The Shifting Demographic Landscape of Pandemic Influenza

Background As Pandemic (H1N1) 2009 influenza spreads around the globe, it strikes school-age children more often than adults. Although there is some evidence of pre-existing immunity among older adults, this alone may not explain the significant gap in age-specific infection rates. Methods and Findings Based on a retrospective analysis of pandemic strains of influenza from the last century, we show that school-age children typically experience the highest attack rates in primarily naive populations, with the burden shifting to adults during the subsequent season. Using a parsimonious network-based mathematical model which incorporates the changing distribution of contacts in the susceptible population, we demonstrate that new pandemic strains of influenza are expected to shift the epidemiological landscape in exactly this way. Conclusions Our analysis provides a simple demographic explanation for the age bias observed for H1N1/09 attack rates, and suggests that this bias may shift in coming months. These results have significant implications for the allocation of public health resources for H1N1/09 and future influenza pandemics.

1 Supplementary Methods

Derivation of Residual Degree Distribution
The full derivation for the degree distribution of the residual network model proceeds as follows: As the extended residual network is dened by the uninfected nodes of the previous epidemic, a proportion α of the infected nodes of the previous epidemic, and all the edges joining them, we dene the degree distribution for the extended residual network as: We nd that the p uninf ected res (k r ) can be found by where p res (k r |k o ) is the probability that a node in the extended residual network will have degree k r given that it had a degree of k in the original network. This condition distribution can Dbe calculated as as discussed in the main text. Following Bayes rule, p uninf ected res (k r ) is then the sum of the product of the probabilities that a node in the extended residual network has degree k r given that it had original degree k o , (p res (k r |k o )), the probability that the node of original degree k o was uninfected in the rst epidemic, (η ko ), and the probability that the node had original degree k o , (p ko ) . In a similar way, p inf ected res (k r ) can be calculated as: Thus, by substituting equations 2 and 4 into equation 1, we have: And lastly, by substituting equations 3 into equation 5, we have:

Incidence Data from Inuenza Pandemics
Here, we list the condence intervals for the odds ratio of attack rates in schoolage children to adults. We note that the denitions of the school-age children and adult age groups vary across the studies and sometimes slightly dier from our own.

Invasion Threshold with Immunity
When invading a population which has previously been (partially) infected by a similar strain, a pathogen faces an epidemic threshold for the a subsequent outbreak, which is a critical value of transmissibility above which a second epidemic is possible in the the same population. This critical transmissibility threshold is a function of the transmissibility of the rst season pathogen, T 1 , and the loss of immunity, α, and can be calculated as: where, k r is dened by k kp res (k) and p res (k) is the residual network degree distribution for our urban network model. T 2 = T c 2 is equivalent to R e = 1. In a naive population (α = 1), the transmissibility of an invading pathogen can be quite low for successful invasion. As you consider scenarios with increasing immunity (i.e. as α decreases), the epidemic threshold for an invading pathogen is increasingly higher. In particular, Figure 1, shows the decreasing value of T c 2 with increasing α for a rst-season epidemic of R 0 = 1.1, 1.6, 2.1.

Sensitivity to Contact Patterns
The contact patterns of the populations in the pandemics of the last century (1918, 1957 and 1968) may have diered from those of today. Here, we explore sensitivity to changes in population structure of the shift that occurs in the risk of infection from an intial pandemic season to a subsequent season. We consider two probability distributions of contacts (degree distributions) dierent than that of our urban network model: a) the Mossong et al study [4] is an empirical survey which reports daily conversational and physical contacts of individuals in European countries, and b) the Eubank et al study [1] is a synthetic population simulated based on data from the city of Portland, Oregon that reports daily contacts. In addition to dierences in contact structure, these studies incorporate dierences in the demographic structure of the population. Specically, the Eubank et al study is made up of 20% children and 51% adults, while the Mossong et al study is made up of 28% children and 48% adults, 6 compared with 23% children and 61% adults in our population model. Figure 2 below shows that both populations still exhibit the shift in infection risk from high contact individuals to moderate contact individuals. Thus given higher contact rates for school-age children than adults, a shift in age-specic incidence can be expected from the prior age group to the latter.  [4], and the Eubank et al study [1]. (B) The degree-based risk of infection for a naive population (no prior immunity) for each network for R 0 = 1.6 (C) The degree-based risk of infection for a subsequent season (of R e = 1.05), following a rst season epidemic (of R 0 = 1.6) with partial immunity (α = 0.05) for each network. The shift in risk of infection from high contact individuals to moderate contact individuals in a partially immune population is still evident. 7

Sensitivity to the Reproductive Number
The reproductive number for the H1N1/09 pandemic has been estimated to be between 1.22 and 2.3 [2,6,5]. Figure 3 shows that our results on the shift in risk of infection are robust to varying R 0 values.

Sensitivity to Demographic Changes
Here, we consider disease consequences in a population with partial immunity from a previous year as well as demographic changes. We ignore births and deaths as they occur at either end of the population age scale and do not impact contact patterns to a signicant degree. Instead, we focus on aging in the population and the impact of this on population contact patterns. Specically, we allow aging of individuals at the boundaries of age (and thus contact) classes: 4 year olds age to 5 year olds (and move from the low-contact class of toddlers to the high-contact class of school-aged children), 18 year olds age to be 19 years old (and move from the high-contact class of school-aged children to the moderate-contact class of adults), and adults of age 64 age to 65 years of age (and move from the moderate-contact class of adults to the low-contact class of elderly.) In Figure 4, we see the dierences in the degree distribution of the contact networks with and without these demographic changes in populations with full, partial and no immunity (α = 0, 0.5, 1). As the left panel shows, aging only impacts the degree distribution in a small way. To quantify the disease impact of these changes, we calculate the size of the epidemic in the populations with and without aging for various levels of immunity and also nd that the impact is not dramatic.

Comparing Stochastic Simulations to Percolation Model
Finally, we compare the predictions from our percolation-based model to stochastic simulations. The standard percolation theory framework is based solely on the degree distribution of the population contact network. Comparison to stochastic simulations allow us to quantify the impact of higher order structural properties (such as clustering in edges, degree correlations, etc.) Below, we compare the predictions from our percolation model to that of stochastic simulations for the risk of infection in the second season by degree and nd congruent results.