population. In other words, counties with the greatest number of infected people are assigned the greatest number of antivirals, ensuring that resources go to the counties that need them the most. This would help check the progression of the pandemic, and result in greater savings in future time periods. This finding is in line with the concept of postponement, which has been well documented in the literature [2,25,27,31,34,37]. Postponement deals with delaying decisions regarding manufacturing and logistics operations as much as possible in order to gain more information on the requirements, thereby reducing risk and uncertainty costs. In other words, delaying the decision of how much to allocate can provide the counties additional options for optimal allocation.

5.2. Scenario 2: Impact of gross attack rate on cost and savings 5.3. Scenario 3: Impact of transshipment on cost and savings Scenario 2 simulates the impact of a higher gross attack rate. Counties in AZRI, AZRV and CARVI have been affected in such a manner that there are not enough antivirals in the stockpile to treat the total infected population across the 19 counties. Essentially, all the counties

Scenario 3 studies the impact of not allowing mutual aid through transshipment. There are enough antivirals to go around; 1% of the population is given prophylaxis; and the SNS pre-allocates 100% of the

Table 11 Summary of results and policy implications of this research. Scenario

Parameter

Savings (million $)

Cost ($)

Policy Implication

1: Impact of pre-allocation

p = 1.0 p = 0.8 p = 0.6 p = 1.0 p = 0.8 p = 0.6 Yes No Population GAR 3% 1% No Yes

4885.9 4885.9 4885.9 6207.2 6207.2 6207.2 4885.9 3769.4 4885.9 4885.9 3360 4885.9 20.7 21.9

2,260,298 999,812 0 0 0 0 2,260,298 0 2,260,298 37,544 175,147 2,260,298 19,747 19,951

Postponement results in lesser transshipment cost.

2: Impact of higher GAR

3: Impact of no transshipment 4: Impact of allocation by GAR 5: Impact of higher prophylaxis percentage 6: Impact of age group analysis (only 5 counties)

Postponement results in allocation by number infected rather than by number susceptible. Resources go to the counties that need them the most. Smaller counties stand to gain most by transshipment. Allocating by GAR helps avoid cost of transshipment. When resources are limited, provide prophylaxis only to highest risk and save the rest for treatment. When reduction of social disruption is the objective, age-group based allocation leads to greater savings.

H. Arora et al. / Decision Support Systems 50 (2010) 304–315

(a) Number of Resources Required versus what was Allocated 5

10

degrees. Note that there is also a drop in cost ($2.26 million v/s $0) since there is no transshipment cost associated with the second case. Also notice in Fig. 5, that it is the smaller counties that stand to gain most from transshipment. Since antivirals have been proportionally allocated by population, a bigger county that has not been hit by the pandemic too severely would end up with a larger cache of excess resources. Therefore smaller counties that have been hit severely have more resources available to them.

p=1

x 10

Required Allocated (Treatment)

0

5

10

15

5

10

0 x 10

10

5

10

15

5

20

5.4. Scenario 4: Impact of allocation by GAR on cost and savings 25

p=0.6 Required Allocated (Treatment)

5

0

5

10

15

20

25

Nodes with Requirement

(b) Proportion of Resources Shared by Nodes that have an Excess p=1

1

Proportion (Treatment) 0.5

Proportion Shared

0

0

5

10

15

20

25

p=0.8

1

Proportion (Treatment) 0.5 0

0

5

10

15

20

25

Proportion (Treatment) 0.5

0

5

10

15

25

20

Nodes with Excess

(c) Proportion for Max Savings versus Proportion of Total Infected 0.7

Proportion of Total Infected (plot)

p=1.0 p=0.8 p=0.6

0.6

Proportion of Max Savings (bar)

5.5. Scenario 5: Impact of prophylaxis on cost and savings

p=0.6

1

0

Scenario 4 studies the impact of allocation by gross attack rates. There are enough antivirals to go around; 1% of the population is given prophylaxis; and counties can rely on mutual aid through transshipment. Two cases are considered: the first one where the SNS preallocates 100% of the stockpile based on population proportions, and the second, where the SNS pre-allocates 100% of the stockpile based on GARs. Since the antivirals supply is sufficient, the total savings in both cases is the same ($4886 million), however, the cost is higher in the first case ($2.26 million v/s $3754). This is because, when there are enough resources, and resources are allocated by GAR, each county is preallocated resources based on need, and therefore there is less of a shortage. Although, it is not really possible to predict the GARs, and preallocate based on the expected infected populations, this scenario does make a case for postponement. It is equivalent to saying that if all the resources are held in the SNS and allocated based on the infected populations in the various counties, the cost of transshipment can be avoided.

0.5

0.5 0.4

0.4 0.3

0.3 0.2

0.2 0.1

0.1 0

0

5

10

15

20

25

All Nodes Fig. 3. Impact of pre-allocation.

stockpile based on population proportions. Two cases are considered: the first one where there is mutual aid through transshipment, and the second case, where there is none. Although there is enough to go around, counties that need resources will not be able to get them in the second case, because counties that have an excess are unable to share their surplus resources, leading to an overall drop in savings ($4886 million v/s $3769 million). This shows that transshipment can be beneficial when counties are affected by the pandemic to different

Scenario 5 studies the impact of increasing the prophylaxis percentage. There is enough to go around. Two cases are considered, the first one where 3% of the susceptible population is given antiviralbased prophylaxis, and the second one where 1% of the susceptible population is given prophylaxis. The SNS pre-allocates 100% of the remaining stockpile for treatment, based on population proportions. Since there is a limited availability of antivirals, and prophylaxis takes 4 courses of antivirals, as against 1 course of antivirals for treatment, what is spent on prophylaxis seriously detracts from what is available for treatment. Therefore, spending 3% on prophylaxis leaves a large population of infecteds with no treatment resources. Whereas spending 1% on prophylaxis leaves enough for the ensuing infected population. This is reflected in the savings of $3360 million v/s $4886 million. In other words, when there is a limited quantity of 0.7 p=1.0 p=0.8 p=0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0

Proportion of Total Infected (plot)

0

25

Required Allocated (Treatment)

5 0

20

p=0.8

x 10

Proportion of Max Savings (bar)

Required versus Allocated

5 0

313

0.1

0

5

10

15

All Nodes Fig. 4. Impact of higher GAR.

20

25

314

H. Arora et al. / Decision Support Systems 50 (2010) 304–315 0.7 groupAnalysis=0 groupAnalysis=1

transship=1 transship=0

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

Proportion of Max Savings (bar)

0.5

0.6

Proportion of Total Infected (plot)

Proportion of Max Savings (bar)

0.6

0.5

0.5

0.4

0.4 0.3

0.3 0.2

0.2 0.1

0.1

0.1 0

0

0

5

10

15

20

25

Proportion of Total Infected (plot)

0.7

0

1

2

3

4

5

6

7

All Nodes

All Nodes

Fig. 6. Impact of age group analysis.

Fig. 5. Impact of no transshipment.

antivirals, prophylaxis should be given only to the people at highest risk. The rest of the antivirals should be saved for treatment purposes.

($20.7 million versus $21.9 million) derived from allocation. In other words, when minimizing social disruption is the objective, age group based allocation can lead to greater savings. 6. Conclusion

5.6. Scenario 6: Impact of allocation by age group on cost and savings Scenario 6 studies the impact of allocation based on age-group analysis. Two cases are considered: while the first case does not take age distribution into account in the optimization problem, the second case does. Since antiviral effectiveness is different for different age groups (Table 5), and savings derived from each group are also different (Table 9), the allocation is expected to be different in the two cases. In order to study the effect of age on allocation, a special case with only 5 of the 19 counties is considered: Maricopa, Pinal, Gila, Pima and Santa Cruz. Let us assume a 0.5 GAR in Gila and Santa Cruz, and 0 GAR elsewhere. The stockpile amount is reduced in order to create a situation of scarcity for the 5 counties. It is also assumed that 0% of the susceptible population is provided with prophylaxis, and 100% of the stockpile is pre-allocated based on population proportion. The population in each county is broken into 3 age groups, 0–19, 20– 64 and 65 and over. The populations and age distributions of these counties are shown in Table 12. As can be seen, Gila has a higher population than Santa Cruz. While the 0–19 and 20–64 age-group populations are comparable in the two counties, the 65 and over age group population is much higher in Gila than in Santa Cruz. In the first case, resources are allocated without taking age groups into account. In this case, since Gila has the higher susceptible population, its requirements are met to a greater degree than Santa Cruz (Fig. 6). In the second case, age groups are taken into account while making allocation decisions. Since antiviral effectiveness is higher for age groups 0–19 and 20–64, and savings from caring for these age groups is higher, some of the resources that were being given to the 65 and over age group in Gila in case one, are now allocated to age groups 0–19 and 20–64 in Santa Cruz (Fig. 6). This increases the saving

Table 12 Age distribution for 5 Counties. County

Maricopa Pinal Gila Pima Santa Cruz

Population

3,768,123 271,059 52,209 946,362 43,080

Age Distribution 0–19

20–64

65 and over

27% 25% 24% (12,530) 24% 32% (13,786)

62% 62% 55% (28,715) 61% 56% (24,125)

11% 14% 21% (10,964) 15% 12% (5170)

There are several conclusions that can be drawn from this work. First, postponement, or delaying the decision of how much to preallocate, will lead to greater flexibility in dealing with the pandemic, thereby providing a more effective response. Second, while mutual aid is beneficial in general to maximize resource utilization, it is most beneficial to the smaller counties, especially when resources are being pre-allocated by population proportion. Third, sub-group analysis based on factors such as age can further optimize the allocation of resources leading to greater savings. And fourth, a different set of objectives (such as preventing deaths regardless of age/risk factors, preventing deaths among those at greatest risk, etc) could lead to a different set of policy implications. The choice of objective function brings a much debated topic to light — the ethical dilemma surrounding allocation objectives. The Public Engagement Pilot Project on Pandemic Influenza (PEPPPI) was undertaken in 2005 to address some of the concerns regarding vaccine prioritization [28]. One of the goals of this pilot project was to demonstrate that policy makers can reach productive outcomes on important policy questions by increasing the level of public engagement. Subsequently, many States have undertaken similar studies to determine policies around allocation objectives. For instance, North Carolina undertook a study in 2007 to determine ethical guidelines for the use of scarce resources during influenza pandemics [26]. The study resulted in two important policy decisions. First, priority for preventive measures such as vaccines must primarily be directed to assure the functioning of society, and secondarily to prevent spread of the disease. Second, priority for treatment measures such as antivirals must primarily be directed to minimize illness, hospitalizations and death, and secondarily to assure the functioning of society. The allocation objective in the resource allocation model can therefore be altered based on the allocation priorities determined by such studies at the regional level. This study also points to the need for a greater understanding of the logistics and supply chain aspects of pandemic planning, and the development of models to understand the implications of resource allocation decisions under various pandemic scenarios. Delayed decision making or postponement strongly hinges on the availability of accurate data from the lower echelons of the supply chain. It is therefore imperative that investments in information gathering and coordination be made as part of the pandemic response plan [5,14,29].

H. Arora et al. / Decision Support Systems 50 (2010) 304–315

Accurate information can increase the ability of the central stockpile to strategically delay its decision of allocating antivirals to different regions, while ensuring the timely availability of antivirals during a pandemic. One of the limitations of this research is that the resource allocation model assumes a static disease environment. Most emergency response contexts encounter dynamic settings and require improvisation [24]. In this study, a single allocation of resources is made, without consideration for how the allocation may affect subsequent spread of the pandemic. In other words, the study evaluates policies based on instantaneous cost–benefit analysis, without regard to the dynamic uncertainties of disease diffusion. The dynamics of the spread of pandemic and its interactions with resource allocation decisions can affect the future spread of the pandemic. Future research should address this limitation by developing a disease model that interacts with the resource allocation model over the duration of the pandemic.

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Hina Arora received her PhD in Information Systems from Arizona State University in 2009. She has over nine years of industry and research experience, and has worked for IBM and the Center for Excellence in Document Analysis and Recognition (CEDAR) at SUNY Buffalo. She is currently a Program Manager at Microsoft. Her research interests include security, resource optimization, and information supply chains. T. S. Raghu is Associate Professor of Information Systems in Arizona State University. He received his Ph.D. in Management Information Systems from SUNY Buffalo in 1999. His research interests are in Business Process Change, Knowledge Management and Collaborative Decision Making. He has also worked as a systems consultant at leading international IT consulting firms. His publications have appeared in refereed international journals such as Information Systems Research, Management Science, Journal of Management Information Systems, Decision Support Systems and Decision Science. He currently serves on the Editorial boards of Information Systems Research and Decision Support Systems. Ajay Vinze is the Davis Distinguished Professor of Information Systems at Arizona State University. Prior to joining ASU, he served on the MIS faculty at Texas A&M University. He received his Ph.D. in MIS from the University of Arizona, Tucson in 1988. Dr. Vinze's research, teaching and consulting interests focus on both IS strategy and technology issues. He has worked extensively on topics related to decision support technologies, knowledge management, computer supported collaborative work, and applications of artificial intelligence technology for business problem solving. His publications have appeared in leading MIS journals including Information Systems Research, MIS Quarterly, Decision Sciences, Journal of MIS and various IEEE Transactions. Dr. Vinze interfaces in various capacities with organizations in the US — NASA, Intel, IBM, US Army — ISEC, St. Luke's Episcopal Hospital, Arizona Department of Health Services, and internationally in Argentina, Australia, India, Mexico, New Zealand, Peru, Russia, Saudi Arabia and Trinidad. He is a member of INFORMS, Association of Information Systems and IEEE Computer society.