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Optimizing Epidemic Protection for Socially Essential Workers

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Optimizing Epidemic Protection for Socially Essential Workers Optimizing Epidemic Protection for Socially Essential Workers C. Barrett R. Beckman K. Bisset J. Chen S. Eubank V. S. Anil Kumar {cbarrett,rbeckman,kbisset,jchen,seubank,akumar}@vbi.vt.edu Virginia Bioinformatics Institute Virginia Tech Blacksburg, VA 24061 B. Lewis M. V. Marathe P. Stretz T. DuBois A. Srinivasan {blewis,mmarathe,pstretz}@vbi.vt.edu {tdubois,srin}@cs.umd.edu Virginia Bioinformatics Institute Department of Computer Science Virginia Tech University of Maryland Blacksburg, VA 24061 College Park, MD 20742 ABSTRACT plans [15] offer guidance to businesses and local agencies Public-health policy makers have many tools to mitigate an for continuity-of-operations planning and identifying critical epidemic’s effects. Most related research focuses on the di- infrastructures essential to the country’s ability to provide rect effects on those infected (in terms of health, life, or pro- necessary services. People who perform these services are es- ductivity). Interventions including treatment, prophylaxis, sential because they are hard to replace quickly and in their quarantine, and social distancing are well studied in this con- absence social organization rapidly deteriorates. Reducing text. These interventions do not address indirect effects due the overall prevalence of infections also affords protection to to the loss of critical services and infrastructures when too critical subpopulations; however, it may not be sufficient. many of those responsible for their day-to-day operations fall Sequestering is a social distancing technique that further ill. We examine, both analytically and through simulation, protects an essential subpopulation by placing its healthy the protection of such essential subpopulations by seques- members into small groups isolated from the general pop- tering them, effectively isolating them into groups during an ulation and from each other. Protective sequestration was epidemic. We develop a framework for studying the benefits used effectively during the 1918 pandemic by a few socially of sequestering and heuristics for when to sequester. We also isolated populations [13] and was used to varying effect for prove a key property of sequestering placement which helps centuries [7], but fell out of use as societies became more partition the subpopulations optimally. Thus we provide a tightly integrated. Nevertheless, protective sequestration first step toward determining how to allocate resources be- continues to be an important tool to consider for ensuring a tween the direct protection of a population, and protection ready national defense force [8]. of those responsible for critical services. The logic of protective sequestering may explain its rela- tive absence in epidemiological and public health policy lit- erature. Sequestering a small subpopulation is costly and Categories and Subject Descriptors may have little direct effect on the progress or ultimate J.3 [Life and Medical Sciences]: Health size of an outbreak. Instead, sequestering’s benefits come from the external and indirect effects on availability of ser- Keywords vices throughout the epidemic. These effects are difficult to quantify inside the purview of contagion control and lie public health informatics, optimization, epidemiology outside the scope of a public health system’s responsibilities to contain an epidemic. Thus, assigning resources to pro- 1. INTRODUCTION tect a small subpopulation is difficult to rationalize using Society’s resilience to a severe infectious disease epidemic the prevailing methodology. Rather, sequestering presents a depends on the continued operation of essential services, e.g. complex problem in which socio-technical interdependencies water, sewer, and power systems, health care, food distri- must be considered in integrated social decision-and policy- bution, law enforcement, leadership, etc. Federal response making. The strategies needed to protect a small group of socially essential individuals are different from those intended to re- duce morbidity and mortality in the general public. Con- Permission to make digital or hard copies of all or part of this work for sider quarantine and isolation, which are standard tools for personal or classroom use is granted without fee provided that copies are controlling outbreaks which protect the uninfected by iso- not made or distributed for profit or commercial advantage and that copies lating the infected. Sequestering is the converse: protec- bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific tion of an uninfected subpopulation from the overall pop- permission and/or a fee. ulation during an outbreak. In this context, small groups IHI’12, January 28–30, 2012, Miami, Florida, USA. could be housed in military barracks, a requisitioned public Copyright 2012 ACM 978-1-4503-0781-9/12/01 ...$10.00. school, industrial facilities, and other socially isolatable lo- threshold level of latent infections in the critical population, cations. The size of a group would include the number of sequestering them into densely connected groups may in fact co-located sequestered individuals and associated logistical- produce more total infections than result from leaving them support personnel. Minimizing logistical connectivity to the in the general population. The group sizes and the time to greater population is a practical challenge, but is outside enforce sequestration are both important logistically as well our scope. We make the simplifying assumption that such as socially – from a logistical standpoint, one would like isolation is total. to keep the group sizes large; from a societal standpoint, The need for protective sequestration is perhaps best high- one would like to sequester individuals as late as possible. lighted by considering a military context [9, 12]. In the Our work provides a method for quantifying the trade-off course of its operations, the military creates circumstances between these competing factors. conducive to the spread of infectious disease, as occurred, Throughout our analysis, we make a number of assump- for example, in the 1918 influenza pandemic. Naval ships tions that we state here. First, we assume that individuals and military bases contain personnel in confined settings, who get infected are removed from the sequestered popu- wherein infectious diseases can be easily transmitted. Per- lation in a predictable way. Removal is possible only after sonnel can be called up and assigned in a way that is care- the individual is symptomatic; in case of flu-like diseases, fully designed to minimize their total infections. This con- this usually happens a couple of days after the individual is text also provides authorities the level of control necessary infectious. We incorporate this time into the transmission to effectively sequester a population. We study how early probability p. Second, we assume that epis can be effec- estimates of individual infection-probabilities can help se- tively estimated. epis capture the uncertainty inherent in quester individuals better in such confined settings. The this complex process; the sensitivity studies reported in the results reported here arose from investigations into seques- experimental results section show that our results are fairly tration feasibility and optimization for DoD. Some of our robust. Third, we assume completely mixed groups with findings are also described in [2, 1]. uniform transmission probabilities. Again while not com- We use the SIR model (Susceptible, Infectious, Recovered) pletely true it appears to be a reasonable assumption for of infection [11] over a contact network to develop a formula- flu-like illness. tion of the sequestration problem. Based upon environment and contact network factors before sequestering, each indi- 2. DEFINITIONS AND MODELS vidual has a probability of being latently infected. Once The Optimal Sequestering Problem (OSP) is defined as fol- individuals are placed into groups, those latently infected lows: Given a set M ′ M of socially essential individuals, will become infectious and may spread the epidemic within an upper bound g on⊆ group size and an upper bound r on the group. The transmission probability over time p for any the number of groups, decide: pair depends upon the amount of time until the infectious 1. Triggered Sequestering (TS): the optimal time topt person recovers or is removed as well as the environmental ′ characteristics of the group and epidemic-specific factors. when the set M of individuals should be removed from the population, isolating the subset M ′ M ′′ of M ′ Using this model we discover and prove an important − property: that an optimal placement of people into groups that are infectious and symptomatic, and places people with others of similar estimated probabilities 2. Optimal Allocation (OA): allocate the asymptomatic of infection (epis). We call this the “well-ordered” property subset M ′′ of M ′ into groups of size at most g with the and use it as the basis for an efficient algorithm for opti- total number of groups being no more than r, so that ′′ mal placement. We explore the computational complexity the expected number of individuals Mexp that eventu- of our sequestering algorithm along with some of the asso- ally become infected is minimized. ciated trade-offs between running time and memory usage. Thus we have two subproblems: (i) finding the best time In addition to providing optimal partitions of the critical when the essential subpopulation
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