Modeling targeted layered containment of an influenza pandemic in the United States

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Fig. 5. Variability of VBI model over 32 runs, scenario 2, with a threshold of 1%, and R₀ = 2.1. The X denotes the mean, and the bar denotes one standard deviation on each side.

Fig. 6. Relative contributions of each activity type in the VBI model to interhousehold transmission, no intervention and scenario 2.

Fig. 7. Sensitivity to school closing threshold. Scenario 2 and three R₀ values, with only the school-closing threshold being varied between 10% and 0.0001% cumulative illness attack rates. Chicago population.

Table 5. UW/LANL model results

Mean of five realizations (standard deviation). Illness attack rates in first row; courses per 1,000 people in the second row.

SI Text

Description of the Three Models

The social structure of each of the three models was constructed differently, with implications for the effects of interventions. In the UW/LANL model, described in ref. 1, the population is geographically distributed among census tracts to closely represent variation in actual age and spatial distribution according to publicly available 2000 U.S. Census data (2). Each tract is in turn organized into 2,000-person communities as in refs. 3-5. The household size distribution corresponds to that in the 2000 U.S. Census data. There are eight mixing groups within which individuals can associate and be infected by others in these groups. Each person belongs to a household, neighborhood, a cluster of four households, and a community. School children go to primary school, middle school, or high school. Preschool children go to small day care centers or play in small play groups. Most adults go to work at workplaces. The neighborhoods and communities in this model provide the source of casual contact, say while shopping or going to a theater, whereas in the other contexts relatively close person-to-person association regularly occurs. Daytime contacts occur in neighborhoods and communities as well as in the age-appropriate setting, whereas nighttime contacts occur only in households, household clusters, neighborhoods, and communities. U.S. 2000 census data on tract-to-tract worker flow is used to configure the commute of working adults to their workplace, thus accurately capturing the short-to-medium distance population mobility important for disease spread. In addition, each individual takes occasional long-distance trips (3 per year on average), lasting between 1 day and 3 weeks (4.1 days on average), matching Bureau of Transportation Statistics data. School closure is implemented by closing the schools, day care centers, and small play groups, and doubling the household transmission probabilities. In the national models, schools are closed according to local illness attack rates, and the other thresholds are national illness attack rates. The model runs in cycles of two 12-hour periods (day and night). The results are reported at day 180 of the epidemic.

In the Imperial/Pitt model, described in ref. 6, individuals are colocated in households, with households being constructed to reflect typical generational structure while matching empirical distributions of age structure and household size for the United States (7, 8). Households are randomly distributed in the modeled geographic region, with a local density determined by the Landscan population density dataset, which has subkilometer resolution (9). Exposure to infection comes from households, schools, workplaces, and random contacts in the community. The last of these depends on distance by using a gravity model, representing random contacts associated with movements and travel, and is the only means by which infection can cross national borders. The spatial kernel used to describe community transmission was parameterized from data on travel patterns in the United States. Air travel was also explicitly represented for national scale model runs. Data on the location, size, and grade composition of every school in the United States (10) were used in initializing the model, and children were allocated to schools with a gravity model so as to match empirical distributions of the journey distance between homes and schools. Similarly, data on commuting distances and the distribution of workplace sizes were used to configure the representation of workplaces in the model and to allocate individuals to workplaces. Thus, the community structure arises naturally from the variation in population density between urban and rural areas. The results at 180 days after the beginning of the epidemic are reported.

The VBI model has smaller mixing groups that are location-specific and determined by the types of activities people do as described in ref. 11, with further background on the concept of location-specific activities in ref. 12. The VBI model was developed by using U.S. Census data from Chicago tracts as well as the Public Use Microdata Samples (PUMS) (13) to build the synthetic Chicago data. Then activity data from the Chicago Area Transportation Study (CATS) (14) were used to synthesize regional travel patterns. The enrollment figures for each school were available from the Department of Education for public schools (10). Each person has a schedule of activities that are matched to activities reported by real people with similar demographics in activity surveys. For example, one person might go from home to work to a restaurant to night school and return home; another might go from home to school, several after-school activities, and home; and yet another might simply stay home all day. In this population, people have on average 4.66 activities per day, ranging from 1 to 22. The locations where these activities take place are chosen based on a form of a gravity model (11, 12) that statistically matches travel distances, by activity type, to those of real people from data gathered by regional transportation surveys. This process generates a realistic distribution of contact durations across age groups, activity types, and locations. For example, each person has from 0 to 262 (mean: 26) person-hours of contact at home and each worker has from 0 to 387 (mean 114) person-hours of contact at work. As each individual's schedule is manipulated to implement nonpharmaceutical interventions as described below, the distribution of contact durations changes, thus changing the population-wide mixing patterns. In this study, the probability of transmission between contacts is based solely on the resulting calibrated contact durations. Given the contact patterns, fitting the transmission rate-a single parameter-to desired values of R₀ yields age-specific attack rates between those of the 1957 and 1968 pandemics as well as the activity-specific attack rates shown in Table 3. The results at day 180 of the epidemic are reported.

In contrast to the other two models, the VBI model has colleges, shopping places, as well as other specific locations where the general community mixes. School closures and keeping children home are implemented on a per-household basis. In compliant households, an adult stays home with the child. In noncompliant households, the child's activity that immediately follows school was prolonged to last the length of school to simulate the mixing that an unsupervised child would do when schools are closed. The degree of interaction at colleges is not affected by social distancing measures.

In these models, R₀ is an output that is computed as part of a complex calibration that includes qualitatively matching to an age-specific illness attack rate pattern between those of the 1957 and 1968 pandemics.

Some Contrasting Implementations. Household transmission and model calibration: All three models base the assumptions about the home transmission on similar sources (15, 16), but used the information differently. The UW/LANL model uses the age-specific household secondary attack rates to calculate the daily probability of an effective contact within the household, then the other effective contact probabilities are calibrated to yield the desired age-specific attack rate pattern corresponding to that used by all three groups as the target pattern. A parameter corresponding to the transmissibility of the assumed pandemic strain is then varied, which yields the different target R₀ values. In the VBI model, the probability of transmission from an infectious person to a susceptible person during a contact (colocation) of duration that is given by the formula ptrans = 1 - (1 - q)^t, where q is a parameter that can be set independently in different locations, for different infectious or susceptible people. For each simulation run here, however, we have used the same value of q for every person and for every location. Differences between within-household and community infection rates in this model are due to different durations and different numbers of contacts in each venue. The overall R₀ is adjusted by changing the value of q. This can in principle affect the proportion of intrahousehold and community infection rates. VBI performed the calibration at a nominal R₀ = 2.1, but the difference in proportions as we change R₀ is minimal. In the Imperial/Pitt model, analysis of household infection data indicates that 30% of influenza transmission occurs in household. In the absence of data to inform the choice, transmission in other contexts was arbitrarily partitioned to give levels of within-place transmission comparable with household transmission, namely 33% of transmission was assumed to occur in schools and workplaces, and 37% in the wider community (i.e., in contexts other than households, schools, and workplaces). The transmission coefficient within schools was assumed to be twice that of workplaces. When varying R₀, the relative proportions of household, place, and community transmission were kept fixed.

Home isolation of cases: In the UW/LANL and VBI models, the compliance percent of cases are completely isolated, whereas the remaining percent continue circulation in the community, as usual. In the Imperial/Pitt model, ascertained cases reduce their contact rates outside the home by the compliance percent.

School closing. In the VBI and Imperial models, an adult stays at home when the children in a household stay home. In noncompliant households in the VBI model, the child's activity that immediately follows school was prolonged to last the length of school to simulate the mixing that an unsupervised child would do when schools are closed. In the UW/LANL and Imperial/Pitt models, when children were not compliant, they continued to mix in the other mixing groups.

Quarantine of household contacts. Either 30 (60,90)% of the contacts are completely compliant with quarantine with the household contact rate doubling (UW/LANL) or contacts outside the home are reduced by 30 (60,90)% for the quarantined individuals (Imperial/Pitt). The VBI model changes the set of activities in which people participate, still providing activities for an entire day. The resulting change in the duration of contacts with other people at each activity determines the probability of transmission associated with that activity.

Workplace social distancing. In the UW/LANL and Imperial/Pitt models, workplace social distancing is implemented by reducing the contact probabilities. In the VBI model, workplace social distancing is achieved by reducing the maximum number of people allowed to be in the same sublocation, such as an office suite, from 50 to 25, representing a reduction in meeting sizes and other contacts.

Community social distancing. In the UW/LANL model, community social distancing is implemented by reducing the contact probabilities by a certain percentage in the community, the neighborhood, and the household clusters. In the Imperial/Pitt model, it is implemented by reducing the contact probabilities in the community. In the VBI model, community social distancing is implemented by removing the specified percentage of the appropriate activities (shopping, social and recreation, visits, and other).

Seeding of infectives. For the UW/LANL U.S. model, the 14 largest international airports act as gateways. Each day a random number (N) of infections is introduced into a random census tract near one of the 14 airports. N is proportional to the number of arriving daily passengers. For the Imperial/Pitt model the infection seeding is based on a global SEIR model that supplies external travelers to the United States at a rate that is based on an external epidemic that has an R₀ = 1.6. This model was used to calculate incidence of infection through time based on 73 million trips per year (Non-U.S. nationals plus U.S. travelers returning from overseas travel). Non-U.S. nationals select destinations proportional to population size. One third (1/3) of the trips are U.S. returnees. Also trips occur at locations according to population density. Returning travelers are expected to have more random destinations than foreign visitors. For the VBI Chicago model five people chosen uniformly at random from the entire population are used to seed the epidemic every day for the duration of the simulation. The same sequence is used on every run. If one of the seeded persons has been prophylaxed or previously infected, no replacement is used.

SI Fig. 5 shows the relative contributions of each activity type in the VBI model to interhousehold transmission in the absence of intervention and in scenario 2. The first person to be infected in each household must acquire the infection outside the household at one of his/her daily activities. Each curve in SI Fig. 5 represents the number of first infections in a household acquired at the indicated activity. In an unmitigated epidemic, (SI Fig. 5a), schools stand out as the driver of interhousehold infection, responsible for 65% of all initial household infections by day 40. When this source is removed by closing schools, other mixing groups, such as colleges in SI Fig. 5b, emerge as the drivers, although at a much reduced level. SI Fig. 5c shows, however, that when schools stay open longer they can force transmission in the workplace, but workplace transmission does not appear to be a driver by itself.

Stochastic variability could explain some of the differences in the absolute illness attack rates under some scenarios among the models, because the number of realizations is relatively small. However, the population of 8.6 million in the Chicago population is relatively large, and the population is seeded regularly with infectives, so that the variability of the effectiveness measures of any given model is relatively small. SI Table 5 contains the mean and standard deviation of five realizations for the UW/LANL Chicago model for several of the scenarios. The variability is considerably under 5% for most scenarios, and nearly zero for some. Variability of the 10 realizations of the Imperial/Pitt model is not shown, but is similar to that of the UW/LANL model. The VBI did 32 runs of one scenario to examine stochastic uncertainty. The histogram based on 32 runs for scenario 2 is in SI Fig. 6. An examination of variability by VBI of 32 runs of scenario 2, R₀ = 2.1, with a 0.1% threshold yielded a mean of 3.75% (standard deviation of 0.35%) illness attack rate, with a range of one standard deviation on either side from 3.4% to 4.1%. With a baseline illness attack rate of 44.7%, the stochastic variability of the effectiveness is from 91.1% to 89.3%. The variability of the UW/LANL and Imperial/Pitt models is less. The VBI model requires much more computing time and resources, and they found it unfeasible to do multiple runs for each scenario and all sensitivity analyses.

Results

In a sensitivity analysis using the UW/LANL model, varying compliance in all three aspects of isolation, quarantine, and children staying home after school closure from 30% through 90%, with scenario 2 at a R₀ of 2.1, the illness attack was only reduced from 2.8% to 2.6%, indicating little sensitivity to the level of compliance in this scenario.

Discussion

Some of the interventions, such as home isolation of identified cases and home quarantine of their household contacts, depend on rapid ascertainment of symptomatic cases, whereas others, such as closure of schools and isolation of children in the home, reducing contacts in the workplace, and reductions in community contacts outside the home, depend on more general triggers.

The effectiveness of the interventions differs more among the three models at the higher R₀ of 3. The UW/LANL model is the most optimistic, likely because of the combination of the social structure and the slightly longer generation time compared with the Imperial/Pitt, so that both the general and the targeted interventions have a strong effect, the combination of which becomes more important at the higher R₀ values.

Models are generally constructed to explore specific scientific questions of interest. Different kinds of models than those presented here could be used to address other questions. Gani et al. (17) use a deterministic, age-structured model with groups at low and high risk of hospitalization to examine the optimal distribution of antivirals in reducing hospitalizations. They do not have households, workplaces, and schools and could not examine the scenarios we consider, but we do not have groups at high risk for hospitalization, so that is not a question that we explored. Colizza et al. (18) used a global model with travel between otherwise homogeneously mixing populations to explore the use of antivirals and travel restrictions to mitigate the pandemic. In contrast, our models assume that global spread has already occurred.

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