We first show the spread of influenza within our unmitigated base case defined with parameters specified above and with ID
chosen to yield an infected attack rate ≈50% to reflect the 1957–58 Asian influenza pandemic (10
). Unless otherwise noted, we report infected attack rates and refer to them as simply attack rates rather than reporting the illness attack rate which is half of this value (pS = 0.5). We then demonstrate the design of effective local mitigation strategies for the base case that focus on targeted social distancing. Finally, we extend these results to design strategies for more infectious strains and for changes to the underlying infectious contact network that deemphasize the role of children and teenagers.
All simulations are initialized by infecting 10 randomly chosen adults with the assumption that adults are first to be infected through business travel or interaction with visitors from outside the community (5
). Some of these initial infections instigate others and grow into an epidemic. Results vary across multiple realizations of the community network and random choice of initially infected adults (controlled by random number seed) not all of which yield an epidemic, defined when the number infected is >1% of the population. For every set of parameters, we conducted >
100 simulations with different random number seeds and collected statistics for all simulations and for only those that result in epidemics ().
Results for base case and miigation strategies*
Unmitigated Base Case
The sequence of infected persons can be represented as an expanding network of infectious transmissions (). The number of secondary infections produced by an infected person, or branching factor, is easily visualized within the infectious contact network. The average branching factor depends on the person's age class and generation during the epidemic (A). The maximum value within the first 10 generations is 2.05 (standard deviation [SD] 0.57) for children, 2.09 (SD 0.72) for teenagers, 1.11 (SD 0.43) for adults, 0.81 (SD 0.47) for older adults, and 1.54 (SD 0.36) for the entire population. Variability (large SD, especially for specific age classes) reflects the heterogeneity inherent within community contact networks of this size (B).
Figure 4 Initial growth of an infectious contact network. Colored rectangles denote persons of designated age class, and colored arrows denote groups within which the infectious transmission takes place. In this example, from the adult initial seed (large purple (more ...)
Figure 5 Branching factor and the approximation of the reproductive number Ro. A) Overall and age class–specific branching factors as a function of generation averaged over 100 simulations. The standard deviations of these averages can be large (< (more ...)
The backbone of infectious contact networks is formed primarily of children and teenagers with infectious transmissions mostly in the household, neighborhood, and schools. Infectious transmissions are highest in households without older adults (39%, SD 3%), followed by extended families or neighborhoods (25%, SD 1%), schools (19%, SD 1%), work (7%, SD 2%), combined random groups (9%, SD 2%), and households with older adults (1%, SD 0.1%). On average, 78% (SD 2%) of children and 71% (SD 3%) of teenagers become infected. Adults (attack rate 44% of adults, SD 2%) get influenza mainly from children, teenagers, and other adults within the family. Older adults, who contact children and teenagers only through the extended family or neighborhoods and the random overall network, are relatively isolated (attack rate 23% of older adults, SD 2%).
Children and teenagers compose only 29% of the population yet they are responsible for 59% (SD 4.5%) of infectious contacts, adults for 38% (SD 7.9%), and older adults for 3% (SD 0.6%) (). Approximately half of infectious contacts of either children or teenagers are within the same age class (19%, SD 0.8% and 9%, SD 0.7%, respectively). Adults get influenza from children or teenagers at approximately the same frequency (24%, SD 1.6%) as from other adults (26%, SD 5.9%). Older adults are equally likely to get influenza from children or teenagers as from adults or older adults (2%, SD 0.3%). Transmission to children or teenagers from adults is 10% (SD 1.8%) and nearly none by older adults. These transmission results are supported by recent field studies that show children who go to preschool or school are more likely to contact influenza and their family members are also more likely to become ill (17,18
) as well as that a person is also more likely to be infected when exposed to children or teenagers than to adults (14
Unmitigated base case infectious contact fractions (% of the total no. of infectious contacts) between age classes*
Reasonable correspondence is observed () between age class–specific attack rates and those of past pandemics (19–21
). Infections transmitted within each environment are also consistent with other simulation studies (10–14
). The maximum value of the overall branching factor () reflects the often-cited reproductive number Ro
. However, how Ro
should be calculated from small-community data such as ours is ambiguous (10,11,14
). To estimate Ro
, we pooled results across 100 communities (simulations) to reflect a population of 1 million (B). The maximum value of the bulk ratio (new infections to old) within the first 10 generations is 1.6, and we choose it as our estimate of Ro
. An Ro
of 1.6 with an attack rate of 50% matches recent pandemic simulation results (10,14
) and lies within the range (1.5–1.7) for the 1957–58 influenza pandemic (B) (10
Figure 6 Comparison of simulated age class–specific illness attack rates with past pandemics. Simulated illness attack rates (half the infectious attack rate) for the unmitigated base case are close to those found in studies of historic pandemics in 1957 (more ...)
Base Case–Targeted Social Distancing
High infectiousness and a high number of contacts, many like-to-like, create a zone of high infectious contact centered on children and teenagers within the community's social network. Targeting this zone can protect the community at large.
First, we examined closing schools. Although contacts in classes are removed, those in all other groups may increase because children and teenagers spend more time at home, in neighborhoods, with friends, and in public spaces. We assume that school closure at a minimum doubles household contacts. Closing schools with 90% compliance the day after 10 symptomatic cases reduces the attack rate to 22% (). However, if we assume that school closure doubles all link contact frequencies for children or teenagers within their nonschool groups, attack rates are increased by 18% ().
Alternatively, we send all children and teenagers home after school closure to remain for the duration of the pandemic. Now contact frequencies are reduced by 90% for all groups that contain only children or teenagers (classes and their random networks) and doubled, as before, for children or teenagers in households. In the extended family or neighborhood and the random overall networks, child or teenager contact frequencies are also reduced by 90%. Thus, although children and teenagers are restricted to the home, adults and older adults go about their day-to-day routines, except that they avoid children or teenagers who are not household members. Imposing this strategy the day after 10 symptomatic cases reduces attack rates by 93% (). Waiting until 80 symptomatic cases reduces attack rates by 73% (A).
Figure 7 Fraction of unmitigated base case attack rate for targeted social distancing of children and teenagers as a function of A) implementation policy threshold given by the number of symptomatic cases (compliance at 90%) and B) compliance with staying at home (more ...)
To evaluate the tradeoff between effectiveness and public compliance, we reduced the percentage of nonschool and nonhousehold contacts that have their frequencies reduced with the child and teenager stay-at-home policy (B). At 50% compliance, attack rates can be reduced by 68% (). Reduction in compliance also increases the time scales for the epidemic. Epidemics lengthen above the base case and reach a factor of ≈1.8 at 40% compliance (B).
Other social distancing strategies can be considered. Because children outnumber teenagers and children are more infective and susceptible, what happens if only children are distanced, while teenagers attend school and follow their usual routines? At 90% compliance, this strategy reduces the attack rates by 47% (). What if all sick persons remain at home when symptomatic? At 90% compliance this strategy reduces attack rates by 20% (<25% of infectious persons are influenced as pS × pH = 0.25 for adults only) ().
More Infective Strains and Contact Networks with Less Emphasis on the Young
We have modeled an influenza strain with an infectivity representative of the 1957–58 Asian influenza pandemic and a social contact network reflective of a stylized US town. Although results for the unmitigated base case yield age class–specific attack rates similar to those for past epidemics (), will the targeted social distancing strategies found above remain effective if 1) the strain is more infective or 2) the importance of the young is deemphasized?
To explore these questions, we considered 3 increases in disease infectivity ID by factors of 1.25 (attack rate ≈66%, Ro ≈ 1.8), 1.5 (attack rate ≈75%, Ro ≈ 2.0), and 2.0 (attack rate ≈86%, Ro ≈2.4). These increases encompass and exceed the 1918–19 Spanish influenza pandemic (Ro 1.8–2.0) (10). We also sequentially removed enhanced transmission by children and teenagers and thus the zone of high infectious contact that we have designed social distancing strategies to target. We created 3 variations: the first removed relative infectivity and susceptibility enhancement of children and teenagers (IA and SA 1.0) (variation 1); the second increased frequency of contact within the work environment by a factor of 4 to give adults the same number of contacts as younger persons (variation 2); and the third combined variations 1 and 2. For each of the resulting set of 4 cases (base, variation 1, variation 2, and variation 1 and 2), ID was altered slightly to maintain the reference of ≈50% infected attack rate for Ro ≈1.6.
As ID increases, age specific–attack rates increase (). As we remove differences in the number of contacts and/or relative infectivity and susceptibility (IA, SA) between young and adults, the infected attack rates systematically shift from young persons to adults (). These results suggest that for such situations, social distancing strategies must be devised that focus on more than children and teenagers alone.
Unmitigated case results for Ro and average attack rates (%) for increasing ID and base case, variation 1, variation 2, and variations 1 and 2 combined*
Figure 8 Unmitigated age-specific attack rate results for disease infectivity (ID) factors of 1.0 and 2.0 and base case, variation 1 (removal of relative infectivity and susceptibility), variation 2 (increase in work group frequency of contact to give all children, (more ...)
To find effective targeted social distancing strategy combinations across the range of disease infectivity and infectious contact networks, we formulated 5 strategies and applied them individually and in combination: 1) school closure (S) where the contact frequency within schools was reduced 90% and children and teenager's household contacts were doubled; 2) children and teenagers social distancing (CTsd) where their contact frequencies in all nonhousehold and nonschool groups were reduced 90% and their household contacts doubled; 3) adult and older adult social distancing (AOAsd) where their contact frequencies in all nonhousehold and nonwork groups were reduced 90% and household contacts doubled; 4) liberal leave (LL), where all children and teenagers and 90% of adults withdraw to the home when symptomatic; and 5) work social distancing (Wsd) where the contact frequency within work groups was reduced 50%. For each combination, we implemented the strategy(ies) the day after 10 symptomatic cases and conducted 100 simulations.
As ID increases, more strategies must be combined to keep the attack rate <10% (, shaded values). As children and teenagers become less prominent, targeting adults becomes important, even at an ID factor of 1. For an ID factor of 1.5 (as infective as the 1918-19 Spanish influenza pandemic) and across all variations, both the young and adults must be targeted and all strategies must be implemented to effectively mitigate the epidemic. However, for an ID factor of 2.0, we can at best reduce the attack rate to 20 –40% through full strategy combination, not ideal but still a significant benefit.
Mitigated case average attack rates (%) for increasing ID and base case, variation 1, variation 2, and variations 1 and 2 combined*