decision support for containing pandemic propagation file23 decision support for containing pandemic...

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Decision Support for Containing Pandemic Propagation

HINA ARORA, Microsoft CorporationT. S. RAGHU and AJAY VINZE, Arizona State University

This research addresses complexities inherent in dynamic decision making settings represented by globaldisasters such as influenza pandemics. By coupling a theoretically grounded Equation-Based Modeling(EBM) approach with more practically nuanced Agent-Based Modeling (ABM) approach we address theinherent heterogeneity of the “influenza pandemic” decision space more effectively. In addition to modelingcontributions, results and findings of this study have three important policy implications for pandemiccontainment; first, an effective way of checking the progression of a pandemic is a multipronged approachthat includes a combination of pharmaceutical and non-pharmaceutical interventions. Second, mutual aidis effective only when regions that have been affected by the pandemic are sufficiently isolated from otherregions through non-pharmaceutical interventions. When regions are not sufficiently isolated, mutual aidcan in fact be detrimental. Finally, intraregion non-pharmaceutical interventions such as school closures aremore effective than interregion nonpharmaceutical interventions such as border closures.

Categories and Subject Descriptors: H.4.2 [Decision Support]; H.1.1 [Value of Information]; K.4.1 [Pub-lic Policy Issues]; I.6.4 [Model Validation and Analysis]; I.2.8 [Plan Execution, Formation andGeneration]

General Terms: Management, Performance, Measurement

Additional Key Words and Phrases: Dynamic decision making, resource allocation, multiagent simulation,public health, pandemics

ACM Reference Format:Arora, H., Raghu, T. S., and Vinze, A. 2011. Decision support for containing pandemic propagation. ACMTrans. Manag. Inform. Syst. 2, 4, Article 23 (December 2011), 25 pages.DOI = 10.1145/2070710.2070714 http://doi.acm.org/10.1145/2070710.2070714

1. INTRODUCTION

Dynamic decision making problems require a series of interdependent decisions in realtime to maximize decision making performance. The problem environment is constantlyevolving as a consequence of the decision maker’s actions and through the influence ofevents otherwise considered independent of the situation at hand. Dynamic decisionmaking is characterized by the following properties [Edwards 1962; Brehmer 1992]: (a)a series of decisions is required to reach the goal, (b) the decisions are interdependent(later decisions are constrained by earlier decisions), (c) the state of the decision prob-lem changes, both autonomously and as a consequence of the decision maker’s actions,and (d) the decisions have to be made in a real-time environment.

Dynamic decision making problems typically involve two sets of evolving factors—one disruptive, the other restorative—and the objective is to minimize the total dis-ruption and facilitate effective situational performance. For instance, in the case of a

Authors’ addresses: H. Arora, Program Manager, Microsoft Corporation; T. S. Raghu and A. Vinze (corre-sponding author), W.P. Carey School of Business, Arizona State University, AZ; email: [email protected] to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrights forcomponents of this work owned by others than ACM must be honored. Abstracting with credit is permitted.To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of thiswork in other works requires prior specific permission and/or a fee. Permissions may be requested fromPublications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or [email protected]© 2011 ACM 2158-656X/2011/12-ART23 $10.00DOI 10.1145/2070710.2070714 http://doi.acm.org/10.1145/2070710.2070714

ACM Transactions on Management Information Systems, Vol. 2, No. 4, Article 23, Publication date: December 2011.

23:2 H. Arora et al.

natural disaster such as Hurricane Katrina, the hurricane constitutes the disruption.Resources such as shelters, water, food, and medical aid constitute restorative mea-sures. And the interaction of disruptive and restorative measures is played out throughthe different states the people go through, such as affected, not affected, sheltered,saved, and dead. Similarly, in the case of large-scale epidemics, the epidemic consti-tutes the disruption. Pharmaceutical interventions such as antivirals and vaccines,and non-pharmaceutical interventions such as school and border closures constituterestorative measures. The interaction of disruptive and restorative measures is playedout through the different states the people go through, such as susceptible, exposed,infected, treated, untreated, dead, and recovered. A third example is a network outage,where the outage constitutes the disruption. Resources such as administrators andbackup/failover servers constitute restorative measures. The interaction of disruptiveand restorative measures is played out through the different states the services gothrough, such as unavailable, restoring, and available.

Characterizing dynamic decision making problems requires modeling the disrup-tive and restorative factors, and their interaction through agent state transitions.While model-based approaches are known to be extremely effective [Gonzalez 2005;Huguenard and Ballou 2006; Sengupta and Abdel-Hamid 1993], tracking and solvingdynamic decision making problems is nontrivial [Radner 2000; Brehmer 1992; Lerchand Harter 2001]. Extant literature focuses primarily on either disruption [Andersonand May 1982; Lloyd and May 1996; Carley et al. 2006], or restoration [Craft et al.2005; Ferguson et al. 2006; Wong et al. 2006; Wee and Dada 2005; Tagaras and Cohen1992; Meltzer et al. 1999]. The typical orientation has been to focus on effectiveness andimpact of a specific restoration strategy. Since decision makers use multiple restorationstrategies it is important that interactions and combined effects be taken into consid-eration. This is particularly the case for dynamic decision making environments dueto the heterogeneity of interactions between agents, entities, and/or regions.

The main objective of this research is to model the disruptive and restorative factorsin the context of an influenza pandemic, and study their interactions in a dynamicdecision making environment. In the influenza pandemic context, the dynamic decisionmaking problem arises due to the complex interactions between disruptive diseaseprogression and restorative measures such as pharmaceutical, non-pharmaceuticalinterventions, and mutual aid. Given the complexity of the problem, the model isdeveloped and validated using both equation-based and agent-based approaches.

This research makes three significant contributions.

—An equation-based mathematical model is developed to capture the interaction be-tween disease propagation, pharmaceutical, and non-pharmaceutical interventions,and mutual aid.

—The equation-based model is extended to a multiagent simulation environment thataddresses the heterogeneity in this dynamic decision making environment.

—Using the results from the two approaches, the following key policy implications forpandemic containment are extrapolated:—Containing the progression of a pandemic is a multipronged approach: a combi-

nation of Pharmaceutical Interventions (PIs) such as antiviral-based treatment,and Non-Pharmaceutical Interventions (NPIs) such as border and school closuresis needed.

—Mutual aid is effective only when regions that have been affected by the pan-demic are sufficiently isolated from other regions through non-pharmaceuticalinterventions. When regions are not sufficiently isolated, mutual aid can in factbe detrimental. It is also noted that the recommended level of mutual aid has an


Decision Support for Containing Pandemic Propagation 23:3

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

interaction effect with the level of effectiveness of non-pharmaceutical interven-tions such as border and school closures.

—Intraregion non-pharmaceutical interventions such as school closures are more ef-fective than interregion non-pharmaceutical interventions such as border closures.

The article is organized as follows: Section 2 reviews the problem context and relevantliterature. Section 3 describes the model development process used in this article.Section 4 describes the equation-based dynamic disease propagation model. Section 5describes the agent-based dynamic decision making environment. Section 6 concludes.

2. DEFINING THE PROBLEM CONTEXT

This research studies dynamic decision making problems in the context of an influenzapandemic. Intervention in a pandemic situation has salient characteristics of dynamicdecision making: (a) a series of interrelated decisions such as vaccination policy, antivi-ral allocation, and relative urgency; (b) complex interactions between the disruptiveaspects of the environment and the restorative actions undertaken by decision makers;and (c) autonomous changes to the decision environment due to virus mutations andother natural factors.

The economic effects of pandemics are dramatic, and could be as great as the averagepostwar recession [Eakin 2005]. The World Health Organization defines six phases(Figure 1) of pandemic influenza [CDC Influenza Pandemic Operation Plan 2006].The H1N1 Flu pandemic, for instance, presents an example of how these phases arecharacterized. This event led to between 14 and 34 million infections and between2,500 and 6,100 deaths in the United States [CDC H1N1 Estimates 2009].

Epidemics are typically classified using the basic SEIR (Susceptible-Exposed-Infected-Recovered) model [Anderson and May 1982]. As illustrated in Figure 2,infectious individuals spread the disease to the (nonimmune) susceptible population.Those in the susceptible population to which the disease is transmitted become exposedand 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



Fig. 2. An antiviral-based intervention program, and its effect on health outcomes.

Table I. Impact of Pandemic Influenza (reproduced from the Department of HHSReport [2005])

Characteristic Moderate (1958/68 –like) Severe (1918-like)Illness 90 million (30%) 90 million (30%)Outpatient medical care 45 million (50%) 45 million (50%)Hospitalization 865,000 9,900,000ICU care 128,750 1,485,000Mechanical ventilation 64,875 742,500Deaths 209,000 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 interventionsnot available during the 20th century pandemics.

recover (with immunity) or die. Since epidemics of contagious diseases such as in-fluenza progress exponentially, once it sets in, it spreads rapidly (Table I). Timelyrestorative measures such as pharmaceutical and non-pharmaceutical interventionscan help contain the epidemic. Pharmaceutical interventions include antiviral-basedprophylaxis or vaccine-based immunization (to reduce the number of susceptible) andantiviral-based treatment (to help the infected recover). Non-pharmaceutical interven-tions include measures such as community closures and border closures in order toreduce the number of contacts between the susceptible and infected population. Theseare typically used in addition to pharmaceutical interventions to combat epidemics.

While vaccines provide the best line of defense, they can only be manufactured whenthe pandemic virus has been identified. One would therefore have to resort to antivi-rals, both to provide immunity (prophylaxis) and for treatment purposes. Based oncurrent clinical evidence, treatment will require one course (10 pills) of antivirals, andprophylaxis will require four courses of antivirals [Department of HHS Report 2005].The U.S. Department of Health and Human Services has made antiviral preparednessrecommendations for the scenario where 30% of the U.S. population will need somekind of medical attention. However, the federal stockpiles have enough to cover only



about 25% of these cases, that is, 44 million courses [Antivirals—State Allocations2008]. Therefore, available federal stock of antivirals is allocated to states based onpopulation. Due to the limited stock of antivirals, it is imperative that they be optimallyallocated to combat the epidemic.

Surge capacity is a health care system’s ability to expand quickly beyond normalservices to meet an increased demand [AHRQ Report 2004]. The healthcare system canimprove surge capacity in two ways: (1) maintaining redundant capacity in centrallylocated stockpiles, and/or (2) reallocating and redirecting existing capacity throughmutual aid (transshipment) at the regional level. The choice between use of redundantcapacity and mutual aid at the regional and local level is not an obvious one. It has beendeemed one of the major challenges in emergency preparedness [Havlak et al. 2002;Bravata 2004]. Also, while extant literature has studied multiple restoration strategiessuch as pharmaceutical and non-pharmaceutical interventions [Ferguson et al. 2006;Mniszweski et al. 2008], the interaction between these interventions and mutual aidhas not been explored.

While DSS technology has evolved significantly in the last three decades [Shim et al.2002], understanding of dynamic decision making tasks and decision support for thisclass of problems remains limited [Gonzalez 2005]. The limitations of bounded ratio-nality are even more pronounced in dynamic decision making tasks. Feedback delaysand poor feedback quality further exacerbate performance. Situational awareness ina dynamic decision making environment requires two overlapping cognitive activities[Lerch and Harter 2001] that compete for the decision maker’s attention: (a) monitor-ing or tracking of key system variables for information regarding present and expectedconditions, and (b) control or the generation, evaluation, and selection of alternativeactions that can change the system. Control can be achieved through feedback (select-ing an action based on current system information), or feedforward (selecting actionsbased on a predicted future state of the system). While feedforward can improve de-cision quality, decision makers are more likely to use available feedback control thanfeedforward, since the former requires less cognitive effort. Monitoring and feedbackcan be improved through improved collection, processing, and delivery of information.Feedforward can be improved through modeling and simulation aids that project futurestates of the system.

Three different decision support mechanisms are therefore prevalent in the literaturefor dynamic decision making [Gonzalez 2005; Huguenard and Ballou 2006]: (a) outcomefeedback, in which decision makers are provided with feedback on the performanceresults of their decisions; (b) cognitive feedback, in which decision makers are giveninstructions on how to perform the decision task; and (c) feedforward, where decisionmakers are provided with the models and tools to perform “what-if” analysis of potentialdecisions. It has been shown that outcome feedback alone is an ineffective form ofdecision support [Lerch and Harter 2001; Gonzalez 2005]. However, cognitive feedbackand feedforward in combination with outcome feedback results in better performance[Sengupta and Abdel-Hamid 1993].

Two feedforward modeling approaches have been used extensively in the study ofpandemics: Equation-Based Models (EBMs) and Agent-Based Models (ABMs). Com-plex problem domains such as pandemics are made up of two kinds of entities [Parunaket al. 1998]: agents and observables. Agents are typically characterized by one or moreof the following characteristics: autonomy, knowledge, beliefs, resources, physical capa-bility, sensory ability, and information processing ability. For instance, viruses, people,and organizations can all be modeled as agents. Observables are measures of interestthat can either be associated with individual agents or with collections of agents. Anexample of an observable at the agent level is the disease stage the agent is in (suchas susceptible, exposed, infected, and recovered) while an observable at the collection



level is the proportion of population that is susceptible. Therein lies the basic differencebetween EBMs and ABMs. EBMs capture the relationship between observables, andthe evaluation of the model produces the evolution of the observables over time. Theagent-level behavior itself is aggregated away. On the other hand, ABMs capture therelationship between agents, and the emulation of agent behavior leads to the emer-gence of the observables over time. In other words, the observables are an output ofthe system, rather than an input to the system.

The choice between the two approaches is based on the requirements of the problemdomain under consideration. While EBMs are mathematically tractable and theoret-ically grounded, they tend to use aggregates of critical system variables and assumehomogeneity. Heterogeneity can be incorporated into EBMs, but only to a limited de-gree, since it increases the complexity of the model and can make it mathematicallyintractable. On the other hand, ABMs allow for heterogeneity in the problem environ-ment; however, they tend to be extremely resource intensive, and are not mathemati-cally tractable. Also, ABMs require the specification of many more parameters in orderto capture agent behavior. While this creates a model that is much more realistic, italso adds to the computation complexity in terms of sensitivity analysis and modelvalidation. In principle, however, EBM and ABM approaches are complementary. TheEBM approach provides a theoretically justifiable baseline from which ABMs can becalibrated, whereas ABMs can extend the insights obtained from EBM by modelinga more realistic decision environment. Therefore, we use a combination of these ap-proaches in our research as detailed in Section 3.

3. MODEL DEVELOPMENT

To model the dynamic decision making environment for influenza pandemics, twodistinct subproblems have to be addressed. The first subproblem is defining an appro-priate human-disease interaction environment (the disruptive factor). The second subproblem is defining an appropriate disease intervention model in a multiregion setting(the restorative factor).

Equation Based Model (EBM). Epidemiological theory provides the basis for studyingcommunicable diseases using the SEIR model [Anderson and May 1982]. A set of dif-ferential equations models the spread of the disease. The basic model is homogeneousin that it considers the spread of the disease over time through the population as awhole. In reality, however, different pockets of a population can have very different lev-els of susceptibility to infected population interactions. Lloyd and May [1996] extendedthe basic SEIR model to include spatial heterogeneity by dividing the heterogeneouspopulation into homogeneous groups. While each homogeneous subpopulation has itsown disease transmission rate, it is also affected by transmission rates of other sub-populations as specified by a coupling coefficient matrix.

Both the homogeneous and heterogeneous SEIR models have been extended in priorliterature to include resource allocation. The homogeneous SEIR model has been ex-tended to include vaccination [Anderson and May 1982], and antiviral-based treatmentand prophylaxis [McCaw and Vernon 2007]. However, because of the homogeneity as-sumption, these studies do not capture the variation in resource requirements acrossthe population resulting from what is in fact a heterogeneous spread of the pandemic.Brandeau et al. [2003] use a simplified optimization model, capturing only the S andI states of the traditional SEIR model, to arrive at an optimal allocation of resourcesbetween a set of noninteracting populations. A quality-adjusted-life-years-based cost-effectiveness measure is used to capture the benefit of health interventions. Addition-ally, a binding budget constraint is used in the optimization model. However, since thefederal government will likely spend as much money as might be required in containing



the spread of the pandemic, the budget constraint is eliminated in our research. Also,since cost-effectiveness measures are known to be subjective and hard to quantify, thecost benefit-based resource allocation approach [Arora et al. 2010] will be applied inthis article.

The disease model developed in this article not only captures all four levels of theSEIR model, but also extends it to include the three treatment stages. This allows usto capture the restorative effects of interventions on the population. Thus, the diseasemodel developed here addresses the dynamics of disruptive and restorative aspectsof the decision environment more completely by integrating both disease propagationand resource optimization in a single model. A six-stage disease model grounded inepidemiological theory [Anderson and May 1982] is the main contribution of the EBMmodel. The six-stage disease model makes certain simplifying assumptions about thehomogeneity of the population and the spread of the disease through it. This simpli-fication allows for the development of an interaction strategy between the resourceallocation model and the disease environment.

Agent-Based Model (ABM). To address limitations of the equation-based model, dis-cussed previously, we extend the model to a multiagent environment that takes theheterogeneous and dynamic aspects of population movement into consideration. Thismultiagent simulation captures the spread of diseases through communities. As in theequation-based model given earlier, agents move through six possible disease stagesin the multiagent environment. However, the spread of the disease happens as a nat-ural consequence of the interaction among the human agents in the population. Theinfected population interacts with the susceptible population based on individual affil-iations to communities (such as schools, colleges, offices) and time of day (day, evening,night), and the disease spreads as a natural consequence of the interaction of thesetwo populations. The disease model periodically interacts with the resource allocationmodel developed over the duration of the pandemic. Allocations are made based on in-stantaneous (dynamic) requirements as determined by the disease propagation model,and policies are evaluated based on the cumulative effect of the allocations. This re-sults in a decision-support prototype for dynamic decision making problems involvingpharmaceutical and non-pharmaceutical interventions during influenza pandemics.

ABMs have frequently been used to study pandemics [Carley et al. 2006; Fergusonet al. 2006; Mniszewski et al. 2008]. In all of these models, human agents are mod-eled as carriers of disease agents. A disease is introduced into the population, andthe disease is carried and spread as a natural consequence of the interaction amongthe human agents in the population. Interaction among the human agents is basedon the concept of communities such as schools, offices, homes, etc. Observables suchas peak infections, cumulative infections, and duration of pandemic are then capturedand compared across different intervention strategies. Carley et al. [2006] created anABM called BioWar to study disease outbreaks resulting from bioterrorist attacks. Thefocus of their research was to capture the attack model, and determine how the diseasespreads. They also modeled the simultaneous occurrence of multiple diseases. Whilemedicine-based treatment interventions were considered, neither multiple interven-tion strategies nor resource shortages were studied. Ferguson et al. [2006] considerthe impact of pharmaceutical and nonpharmaceutical interventions (such as schoolclosures, border closings, and case isolations) during the simulation of an influenzapandemic across the U.S. They found that the effectiveness of these strategies dependson the timing and extent of the intervention. They also found that within-county NPIssuch as school closures were far more effective than between-county NPIs such asborder closures. School closures were applied on a rolling basis—a school closes if itaccumulates a certain number of infected children—remaining closed for three weeks.



Fig. 3. Three different mutual aid regions considered in this research. California Region VI, Arizona RegionsI and V, with a total of 19 counties.

They do not consider the possibility of mutual aid in their model. Along similar lines,Mniszewski et al. [2008] consider the impact of pharmaceutical (such as antiviral treat-ment and vaccines) and nonpharmaceutical interventions (such as schools closures, andfear-based isolations) during the simulation of an influenza pandemic across five coun-ties in California. Like Ferguson et al. [2006], they also found that school closures canbe an effective strategy in combating the pandemic. School closures were applied at oneshot—all schools close if the total number of infections is above a threshold—remainingclosed for five months. Again, they do not take into account mutual aid in their model.The ABM model developed in this research considers multiple intervention strategies(pharmaceutical and non-pharmaceutical), mutual aid, and both one-shot and rollingschool closures. In summary, ABM provides a more realistic representation of the prob-lem, and better control over the disruptive and restorative aspects of the decisionproblem. The equation-based model and the agent-based extensions are discussed inthe next two sections.

4. EQUATION-BASED DYNAMIC DISEASE PROPAGATION MODEL

The disease propagation model extends the heterogeneous SEIR model to consider sixdisease stages, including three different treatment stages. A schematic of the modi-fied SEIR model is shown in Figure 4 and Table II lists the variable definitions. Themodel captures the spatial heterogeneity of the population by dividing the popula-tion into multiple, interacting regions. The model starts with most of the populationin a susceptible state, with a few of them in the infected and exposed stages. Theinfected population interacts with the susceptible population and exposes them tothe influenza virus. Some proportion of the (untreated) infected population opts for



Fig. 4. Heterogeneous SEIR model with treatment intervention.

Table II. List of Variable Definitions for Equations Based Disease Propagation Model

Variable DefinitionNj Total number of individuals in subpopulation jSj (t) Number of susceptible individuals in subpopulation j at time

instant tEj (t) Number of exposed individuals in subpopulation j at time

instant tIj,U (t) (similarly for Ij,IT and Ij,ET ) Number of untreated infected individuals in subpopulation j at

time instant tC j,U (t) (similarly forC j,IT and C j,ET )

Cumulative number of untreated infected individuals insubpopulation j until time instant t

Rj (t) Number of recovered individuals in subpopulation j at timeinstant t

1/κ Incubation period1/γU (similarly for 1/γIT and 1/γET ) Infectious period for untreated populationR0 j Reproduction number for subpopulation jβii,U = R0i ∗ γU /Si(0) Transmission rate of untreated individuals in subpopulation iβii,IT = βii,U ∗ γU /γIT Transmission rate of ineffective treated individuals in

subpopulation iβii,ET = βii,U ∗ γU /γET Transmission rate of effective treated individuals in

subpopulation iεi j Coupling coefficient between subpopulation i to subpopulation jβi j,U = εi jβii (similarly forβi j,IT and βi j,ET )

Transmission rate of untreated individuals from subpopulationi to subpopulation j

μ j,U (t) =n∑

i=1βi j,U Ii (t) (similarly for

μ j,IT and μ j,ET )

Force of infection for untreated individuals in subpopulation jat time instant t (similarly for ineffective and effective treated)

δU Death rate for untreated populationδIT = δU ∗ γU /γIT Death rate for ineffective treated populationδET = δU ∗ γU /γET Death rate for effective treated populationa0 j Antiviral allocation at beginning of review period for

subpopulation jntix Number of time steps in a review period (a review period is

time between allocations)λ j (t) ={

a0 jntix if a0 > C j,T (t) and I j,U (0) > 00 otherwise

Maximum possible rate from untreated to treated state forsubpopulation j in a review period

p Proportion of population that goes for treatmentq Effectiveness of antiviral



treatment. If resources are available (as determined by the resource allocation model),and the treatment is effective, the infected individuals move to the effective treatedinfected stage. If the treatment is ineffective, the infected individuals move to the in-effective treated infected stage. Individuals can move from any of the infected stagesto the recovered or dead stages at rates that are dependent on the infectious periods ofeach of those states. The disease propagation model therefore captures the spread ofthe pandemic, creating a demand for resources.

The disease model periodically interacts with the resource allocation model overthe duration of the pandemic. Allocations are made based on instantaneous (dy-namic) requirements as determined by the disease propagation model. In additionto pharmaceutical interventions such as antiviral-based prophylaxis and treatment,non-pharmaceutical interventions such as border closures and their interaction withmutual aid are also explored.

The disease propagation model and the resource allocation models interact throughthe maximum rate of transfer from the untreated infected state to the treated infectedstate, which is limited by resource availability. As mentioned before, one unit of resourceis required for the treatment of one untreated infected individual. If allocations aremade periodically every ntixtime steps, and a0 j units of resources are allocated eachperiod, then, the maximum rate of transfer from the untreated infected state to atreated infected state is λ j = (a0 j/ntix). Also, the effect of the treatment is to reducethe infectious period. The reduction is greater for those individuals for whom thetreatment was effective. This in turn affects the rate of recovery, rate of death, andforce of infection from these different states.

The set of differential equations for each interacting region (subpopulation) j areshown next. For a problem space of 20 interacting regions, the 20 × 10 differentialequations would need to be simultaneously solved.1

dSj(t)dt

= −μ j,U Sj(t) − μ j,IT Sj(t) − μ j,ET Sj(t) (1)

dEj(t)dt

= μ j,U Sj(t) + μ j,IT Sj(t) + μ j,ET Sj(t) − κEj(t) (2)

dIj,U (t)dt

= κEj(t) − pλ j(t) − γU Ij,U (t) − δU Ij,U (t) (3)

dIj,IT (t)dt

= pλ j(t)(1 − q) − γIT Ij,IT (t) − δIT Ij,IT (t) (4)

dIj,ET (t)dt

= pλ j(t)q − γET Ij,ET (t) − δET Ij,ET (t) (5)

dRj(t)dt

= γU Ij,U (t) + γIT Ij,IT (t) + γET Ij,ET (t) (6)

dDj(t)dt

= δU Ij,U (t) + δIT Ij,IT (t) + δET Ij,ET (t) (7)

dCj,U (t)dt

= κEj(t) (8)

dC j,IT (t)dt

= pλ j(t)(1 − q) (9)

1The disease propagation model was implemented in Mathematica. To address resource allocation inter-action, the solution states of the differential equations were stored in memory before calling the resourceallocation module and then reinitiated based on the expected impacts of the allocations.



dC j,ET (t)dt

= pλ j(t)q (10)

Sj(t) + Ej(t) + Ij,U (t) + Ij,IT (t) + Ij,ET (t) + Rj(t) + Dj(t) = Nj (11)

Results and Discussion. The SEIR model assumes exponential distributions of the incu-bation and infectious periods. The influenza virus has been found to have an incubationperiod of between 1 and 3 days, with a mean of 1.9 days, and an untreated infectiousperiod of between 3 and 6 days, with a mean of 4.1 days [Longini et al. 2005]. Basedon extant literature [Longini et al. 2005; Alexander et al. 2008], the mean effectivetreated infectious period is set to 1.5 days, and the mean ineffective treated infectiousperiod is set to 3 days. The death rate for the Influenza virus is 0.5%. Three differentReproduction Numbers are considered: 2 (high), 1.7 (medium) and 1.4 (low) [Fergusonet al. 2006]. As in Meltzer et al. [1999], it is assumed that 47.1% of the untreatedinfected population seeks treatment.

The pandemic is simulated in the 19 counties by setting different Reproduction Num-bers and preliminary infected and exposed numbers in the three emergency regions.Tamiflu is considered as the antiviral of choice to combat the influenza pandemic. Theinfluenza pandemic is assumed to start in Maricopa County, with 100 infected indi-viduals and 4000 exposed individuals. All the other counties have 1 infected personeach. Data sources used for modeling antiviral allocations for counties, effectivenessof Tamiflu by age group, distances between counties, FedEx shipping rates, expectedhealth outcomes, and dollar savings per health outcomes are replicated from Aroraet al. [2010].2

The coupling coefficient matrix (Appendix Table I) captures inter-county populationinteraction, and is based on worker population movement (Appendix Table VII) andflight data (Appendix Table VIII) from the census. The coupling coefficient matrix wasarrived at by adding the worker movement and flight data numbers, scaling the off-diagonal elements by the diagonal elements, and setting the diagonal elements to 1(to capture homogeneous mixing within counties).

Four different scenarios were investigated using the theoretical model describedbefore (Table III). Scenarios 1, 2, and 3 focus on the restorative strategies, Scenario 4addresses the impact of a more severe epidemic (disruption). The results for each of thescenarios and the key policy implications are summarized in Table IV. The benefits ofsharing resources are largely dependent on which regions are affected by the pandemic.If the smaller counties are affected the most, then having the large counties share theirexcess resources can have a positive effect on total benefits. However, if larger countiesare affected the most, the smaller counties should hold on to their resources. Second,the extent of how much counties should share is dependent on how soon a vaccinecan be made available, or how effectively non-pharmaceutical interventions can beimposed; there is an interaction effect (Figure 5). The counties can afford to share alarge proportion of their resources if they are guaranteed to be rescued by other PIs suchas vaccines, or, if effective between-county NPIs such as border closures are put intoplace so as to isolate the pandemic, and keep it from spreading to their counties. Third,within-county NPIs such as school closures were found far more effective than between-county NPIs. In summary, the theoretical model points to the need for a multiprongedapproach to combating pandemics that includes a combination of pharmaceutical andnon-pharmaceutical interventions. This is in line with what other simulation-basedstudies have found in the past [Ferguson et al. 2006; Mniszewski et al. 2008]. An

2Tables of numbers for each of these variables are available in Arora et al. [2010]. For convenience, thesenumbers will also be available as an online supplement.



Table III. Different Scenarios that Were Simulated for the Equation-Based Model.

FirstReproduction Intervention Prophylaxis Share Treatment Non-Pharma

Scenario Number Time Percentage Proportion Effectiveness Intervention1: Impact of

shareproportion

AZRI: 2.0,AZRV: 1.7,CARVI: 1.4

2 days 0.01 0.0, 0.10,0.20,0.30,0.40, 1.0

0.44 none

2: Impact ofdelayedintervention


20 days 0.01 0.10 0.44 none

3: Impact ofnon-pharmainterventions


2 days 0.01 0.10 0.44 Within couplingreduced by 10%;Between couplingreduced by10–50%;Interaction withmutual aid

4: Impact ofepidemicseverity


2 days 0.01 0.10 0.44 none

important contribution of this theoretical (EBM) model is to show the interaction effectbetween these intervention strategies using a deterministic model.

One of the limitations of the equation-based disease propagation model is that it isstill largely homogeneous within regions. In reality though, interactions between peopleare known to be extremely heterogeneous, and largely dictated by the social networksthat people are affiliated to. Also, the movement of people is based on demographicsand time of day, and is not consistent across time. These limitations are addressed byextending the equation-based disease propagation model to a multiagent environmentthat interacts with the resource allocation model over the duration of the pandemic.Human agents are affiliated to communities and movement across communities isbased on demographics and time of day. Disease transmission varies by communitiesand demographics. Using a multiagent community-based model also allows for thestudy of within-county non-pharmaceutical interventions such as school closures. Thisfinal iteration of the model is discussed next.

5. AGENT BASED DYNAMIC DISEASE PROPAGATION MODEL

A conceptual schematic of the ABM3 model developed in this research is presented inFigure 6. The model includes 7 modules: the simulation space, simulation time, popu-lation model, resource model, disease model, intervention model, and decision support.The simulation space is a two-dimensional space for the movement of agents. Thetwo-dimensional space is divided into grids, where each grid represents a geograph-ical area. While the geographical areas can be at any level of hierarchy: zip codes,counties, states, countries, etc., in this research, grids represent counties. Each countyis assigned a population of agents, households, communities, hospitals, and resourcesbased on census and CDC data.

Simulation time is represented as a tick. A tick can represent any level of hierarchy:seconds, minutes, hours, days, etc. In this research, each tick represents an 8-hourduration, and three ticks make up a day. Agent behavior (what the agent is most likely

3The ABM model in this article was developed using Repast, a Java-based open-source multiagent simulationtoolkit. It provides a library of objects for representing space and time, creating agents, running simulations,data collection, and displaying the environment and results [Collier 2006].



Tabl

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Fig. 5. Interaction effect how much between-county interaction is curtailed and what percentage of excessresources counties are willing to share.

to be doing) is largely dictated by the time of day and demographics. Most children, forinstance, will be in school during the day and at home at night.

The population model captures both communities of human agents, and movementof human agents between communities and grids. Five different communities are con-sidered in this research: households, schools, colleges, offices, and recreation facilities.Human agents are affiliated to certain communities based on their location and demo-graphic. Human agents move between communities based on the time of day, wherethe day is divided into three 8-hr. segments: morning, evening, and night. For in-stance, human agents would mostly spend the morning at work or school, evening inrecreation facilities, and nights at home. Human agents can also move between grids,either for their daily commute or travel. The movements of agents can be confined tocertain communities externally through community shutdowns, border closings, andquarantines.

The resource model captures both the healthcare entities and the usage of resourcesby the human agents. The simulation space has one stockpile. Each grid in the sim-ulation space has a number of hospitals assigned to it based on census data. All thehuman agents in the simulation are affiliated to a hospital based on the proximityof their home location to the hospital. Each hospital is assigned a certain number ofresources based on the preallocation policy from the stockpile. When a human agentgoes to the hospital for treatment, a unit of resource (one course of Tamiflu) is assignedto the human agent. If the hospital does not have any resources, the agent is put on awaiting list. The transshipment policy is set in terms of the frequency and quantity of



Fig. 6. Conceptual model of agent based model.

resources shared by grids that have an excess of resources. As before, transshipmentis done using the optimization model and resources are transferred between hospitalsaccording to the results of the optimization model.4

Disease transmission between human agents follows the SEIR stages. Building fromour theoretical model, the multiagent disease model is heterogeneous even within eachgrid. The likelihood of disease transfer from one human agent to another depends onthe communities they are affiliated with. Within each community, the disease trans-mission rate is based upon the total number of infectious people in that community,and the observed disease spread probabilities within those communities. This allowsfor a much more realistic disease transmission model, and greater control over non-pharmaceutical interventions.

As in our EBM (theoretical model), both pharmaceutical and non-pharmaceuticalinterventions are implemented in the multiagent simulation environment. Pharma-ceutical interventions are done through prophylaxis and treatment using an antiviral.Non-pharmaceutical interventions are done through community closures and quaran-tines (within-county NPI) and border closures (between-county NPI). The multiagentenvironment allows for greater flexibility in studying within-county NPIs since specificcommunities such as schools, colleges, work places, or recreational facilities can be shutdown to understand their effect on pandemic containment.

4The Repast environment sends the optimization model parameters to the Matlab environment. The opti-mization model is then solved in the Matlab environment, and the allocation data is sent back to the Repastenvironment.



Model Instantiation

One grid is assigned per county in the simulation space (except for LA County which isassigned two grids due to its larger population). The simulation space is therefore madeup of 20 grids. Note that the simulation environment is generic enough to support moregrids at any level of hierarchy. We have chosen these 19 counties in order to compare theresults of ABM with the theoretical model. The disease characteristics data from EBMis retained here as well. The incubation and infectious periods are assumed normallydistributed.

Community data was collected from the census. Appendix Tables II through VIpresent statistics for year 2000 pertaining to households, schools, colleges, offices, andrecreation facilities. Total number of households and the average number per householdfor the year 2000 were collected for each of the 19 counties from the census quick factssite [Census: State and County Quick Facts 2011d]. Total number of schools and uni-versities for the years 2000–2001 was collected from the National Center for EducationStatistics [NCES: The Integrated Postsecondary Education Data System 2011]. Thetotal number of schools is the sum of the total number of public and private schools.The total school and university enrollment for each of the 19 counties for the years2005–2007 was collected from the census fact finder site [Census: School Enrollment].Worker data, in terms of the number of establishments and total number of workers,for the year 2000, was collected from the census fact finder site [Census: Business andGovernment Economic Fact Sheet 2011a]. The numbers for the different industrieswere summed to yield a single number per county. Recreation data was collected fromthe census site [Census: Parks, Recreation and Travel 2011c]. Three kinds of recreationwere considered: sports, movies, and eating out. National monthly estimates for sports,movies, and eating out were used to estimate per-county daily numbers for recreation.Movement data was collected from the department of transportation sites. AppendixTables VII and VIII present data for year 2000 pertaining to worker movement andair traffic between the counties. All of the population, community, and resource datais scaled by 100 to accommodate the agents in the simulation space and reduce thecomputational complexity of the model.

Scenarios Considered

Two different “what-if” scenarios pertaining to school closures are explored to demon-strate the viability of the ABM. Recall that we were unable to explore within-countyNPIs such as school closures in the theoretical model because EBMs do not lend them-selves to that level of heterogeneity. The two scenarios explored in our simulation areas follows: (a) Are rolling school closures as implemented in Ferguson et al. [2006]more effective, or are one-shot closures as in Mniszewski et al. [2008] more effective?(b) What is the optimal duration of school closures; are short, frequent closures aseffective as long, one-shot closures?

The Closure Policy captures the community closure policy for within-county non-pharmaceutical interventions. It is used to simulate the two “what-if” scenarios de-scribed before. The policy has four constituent parts.

(a) Closure Threshold is used to capture the number of untreated infected people in agrid that will trigger community closures.

(b) Closure Duration is used to capture how long the communities will remain closedonce triggered to close.

(c) Community Closure Participants is used to capture which communities will partic-ipate in the closures.

(d) Grid Closure Participants is used to capture which grids will participate in theclosures, just the grid that exceeded the threshold (trigger), or all grids.



For instance, the closure policy could be that when a grid exceeds a threshold of 100untreated infected people, schools and colleges in that grid will be closed for a two-weekduration (trigger). Another policy could be that when a grid exceeds a threshold of 100untreated infected people, schools and colleges in all grids will be closed for a two-weekduration (all). Table V details the different scenarios that were simulated. For eachscenario, a pandemic is simulated in the 19 counties by introducing a small number ofexposed individuals in Maricopa County (each scenario is replicated 50 times to enablestatistical analysis). The simulation is then allowed to run, and the disease starts tospread as a natural consequence of the interaction among the human agents. Based onthe closure policy, the ABM closes and reopens communities in different grids.

Results and Discussion

Table V summarizes the results for each scenario, and policy implications of the dif-ferent scenarios. The first one, “Closure Policy—Trigger”, is set such that when a gridexceeds a threshold of 100 untreated infected people, it will close all schools and col-leges for the specified duration. In other words, different grids will close at differenttimes. Note that schools and colleges could close and reopen multiple times during thecourse of the pandemic depending on how often the threshold is exceeded when thecommunities are open. Three different closure periods were considered for this policy:2 weeks, 4 weeks, and infinite weeks (indefinite closure until pandemic dies out com-pletely). As can be seen, the greater the number of weeks the closure happens for, thelower the cumulative number of infections. In other words, indefinite closure is betterthan 4 weeks; and 4 weeks is better than 2 weeks. However, since schools may closeand reopen and close again based on how often they cross the threshold, it is importantto look at the total amount of closure duration. The table tracks the total number ofweeks that schools remained closed in Maricopa County (note: all the other countieswere closed for a lesser duration than Maricopa). As can be seen, when the total dura-tion of school closures is taken into account, the indefinite closures actually ended upwith the lower total duration than the 4-week closure case (20 weeks when comparedto a total of 40 weeks in the 4-week closure case), and it has the lowest cumulativenumber of infections. In other words, a more stringent policy of school closures wouldactually help bring down the cumulative infections dramatically. Therefore “rolling”school closures as suggested by Ferguson et al. [2006] is really effective only if theclosures are done over a five-month duration.

The second policy, “Closure Policy—All”, is set such that when a grid exceeds athreshold of 100 untreated infected people, all grids will close their schools and collegesfor the specified duration. In other words, all the grids will close at the same time. Notethat schools and colleges could close and reopen multiple times during the course of thepandemic depending on how often the threshold is exceeded when the communities areopen. Three different closure periods were considered for the second policy: 2 weeks,4 weeks, and 8 weeks. Three things become immediately apparent from the results.First, the “Closure Policy—All” is more effective than the “Closure Policy—Trigger” fora given closure period. In other words, it is essential that all counties take precautionsin tandem for the closure policies to be most effective. Second, the greater the number ofweeks the closure happens for, the lower the number of cumulative infections. Third, aclosure of 8 weeks ends the pandemic almost immediately. Therefore, taking stringentaction early in the pandemic is the most effective. Again, total closure durations aretracked for Maricopa County. As can be seen, when the total duration of school closuresis taken into account, the 8-week closures actually ended up with the lowest totalduration (8 weeks when compared to a total of 96 weeks in the 4-week closure case),and it has the lowest cumulative number of infections. Therefore, a more stringentpolicy of school closures would actually help bring down the cumulative infections



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Fig. 7. School closures for a 4-week duration results in multiple peaks. The peaks are more pronouncedwhen the schools are closed all at once (top) rather than on a rolling basis (bottom).

dramatically, and result in the lowest total closure duration. The simulation thereforesuggests that it is far better to have a stringent policy of 8-week long closures, wherethe closures happen all at once, rather than having rolling closures as suggested byFerguson et al. [2006]. Results from the ABM also suggest that 8 weeks of closuresall at once is sufficient in ending the pandemic, rather than the 5 months of closuressuggested by Mniswezski [2008].

Notice that peak infected numbers are reported but are not used to compare resultsacross the different cases in this scenario. This is because school closures result inmultiple peaks; the pandemic dwindles down as a school is closed, and then picks backup again as the schools are reopened. The peaks are more pronounced for the casewhere all schools are closed all at once, when compared to the case where schools areclosed on a rolling basis (Figure 6). It is therefore infeasible to compare results basedon peaks.

We also explored the interaction effect between school closure NPI and mutual aid.Transshipment was considered for the 2-week and 4-week closure periods for both



closure policies. Indefinite closures and 8-week closures are not considered in this sce-nario, since the pandemic in these cases does not get severe enough to require mutualaid, thereby preventing an interaction effect. There is an interaction effect betweenwithin-county non-pharmaceutical interventions and transshipment. At lower levelsof community closures, transshipment can actually be detrimental, since other coun-ties remain vulnerable to pandemic spread. However, at higher levels of communityclosures, transshipment is beneficial or not detrimental, since other counties are thensufficiently isolated from the pandemic. This is borne out in the results, where trans-shipment is detrimental (higher cumulative infections) for lower levels of communityclosures (closure period of 2 weeks), whereas transshipment is beneficial (lower cumu-lative infections) for higher levels of community closures (closure period of 4 weeks).

While Arora et al. [2010] provided a baseline for comparison by focusing on the devel-opment of a resource allocation model, the theoretical (EBM) model developed in thisarticle focused on the interaction between the resource allocation model and a partiallyheterogeneous disease model. The simulation produces an instantiation of an ABM thatcan eventually be developed into a decision support system that decision makers canuse to make resource allocation decisions in a fully heterogeneous and dynamic dis-ease environment. An interesting finding from the ABM is that contrary to what issuggested by extant literature, persistent and extended use of non-pharmaceutical in-terventions is much more effective than the use of non-pharmaceutical interventionsover shorter durations of time.

It is important that model results be interpreted in light of its comparisons to real-lifecontext and other related studies. Since it is difficult to validate results of the simulationagainst a real pandemic for obvious reasons, we provide the following clarifications onexternal and internal validity of the model results.

Internal Validation. Validity of multiagent models can be a major concern because ofthe large number of model parameters. While we based parameters on extensive datacollected from Census bureau and other sources, we also ensured that the multiagentmodel was internally consistent with the theoretically-based modified SEIR model forbaseline scenarios as described below shortly.

The agent-based model was validated against the theoretical model by running thebaseline case of no non-pharmaceutical interventions and no mutual aid across the 20grids. The peak number of infected people in the agent based model was approximately0.7 million and occurred around the 130th day, with a cumulative number of infected atthe end of the epidemic being approximately 8 million. By contrast, the peak numberof infected people in the theoretical model was approximately 0.6 million and occurredaround the 90th day, with a cumulative number of infected at the end of the epidemicbeing approximately 14 million. In other words, while the peak numbers of infectedpeople were comparable, the pandemic seemed to produce fewer infected people pertime step in the agent-based model. This is partly expected since the theoretical modelis largely homogeneous; that is, people within a county were equally likely to infectother people in the county. In reality, However, people are likely to spread the infectionwithin their own communities (as in the agent-based model). The agent-based modeltherefore would have slower spread of the pandemic, and brings down the number ofnew infections per time step. Increasing the community-based infection rates escalatedthe cumulative infection numbers and made them comparable to those from the theo-retical model. The next step in model validation was external validity. We describe ourapproach to ensuring external validity next.

External Validation. Only a limited amount of data is available from the 1918, 1957,and 1968 influenza pandemics. Based on this historical data and estimated diseasetransmission rates, it is predicted that a modern day influenza pandemic in the UnitedStates 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 couldbe as great as the average postwar recession [Eakin 2005]. In a separate report, Depart-ment of Health and Human Services in 2005 estimated that the pandemic could resultin 90 million (i.e., about 30% infection rate) infected and 206,000 to 1.9 million deaths.More recently, Andradottir [2011] simulated pandemic scenarios for a Canadian cityand estimated a 34% infection rate when no interventions were attempted. Reed et al.[2009] attempted to evaluate the impact of 2009–2010 H1N1 epidemic and concludedthat the number of infections in the U.S. was in the range of 1.8 million to 5.7 million.Thus, with all the interventions diligently applied, the 2009 infection rate in the pop-ulation was within a 2% rate for the U.S. population. In addition, the attack rate andvirility of H1N1 was considered quite moderate when compared to the virility of earlierpandemic strains. Interestingly, the 2% rate estimation is similar to the simulationestimations of Andradottir [2011] for a single city, where they predicted that infectionrates can be brought down to 0.2%–2% range when a full range of interventions areapplied.

Our simulation results are well within these ranges and therefore there is strongsupport for external validity of the model. For example, the theoretical model predicts14 million cumulative infections when there are no non-pharmaceutical interventionsand the agent-based model predicts an average of 8 million. This is equivalent toapproximately 12%–20% infection rate. This is comparable to the ranges reported byDepartment of HHS study and Andradottir [2011], especially considering the multigridapproach developed in this manuscript. When the full range of nonpharmaceuticalinterventions are applied, our estimations for infection rates are near the 1% range;this is consistent with both Andradottir [2011] and Reed et al. [2011] estimates.

6. CONCLUSION

This article provides a novel approach for modeling the interaction between disruptivefactors and restorative measures in a dynamic decision making environment such asa pandemic.

We provide three significant extensions to the existing literature.

—First, extant literature predominantly focuses on either disruption [Anderson andMay 1982; Lloyd and May 1996; Carley et al. 2006], or restoration [Craft et al. 2005;Ferguson et al. 2006; Wong et al. 2006; Wee and Dada 2005; Tagaras and Cohen 1992;Meltzer et al. 1999]. A limited amount of literature attempts to study the interactionof disruptive and restorative factors [McCaw and Vernon 2007; Brandeau et al. 2003],but explores these factors and interactions using equation-based models which makecertain simplifying assumptions about the heterogeneity of interactions. In contrast,in this article, while the equation-based model provides the theoretical underpin-nings of a six-stage disease model, the multiagent extension provides a completelyheterogeneous setting for the interaction between restorative and disruptive factors.

—Second, while prior works have studied multiple restoration strategies such as phar-maceutical and non-pharmaceutical interventions [Ferguson et al. 2006; Mniszweskiet al. 2008,; McCaw and Vernon 2007], the interaction between these interventionsand mutual aid [AHRQ Report 2004] has not been explored. In this article, we ad-dress this interaction and derive important policy implications from it.

—Third, we extend earlier efforts which typically take either the EBM approach [Lloydand May 1996; Meltzer et al. 1999; Brandeau et al. 2003; McCaw and Vernon 2007],or the ABM approach [Carley et al. 2006; Ferguson et al. 2006]. Leveraging thecomplementary nature, we present an approach to combine the benefits of EBM andABM. We first develop the EBM, thereby grounding the choice of our disease model



in theory. Next we use an agent-based environment to address the heterogeneity inthe decision space.

Three significant contributions result from our efforts.

—First, an equation-based mathematical model is developed to capture the interactionbetween disease propagation, pharmaceutical, and non-pharmaceutical interven-tions, and mutual aid.

—Second, the equation-based model is extended to a multiagent simulation environ-ment that captures the heterogeneous and dynamic decision making environment.

—Third, the key policy directives for pandemic containment emerge.—The most effective way of checking the progression of a pandemic is a multipronged

approach: a combination of Pharmaceutical Interventions (PI) such as antiviral-based treatment, and Non-Pharmaceutical Interventions (NPIs) such as borderand school closures.

—The recommended level of mutual aid has an interaction effect with the levelof effectiveness of non-pharmaceutical interventions such as border and schoolclosures. Mutual aid is effective only when regions that have been affected by thepandemic are sufficiently isolated from other regions through non-pharmaceuticalinterventions. When regions are not sufficiently isolated, mutual aid can in fact bedetrimental.

—Intraregion non-pharmaceutical interventions such as school closures are more ef-fective than interregion non-pharmaceutical interventions such as border closures.

Future research will have to focus on improving the population model, especially interms of agent movement between counties. More research also needs to be done inorder to determine better estimates of transmission probabilities within various com-munities. School closures would result in children spending more time in other com-munities, and those patterns need to be studied and incorporated into the model. Ingeneral, communicable diseases share a number of characteristics that can be pa-rameterized. For example, incubation period, spread probability, length of infection,vaccination, antiviral effectiveness, and mortality rates have been studied and docu-mented for a number of different communicable diseases. We have built the models tobe parameterized against these important characteristics. Therefore, it is potentiallypossible to study different disease characteristics using this model. However, the find-ings reported in the manuscript are specific to influenza. Therefore, we caution againstgeneralizing the study findings to other communicable disease contexts.

Several extensions to both the disruptive and restorative aspects of the model arealso possible. The resource allocation model can be extended to include the injection ofadditional resources at certain points in the simulation. Additional restorative mecha-nisms with a different impact on the disruptive aspects of the model can be introduced.For example, vaccines can be introduced after a 4-6 month time period to alter diseaseprogression through the population. A third extension can explore the allocation ofresource bundles. For instance, transferring food from one region to another requirestrucks and drivers, just as administering vaccinations requires nurses. The allocationof one element of the bundle without the availability of the other is futile. Since de-mand surge situations are invariably accompanied by resource and labor scarcity, suchbundling constraints can add another layer of complexity to resource allocation deci-sions and affect decision-making performance. Finally, resource distribution challengeson decision making performance can also be explored.

ELECTRONIC APPENDIX

The electronic appendix to this article is available in the ACM Digital Library.



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Received November 2010; revised September 2011; accepted November 2011


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