Description of participant data

We measured 67,683 HI titres against 20 A/H3N2 strains isolated between 1968 and 2014 at 2 to 3 year intervals from serum samples collected between 2009-12-22 and 2015-06-02 (S1 Fig) as part of the Fluscape cohort study (1,130 individuals, 2 samples each) (S2 and S3 Figs). Sampling was done over 4 study rounds, with a mean time between serum sample collection of 3.87 years (standard deviation: 0.780; 95% quantiles: 1.68 to 4.74 years). This cohort covered 40 unique locations and 651 unique households in a 60 kilometre transect from Guangzhou, China. Participant ages at the most recent sampling round ranged from 6 years to 97 years, with a median of 50 years. Vaccination rates were low in this cohort (S1 Table), consistent with low reported vaccine coverage rates in mainland China, particularly in older individuals [39,40]. We refer to an individual’s set of antibody titres against all strains in the panel as their antibody profile. All individual antibody profiles and observed changes in titre are shown in S20 and S21 Figs. Summary statistics of these profiles and full study details have been described elsewhere [38,41].

Antibody titres vary by age and in space

We saw 5 broad patterns of seropositivity (HI titre > = 1:40) when stratifying the antibody data by age group and strain (Fig 1). First, individuals had mostly low or undetectable titres against strains that circulated before they were born (cells below the black lines in Fig 1), though many individuals were seropositive against the strain which circulated in the years immediately prior to birth. Second, in the youngest age group (0 to 10 years old), many individuals were also seropositive to pre-birth strains that circulated further back in time (A/California/2004, A/Fujian/2002, A/Fujian/2000, and A/Victoria/1998), indicating the presence of cross-reactive antibodies as these individuals could not have been exposed to those strains (Fig 1C). Third, seroprevalence tended to be high among individuals who were young when a strain was first isolated (cells just above the black line in Fig 1) compared to individuals who were older at the time that the strain was first isolated. Fourth, the total proportion of individuals who seroconverted between sample collection dates was high against recent strains at nearly 50% across all age groups. Finally, some strains exhibited systematically higher titres than others, for example, titres against A/Fujian/2002 and A/Mississippi/1985 were higher than all other strains and particularly high for individuals who were under 10 years old at the time of their first circulation.

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Fig 1. Proportion of individuals seropositive and seroconverted to 20 A/H3N2 strains circulating from 1968 to 2014 stratified by age.

Solid black line divides age groups that were alive or not at the time of strain circulation. Seropositivity was defined as having an HI titre of ≥1:40 (a log titre of 3). (A) First serum sample. (B) Second serum sample. (C) Seroconversion between samples, defined as a ≥4-fold increase in HI titre. The data underlying this figure can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.g001

Inferring antibody kinetics and individual infection histories from antibody profiles

To infer the set of A/H3N2 influenza strains each individual was infected with, we further developed and used the serosolver R-package to fit an infection history and antibody kinetics model to all individual-level antibody profiles [35]. Briefly, the method finds the combination of influenza strains which an individual is most likely to have encountered conditional on their antibody profile, accounting for cross-reactive, transient antibody boosting, and antigenic seniority arising from repeated exposures to antigenically related strains (see Materials and methods). Individuals can be infected with the strain assumed to be circulating in each time period provided they were alive at the time of circulation. Individuals have distinct infection histories, but the parameters governing post-infection antibody kinetics were assumed to be universal, and individual infection probabilities were assumed to arise from a single population-level infection probability parameter per time period, regardless of location. A crucial component of the model is the observation level, which accounts for the fact that some strains elicit systematically higher or lower titres than others in the HI assay (see S1 Text).

We routinely fitted the same model to both the Fluscape cohort data and to previously published antibody profile data from a cohort of 69 individuals in Ha Nam, Viet Nam (see Materials and methods and [37]), and compared the inferred antibody kinetics and epidemic dynamics. Posterior distributions for the model predicted titres and infection histories compared to observed titres are shown for 5 randomly selected individuals from the Fluscape cohort in S4 Fig, and from the Ha Nam, Viet Nam cohort in S5 Fig. Overall, we imputed a total of 10,558 (posterior median; 95% credible interval (CrI): 10,394 to 10,750) distinct infections (see Post-processing of infection history posteriors in Materials and methods) across all individuals and times from the Fluscape data and 547 (posterior median; 95% CrI: 519 to 574) infections from the Ha Nam, Viet Nam data. We also ran varied scenario analyses using simulated data closely resembling the Fluscape data, demonstrating that our inference system was able to accurately recover infection histories, attack rates, and antibody kinetics parameters under a range of assumptions and model misspecifications (S2 Text).

Although there were no virologically confirmed infections reported in the Fluscape study with which to validate our infection history estimates, individuals did self-report influenza vaccination at each study visit, but only within a window of time rather than on a specific date (e.g., in the preceding calendar year; see Materials and methods). We estimated antibody boosting events (infections) with >25% posterior probability for 68.9% of time windows in which individuals self-reported influenza vaccination (62 of 90 windows from 77 individuals; S22 and S23 Figs), compared to 43.1% (mean of null simulations; 95% quantiles: 32.8% to 51.7%) of randomly selected time windows of the same duration from randomly selected individuals, suggesting that our model was more likely to infer antibody boosting events that coincided with self-reported vaccination than in randomly selected time periods. Vaccination windows in which no vaccination/infection was inferred tended to either have antibody boosts identified soon after the reported vaccination, suggesting either delayed boosting or recall error, or very low antibody titres to recent influenza strains, suggesting either inaccurate recall or no vaccine-induced antibody boosting (S23 Fig).

Antibody kinetics parameters

We obtained estimates for the antibody kinetics model parameters assumed to underlie the generation of observed influenza antibody titres. We estimated parameter values consistent with an initial, broadly reactive antibody response that decays within approximately 1 year to leave an antigenically narrow, persistent antibody response. These estimates are in line with previous estimates using a similar method applied to a smaller pilot serosurvey from the Fluscape cohort [36], as well as from model fits to the Ha Nam, Viet Nam cohort data (Table 1). However, there were some differences in the estimated magnitude of the antibody response between the Fluscape and Ha Nam data sets. Estimates from the Fluscape data suggested that infection elicited an antigenically narrow, long-term antibody boost of 1.42 log HI units (posterior median; 95% CrI: 1.39 to 1.44) compared to 1.96 log HI units (posterior median; 95% CrI: 1.89 to 2.03) from the Ha Nam data set, and an additional antigenically broad, short-term boost of 2.06 log HI units (posterior median; 95% CrI: 1.99 to 2.14) compared to 2.65 log HI units (posterior median; 95% CrI: 2.43 to 2.92). The estimated waning rate of the short-term response was similar from the 2 data sets, suggesting that the short-term response took 0.995 years (posterior median; 95% CrI: 0.963 to 1.03) to fully subside based on the Fluscape data and 1.33 years (posterior median; 95% CrI: 1.22 to 1.45) based on the Ha Nam data. Finally, the observation error standard deviation was estimated to be larger for the Ha Nam data set at 1.29 (posterior median; 95% CrI: 1.27 to 1.31) versus 0.636 (posterior median; 95% CrI: 0.631 to 0.641) (posterior medians and 95% CrI), though this is likely attributable to the different assumed observation model and measurement of more influenza strains in the latter data set.

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Table 1. Estimated antibody kinetics parameters.

The data underlying this table can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.t001

Inferred historical and contemporary attack rates

By combining all individual-level inferred infection histories, we obtained A/H3N2 incidence estimates in the Fluscape cohort for each 3-month window since 1968 (Fig 2A). The estimated median quarterly sample attack rate was 3.54% (median across all posterior samples; 95% CrI: 3.08% to 4.01%). This corresponded to a median annual attack rate of 19.1% (posterior median; 95% CrI: 17.2% to 20.9%), defined as the proportion of alive individuals who experienced at least 1 infection within a calendar year. This was comparable to estimates from the Ha Nam, Viet Nam data set, which gave an estimated median annual attack rate of 18.6% (posterior median; 95% CrI: 14.5% to 22.9%).

Quarterly A/H3N2 attack rate estimates in the Fluscape cohort varied over time, ranging from a minimum of 0.459% (posterior median; 95% CrI: 0.00% to 2.53%) in Q1-1977 (during the re-emergence of A/H1N1) to a maximum of 64.8% in Q1-1968 (posterior median; 95% CrI: 54.2% to 71.3%) (at the beginning of the A/H3N2 pandemic). The attack rate estimate for Q1-1985 was unusually high (51.8% posterior median; 95% CrI: 0.207% to 63.8%), suggesting that there might be residual bias from systematically higher titres for the A/Mississippi/1985 virus not captured by the antibody kinetics and measurement models. Periods of high and low attack rates were reasonably well synchronised between the Fluscape and Ha Nam data sets, though attack rates were estimated to be significantly higher in Ha Nam from 2000 to 2003, and significantly higher in the Fluscape cohort from 2010 to 2012 (S6 Fig). For example, both data sets gave very high attack rates for 1968, 1989, and 2009, and similar attack rates from 1970 to 1985 (though with substantial uncertainty in the Ha Nam estimates due to the much smaller sample size). There were also periods with clear differences—attack rate estimates were much lower in the Fluscape cohort in the early 2000s, and higher during 2010 to 2012.

Quarterly attack rate estimates were substantially higher in the Fluscape cohort after the first serum sample in Q4-2009 than the overall average, with the median quarterly attack rate since Q4-2009 estimated to be 7.79% (median of posterior samples for median per-quarter attack rate; 95% CrI: 3.46% to 21.7%) versus 2.98% before (95% CrI: 0.748% to 13.7%). Annual attack rates (defined as the proportion of individuals who were inferred to have been infected at least once per year) fluctuated, with lower attack rates in 2010, 2012, and 2013, and high attack rates in 2011 and 2014 (Table 2). These attack rate estimates were of a similar magnitude to the proportion of individuals that seroconverted between study visits against strains that circulated during that time. Surveillance data collected from the Guangdong Provincial Centre for Disease Control and Prevention influenza surveillance system during the same time period reported slightly different dynamics, with an increased number of A/H3N2 virus isolates in Q2/Q3-2009, Q3-2010, Q4-2011, and Q1/Q2-2012, as well as a higher proportion of A/H3N2 isolates in 2011 and 2012 [42]. However, direct comparison to our estimated attack rates is difficult, as the influenza surveillance network only collects nasopharyngeal swab samples from patients presenting to sentinel hospitals with ILI, whereas our estimates relate to all exposure events regardless of symptoms.

A substantial proportion of people were estimated to have been reinfected within a single year, with higher reinfection rates in years with higher overall attack rates; 33.9% (posterior median; 95% CrI: 27.0% to 41.0%) of all reinfections between 1968 and 2015 occurred since 2008, whereas only 14.9% of possible infection events were in this time frame, suggesting that reinfections were disproportionately more likely in recent time periods. Annual reinfection rates, defined as having been infected at least once in a year for multiple years, were also high for recent years (Table 3).

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Fig 2. Quarterly incidence and individual infection histories from the Fluscape data set.

(A) Model predicted per-capita incidence per quarter. Attack rates were estimated by dividing the number of inferred infections by the number alive in each 3-month period. Red line shows the posterior median estimate from 1,000 posterior samples. Dark and light red shaded regions show 50% and 95% credible intervals respectively from 1,000 posterior samples. Grey shaded box shows duration of the Fluscape study—the improved precision is due to the inclusion of sera bracketing this time period. Asterisks mark times from which a strain included in the HI panel was first isolated. (B) Inferred infection histories for each individual. Each row represents an individual ordered by increasing age in years. Each column represents the time of a potential infection. Cells are shaded based on the proportion of posterior samples with an infection at that time. The grey areas show time periods prior to each individual’s birth. The data underlying this figure can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.g002

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Table 2. Estimated attack rates and infection patterns 2010–2014.

Percentages shown are posterior median and 95% credible intervals. “Attack rate” was defined as the proportion of individuals who were infected at least once in that year. “Seroconverted” gives the percentage of individuals that seroconverted to the measured strain, with ranges showing 95% binomial confidence intervals. “Reinfected” gives the percentage of people that were infected more than once in a year. The data underlying this table can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.t002

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Table 3. Percentage of individuals who were infected at least once per year in 0, 1, 2, 3, 4 or 5 years between 2010 and 2014 inclusive.

The data underlying this Table can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.t003

Attack rates varied in space but were not clearly associated with proximity

Previous analyses from this cohort found that antibody titres varied significantly between study locations after accounting for differences in demographics, suggesting that there may be differences in influenza A/H3N2 epidemiological dynamics between locations [1]. We grouped the posterior draws for the Fluscape infection histories by study location to investigate spatial patterns in attack rates. Attack rates exhibited variation between locations and over time (S7 Fig), though the timing of high attack rate periods was synchronised. S8 Fig shows snapshots from a video of attack rates over time across the study region, demonstrating that the periods of high influenza incidence are similar across the study locations, but that there is some variation in the timing and magnitude of incidence (full video in S1 Video). The overall coefficient of variation (CoV) for per-quarter attack rates across all locations and times was 1.12 (posterior median; 95% CrI: 1.04 to 1.19). This was reduced to 0.493 (posterior median; 95% CrI: 0.458 to 0.535) after aggregating the infection histories into per-year attack rates, defined as the proportion of individuals infected at least once in a calendar year. There was a strong negative correlation between the estimated posterior median CoV and per-quarter attack rate (Pearson correlation coefficient of −0.829).

To understand if this variation reflects epidemiological differences between locations or simply sampling variation, we generated a comparable null simulation where no significant spatial variation in incidence would be expected (see Spatial correlation in inferred attack rates in Materials and methods). The mean CoV for these simulations was 0.300 (95% quantiles: 0.0307 to 1.22). Given the overlapping uncertainty intervals of the model estimates with the simulations, we cannot exclude the possibility that the estimated attack rates from the Fluscape data exhibited no more variation overall than would be expected by chance if all attack rates were drawn from the same binomial distribution. S9 Fig demonstrates that this pattern is maintained across time, though with higher variation in time periods with low infection rates. Fitting a spatial nonparametric correlation function—a model describing correlation of different spatial units over time as a function of their distance to each other—revealed high correlation in attack rates between locations which showed no clear association with increasing distance from one another (S9D Fig). This consistent correlation across space was also observed when subsetting attack rates by recent (from Q1-2009 onward) or historical times (from Q1-1968 to Q1-2009), when considering either per-quarter or per-year attack rates, and also when considering the proportion of individuals in a population who seroconverted against all or only recent strains.

Age-specific infection patterns

Periods of high infection probability were largely synchronised across all individuals regardless of age, though individuals typically experienced more frequent infections in the years immediately following birth (S10 Fig). Two age-specific patterns emerged. First, almost all individuals who were alive in 1968 were almost certainly infected in or around 1968, demonstrated by the high posterior probability of infection across all individuals alive at that time. Second, the posterior probability that an individual was infected soon after birth was consistently high, demonstrated by the lower edge of the heatmap in Figs 2B and S10. The posterior mean, median, and 95% CrI on the age of first infection was 1.36, 0.75, and 0.00–6.00 years, respectively (estimates using 1,000 posterior samples for all individuals born since 1968). We note that although these augmented infection histories suggest most individuals were infected soon after birth, our model requires a number of assumptions regarding cross-reactivity to pre-birth strains and antigenic distance between strains which limits our ability to draw this conclusion, particularly given contrasting previous findings from longitudinal serological studies in children [43].

We calculated each individual’s age at the time of each infection and found that the number of infections per 10 year period decreased through childhood and became stable in adulthood (Figs 3C and S11). Overall, individuals were estimated to be infected 2.12 times per 10 year period (posterior median; 95% CrI: 1.06 to 7.86), in line with previous estimates (Fig 3D) [36]. Infection frequency patterns and trends with respect to age were similar under different assumptions for the infection history model, suggesting that these findings were driven by features of the data and not an artifact of the model structure (S12 Fig; [35]).

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Fig 3. Age-specific patterns of infection.

(A) Pointrange plot shows posterior median and 95% CrI on the total number of lifetime infections for each individual in the Fluscape cohort, ordered by increasing age at time of sampling. (B) Distribution of the total number of infections across all individuals based on the posterior median total number of infections. (C) Posterior median number of infections per 10 year period stratified by age group at the time of infection, excluding individuals who spent less than 2 years in that age group and including only time periods prior to the first serum sample in Q4-2009 (see S11 Fig for explanation and comparison using all time periods). Text shows sample size within each age group—note this does not sum to the number of individuals in the sample, as individuals contribute to multiple age groups during their lifetime. (D) Distribution of individual posterior median number of infections per 10 years alive. The data underlying this figure can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.g003

Relationship between titre and probability of infection

Our model fits estimated not only infection histories, but also each individual’s expected HI titre against all strains at each point in time, and thus we were able to estimate the relationship between probability of infection and model-predicted latent HI titre (Fig 4). For each sample from the posterior, we found the proportion of time periods (across all times and individuals) where infection was estimated to have occurred, stratified by log HI titre against the circulating strain in that time period. There was a clear pattern of decreasing relative risk of infection as a function of increasing titre (Fig 5). An HI titre of 1:40 (log titre of 3) corresponded to a risk of infection of 0.559 (posterior median; 95% CrI: 0.514 to 0.605) relative to individuals with no detectable titre, consistent with prior evidence from deliberate infection experiments [44]. This pattern appeared to vary with age. An HI titre of 1:40 gave a relative risk of infection of 0.374 (posterior median; 95% CrI: 0.312 to 0.427) in the 0 to 10 age group, but only 0.789 (posterior median; 95% CrI: 0.638 to 0.955) in the 60+ age group.

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Fig 4. Model predicted titres against circulating strains since birth.

Each subplot shows one randomly selected individual. X-axis shows time since birth. Blue line and shaded region show model-predicted, true latent titre against the strain assumed to be circulating at each time period (posterior median and 95% CrI). Note that titres are continuous and represent latent, true values, not observations. Orange lines indicate times of high posterior probability of infection. Grey regions show times before birth and after the last serum sample. The data underlying this figure can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.g004

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Fig 5. Estimated relationship between HI titre and probability of infection.

Top left panel shows the relative risk of infection at all time points stratified by model-predicted HI titre against the circulating strain just before infection for all age groups. Remaining plots show the same relationship but stratified by age group in 10-year bands at the time of infection. Solid lines and shaded regions show posterior median and 95% CrI. Note that the uncertainty intervals reflect uncertainty in the imputed infection states and latent antibody titres; the relationships shown here are empirically calculated from the serosolver estimates. Wide uncertainty intervals at higher titres reflect limited data as few individuals reach such high titres; the posterior median for number of individuals with HI titre of 1:640 or greater at the time of infection was less than 10 for all but the youngest age group. Vertical dashed line shows HI titre 1:40. Horizontal grey dashed line shows 50% protective titre. The data underlying this figure can be found at https://doi.org/10.5281/zenodo.12795911.

https://doi.org/10.1371/journal.pbio.3002864.g005

Ethics statement

Study protocols and instruments were approved by the following institutional review boards: Johns Hopkins Bloomberg School of Public Health, University of Hong Kong, Guangzhou No. 12 Hospital, and Shantou University. Written informed consent was obtained from all participants over 12 years old, and verbal assent was obtained from participants 12 years old or younger. Written permission of a legally authorised representative was obtained for all participants under 18 years old.

Cohort description—Fluscape

Influenza haemagglutination inhibition (HI) titres were obtained from a previously described cohort in Guangzhou, China, called the Fluscape study (described in detail in a cohort profile [38]) [41]. The Fluscape study is a serological, contact, and demographic survey in and near Guangzhou, China. The study covers 40 locations randomly selected from a 60 km cone-shaped transect extending from Guangzhou city center into the surrounding rural area. Latitudes and longitudes of each study location were assigned based on a central place (e.g., a street or village committee center). A total of 60 households within 1 km of each chosen location were randomly selected and contacted one-by-one until 20 households with at least 1 member willing to provide a blood sample and answer the survey questionnaire were contacted. The majority of locations were classified as rural (30/40), were between 20 and 80 min travel time from Guangzhou city center, and had a population density between 256 and 367,346 persons per 9 km grid cell. All household members aged 2 or above were eligible to participate, and 5 ml of blood was taken for each blood sample from each visit. Here, we used data from a subset of serum samples, capturing 1,130 individuals who had 2 serum samples taken and analysed from 2 different rounds of sampling between 2009 and 2015 inclusive. The month of sampling, age of participant, vaccination history, and other socioeconomic covariates were available for all individuals.

Serological data

HI assays were performed for each sample to measure antibody titres against 20 A/H3N2 strains that circulated between 1968 and 2014 inclusive with approximately 2 year spacing between circulation years (A/Hong Kong/1968, A/England/1972, A/Victoria/1975, A/Texas/1977, A/Bangkok/1979, A/Philippines/1982, A/Mississippi/1985,A/Sichuan/1987, A/Beijing/1989, A/Beijing/1992, A/Wuhan/1995, A/Victoria/1998, A/Fujian/2000, A/Fujian/2002, A/California/2004, A/Brisbane/2007, A/Perth/2009, A/Victoria/2009, A/Texas/2012, A/Hong Kong/2014). The 50% tissue culture infectious dose (TCID50) for each virus was determined using Madin-Darby canine kidney (MDCK) cells, calculated using the Reed–Muench method [94]. Blood samples were kept at 4°C on the day of collection until extracted sera was frozen at −80°C until testing. Sera were thawed and treated with receptor-destroying enzyme (RDE) derived from Vibrio cholerae to remove nonspecific inhibitors prior to incubation at 56°C for 30 min, and then absorbed with Turkey red blood cells to remove substances which might lead to nonspecific agglutination. HI assays were conducted in 96-well microtiter plates with 0.5% turkey erythrocytes using 4 haemagglutination units. Sera from each individual’s first and second sample were tested side-by-side on the same plate. Repeat titres were generated for all of the strains tested from the second serum sample on a separate plate (23,686 repeat measurements in total), but were not run for the first sample due to insufficient sample volume. Titres were tested in serial 2-fold dilutions from 1:10 to 1:1,280, with the reciprocal of the highest dilution at which haemagglutination was inhibited recorded as the titre, and undetectable titres recorded as <1:10. The full laboratory protocol is described in [95].

For all analyses, titres were transformed to a log₂ scale (i.e., 2-fold dilutions), where y = log₂(D/5), giving log titres between 0 and 8 (undetectable titres were treated as a 0 log titre). Seroconversion between study visits was defined as a 4-fold rise in titre, equivalent to a ≥2 unit increase on the log scale. Seropositivity was defined as having a titre of ≥1:40 (log titre ≥3). Further details on laboratory testing have been described previously [41].

Summary of model

The overall inference task is to obtain estimates for the joint posterior distribution of antibody kinetics parameters (Θ), infection histories (Z) for all n individuals, and the attack rate within each of m possible discrete infection periods conditional on the set of observed HI titres (Y). Crucially, only Y is observed, so we must infer (or augment) the values of Z as latent features. Throughout the remainder of the methods, we use capital letters to represent random variables and bold letters to represent vectors of random variables. A detailed description of the inference problem and approach is described in [35], but can be summarised as sampling from the posterior distribution: where θ is the vector of antibody kinetics parameters that describes the link between Z and Y. The set of times t_1,i to t_max,i gives the time periods when a serum sample was obtained from individual i. P(Y |Z, θ) is defined by the antibody kinetics and observation model, and is the infection history prior, described below.

An individual’s entire infection history is given as a vector of unobserved binary variables, Z_i = [Z_i,1, Z_i,2, Z_i,j]. Each infection event, Z_i,j, is the outcome of a single Bernoulli trial, where Z_i,j = 1 indicates that individual i was infected with the strain circulating in discrete time period j, Z_i,j = 0 indicates that they were not. The entire infection history matrix Z for all n individuals across all m time windows was therefore represented by an n by m binary matrix. We estimated infection histories at a 3-monthly resolution, such that each individual had an unobserved infection state for each 3-month period j since birth. Individuals could be infected from the first quarter after they were born. We did not attempt to impute infection states occurring after an individual’s last serum sample. The attack rate is then given as , where N_j gives the number of individuals alive in time period j.

Infections lead to the production of antibodies (“seroresponses”) that undergo longitudinal and cross-reactive kinetics. The vector of true, latent antibody titres across all time periods is given as A_i = [A_i,1, A_i,2,…, A_i,j], which is generated from an antibody kinetics process with parameters θ described below. The process generating measured titres from the latent antibody titres is modelled through an observation level. The vector of observations is given as Y_i = [Y_i,1, Y_i,2,…, Y_i,t]. Note that the time index for Y_i is different to A_i and Z_i, as observations are only made at a subset of t times, whereas latent infection states and antibody titres must be represented at all j times. Y_i,t is also itself a vector, as it contains HI titres against all strains measured from that serum sample. A schematic of the full model is shown in S13 Fig and an example simulated antibody landscape over time is shown in S14 Fig.

Note that the estimated attack rates and infection histories incorporate uncertainty in whether elevated titres against a particular strain result from infection with that strain and/or from cross-reactive antibodies from infection with a different, antigenically related strain. For example, a high titre measurement might be explained by a single infection with that strain (leading to a large boost) or by multiple infections with other, antigenically distant strains (leading to multiple small boosts). The antibody kinetics model accounts for both possibilities by including a homologous antibody boosting parameter and a model for cross-reactive antibody boosting as a function of antigenic distance (see below). This structure also allows us to estimate infection states during years from which we do not have a representative influenza strain in the HI panel. For example, in the time period 2010 to 2014 (Table 2), the model samples possible infection histories for that time period where elevated titres against A/Perth/2009, A/Victoria/2009, A/Texas/2012, and A/Hong Kong/2014 could reflect strain-specific antibody boosting from infections in 2010, 2012, and 2014, or cross-reactive antibodies from infection in 2011 and 2013. Hence, we do not present a single estimate for the most plausible infection history, but rather incorporate this uncertainty into the attack rate estimates by using multiple samples from the posterior for each individual’s infection history.

Antibody kinetics, antigenic map, and observation model

We used an existing deterministic model to describe the generation of observed antibody titres following exposure to an influenza strain [35,36]. The model has 3 components: (i) the antibody kinetics model; (ii) the antigenic map; and (iii) the observation model. The antibody kinetics model describes linear short- (μ_s) and long-term (μ_l) antibody boosting on the log scale immediately following infection. The short-term boost wanes over time such that the remaining short-term boost j − k time periods after infection at time k is given by μ_sw(k,j) = μ_smax{0, 1 − ω(j-k)}. Long-term boosting is persistent. In addition to boosting antibodies against the infecting strain, infection also elicits the production of cross-reactive antibodies against antigenically related strains. Cross-reactivity is assumed to decrease linearly with antigenic distance by a factor of d_l(k, j) = max(0, 1 − σδ_k,j), where δ_k,j represents the antigenic distance between the infecting strain k and the measured strain j. The short- and long-term boost for a given strain is therefore given by μ_ld_l(k, j) = max{0, μ_l(1 − σ_lδ_k,j)} and μ_sd_s(k, j) = max{0, μ_s(1 − σ_sδ_k,j)}, respectively. Finally, antigenic seniority by suppression was included, wherein the full amount of boosting decreased linearly by a proportion τ for every infection following the first. Boosting was scaled by s(Z_i,j, j) = max{0,(1 − τ(N_j − 1))} after each infection, where N_j − 1 is the number of previous infections before strain j. The full model, f(A|Z,θ) (where θ = {μ_s, μ_l, ω, σ_l, σ_s, τ}), for expected titre for individual i measured at time t against strain j, A_i,j,t, is given by:

A key component of the model is the antigenic map specifying the antigenic distance between A/H3N2 strains and thus their cross-reactivity measured by the HI assay. That is, infection with strain A generates cross-reactive antibodies which also recognise epitopes on strain B, where the degree of cross-reactivity can be modelled as the antigenic distance between the 2 strains. In the serosolver model, antigenic distance between strain k and strain j, δ_k,j, was given by the Euclidean distance between them on the antigenic map. In theory, we might jointly estimate the antigenic map alongside the other model parameters, but at present this is computationally infeasible and thus we assume a fixed antigenic map for model fitting.

We used the antigenic coordinates of the 20 strains measured in the Fluscape study to represent the strains circulating in each time period (S15 Fig). We fit a cubic smoothing spline with low amounts of smoothing (smoothing parameter = 0.3) through the coordinates to provide a comparable model to previous analyses and to smooth over large jumps in the posterior surface, which greatly aids in model convergence by smoothing over multiple posterior modes (see S2 Text) [36]. We also attempted to fit a version of the model assuming punctuated rather than continuous evolution through antigenic space, placing strains into clusters based on previous analyses of A/H3N2 strains in China [96]. Although this punctuated version of the cross-reactivity model may be more realistic [13], the posterior distribution under this model was multimodal and thus we were unable to produce reliably converged model fits.

We assumed that log HI titres observed at time t were normally distributed with mean A_i,j,t and variance ε as in [35,36], with censoring to account for the upper and lower bounds of the assay. The probability of observing an empirical titre at time t within the limits of a particular assay Y_i,j,t ∈ {0,…, q_max} given expected titre A_i,j,t is given by: where q_max = 8 and , the probability density function of the normal distribution and ε is the standard deviation.

Fitting the model as described so far to the HI titre data led to systematic under- or overestimation of titres to certain strains (S16 and S17 Figs). We therefore introduced fixed, strain-specific measurement offsets similar to [97]. The procedure for estimating these offsets is described in S1 Text. In short, we added fixed offsets to each predicted titre as: , where χ_j is the measurement offset for strain j. These additional offset parameters aim to capture the residual observation error not explained by the estimated infection histories, antibody kinetics, and normally distributed observation error.

Infection history prior

The choice of prior for the infection history matrix Z is discussed in detail in [35]. The crux of the problem is that the choice of prior P(Z) determines not only P(Z_i,j = 1), but also the prior distribution of total number of lifetime infections, attack rate in a given time period, and time between infections. As we are interested here in reconstructing historical attack rates, we chose to place a Beta prior on the probability of infection in a given time window (prior version 2 in serosolver). The infection history matrix Z is then a Beta-Bernoulli distributed variable such that: where B is the Beta function; k_j = ∑_iZ_i,j is the total number of infections across all individuals during time period j, and n_j is the number of individuals that could be infected during time period j. Values for α and β can then be set to give known priors and variance on the total number of infections as and . When α = β, the attack rate prior has an expectation of 0.5n, and the variance may be decreased by increasing α and β. Here, we set α = β = 1. This choice of prior also implicitly assumes that the total number of lifetime infections for an individual is binomially distributed with success probability p = α/(α+β) and N = m_i, where m_i is the number of time periods that individual i could be infected.

Inference using Markov chain Monte Carlo

All models were fitted using the Markov chain Monte Carlo (MCMC) algorithm implemented in the serosolver R package. This is a custom, adaptive Metropolis–Hastings algorithm with alternating univariate normal proposals for the model parameters θ and custom proposals Z for the infection history states. Step sizes for all antibody kinetics parameter proposals were also scaled automatically during the burn in to achieve an acceptance rate of 0.44 for all parameters. Uniform priors were placed on all antibody kinetics parameters θ shown in S4 Table. Infection history state proposals were randomly chosen between one of 2 options: (i) select 2 potential infection times 12 time periods apart and swap infection states for all individuals that could have been infected at these times; (ii) randomly select 20% of individuals and for each individual, with 50% probability, either sample new infection state values for all possible infection times, or choose 2 potential infection times 12 time units apart and swap their values.

For the main model results, 5 chains were run for 50,000,000 iterations, with the first 20,000,000 discarded as burn-in, to achieve an effective sample size of >200 for all inferred parameters. Convergence was assessed visually and using the Gelman-Rubin diagnostic criteria () with the coda R-package [98]. Note that the huge number of iterations is due to the need to impute nearly 4*N*T = 4*1130*190 = 858,800 latent binary variables in Z (i.e., the infection state for each individual in each 3-month window since 1968, slightly less as individuals could not be infected prior to birth). The ability for the model to recover ground-truth parameters was explored through simulation-recovery, described in S2 Text.

Post-processing of infection history posteriors

When fitting to the Fluscape data set, the serosolver model occasionally imputed continuous runs of repeated infections in adjacent 3-month windows (e.g., imputing infections in Q1-1968, Q2-1968, and Q3-1968), reflecting either genuine repeat infections or the model explaining titres that were higher than a single antibody boost could explain. These infection runs were relatively rare, but were most common for infection in young children and occasionally for recent time periods (S18 and S19 Figs). Runs of repeat infections were not estimated when fitting to simulated data, suggesting that their occurrence in the Fluscape data represents either genuine repeat infections early in life or the model explaining titres that were higher than a single antibody boost could explain. Although these reinfections may be real, particularly among children [99,100], this may be a limitation of the model, which sometimes explains high antibody titres through multiple infections rather than from a single infection eliciting a large boost. Our pipeline therefore included a post-processing step to count these runs as single infection events starting on the date of the first infection in the run. This reduced the total number of distinct infection events from 11,544 infections (posterior median; 95% CrI: 11,358 to 11,736) to 10,558 (posterior median; 95% CrI: 10,394 to 10,750). The attack rate estimates and reinfection rates from the posterior samples prior to removing the runs of consecutive infections are shown in S2 Table, showing that although the post-processing step only slightly reduced overall attack rate estimates, it substantially reduced the reinfection rate estimates, which were as high as 8% in 2013 prior to removing consecutive infections.

We also carried out an additional data augmentation step when calculating annual attack rates for more recent time periods, as we are unable to infer infection states after the latest serum sample for each individual. For example, if an individual’s final serum sample is in Q3-2014 and they have been estimated to have no infections in Q1 or Q2, it is not known if the individual should contribute to the numerator and/or denominator when calculating the annual attack rate for 2014. However, because the model assumes that the per-time infection probability is Beta-distributed, it is straightforward to sample new infection states from the model prior to seeing any antibody data using the probability of infection given by (described in [35]):

Where Z_−i,j gives the infection states of all individuals other than i at time j, α, and β are parameters of the Beta prior (assumed to both be 1), k_j is the number of infections in Z_-i,j and n is the number of alive individuals at time point j. For every individual with an unknown infection state following their final serum sample, we resample a new infection state using the above formula for each posterior draw.

Fitting to longitudinal HI titre data from Ha Nam, Viet Nam

To provide a comparison for the model fits to the Fluscape data, we fit the same model to a publicly available data set from Ha Nam, Viet Nam. This data set consisted of 69 participants, each with between 1 and 6 (inclusive) serum samples taken annually from 2007 to 2012 as described previously [37]. In this cohort, HI assays were performed against a panel of up to 57 A/H3N2 strains isolated between 1968 and 2008, with greater sampling of titres against more recent strains. This represents a comparable data set with different dimensions: a much smaller sample size, but far more titres and samples tested per individual. Unlike the Fluscape data, we only had access to the year of sample collection, and thus we could only estimate infection histories at an annual resolution. Furthermore, we did not have dates of birth available for the individuals, and thus we assumed that all individuals were born prior to 1968, noting that age signals will still be detected based on the individual’s antibody profile. For this analysis, we did not include the strain-specific measurement offsets in the observation model, as there were multiple strains tested for each time period. We ran the MCMC algorithm for 5 chains each for 1,500,000 iterations and discarded the first 500,000 iterations as burn-in, which was sufficient to achieve effective sample sizes of >200 and upper 95% confidence intervals of the values of less than 1.1 for all estimated parameters.

Spatial correlation in inferred attack rates

To test for patterns of influenza incidence in space, nonparametric correlation as a function of distance was tested using the Sncf function from the ncf R-package [101], where observations were the quarterly attack rate estimates stratified by location ID (40 locations, 190 observations per location). We fit these spline correlograms to 100 samples from the posterior distribution of attack rates, resampling the 40 locations with replacement for each sample to generate posterior medians and 95% CrIs. This analysis was repeated using estimated attack rates since 1968, since 2009, and pre-2009. To provide a comparable null simulation where no significant spatial variation would be expected, we simulated 40 draws from a binomial distribution with n = 25 (i.e., 1,000 individuals across 40 locations) and success probability drawn from a uniform distribution between 0 and 1, and repeated this process 10,000 times to calculate the mean and 95% quantiles of the resulting coefficients of variation.

Relationship between titre and probability of infection

For the analyses investigating the relationship between titre at time of infection and probability of infection, we drew samples from the estimated posterior distribution of antibody kinetics parameters and infection histories, and calculated model-predicted preinfection latent antibody titres against the circulating strain for each individual at each possible infection time. Predicted titres were converted to integers, with titres ≥8 assumed to be 8 to match the observation process. We then calculated the proportion of time periods where infection occurred (Z_i,j = 1) (the overall probability of infection) stratified by log titre at the time of infection relative to the overall probability of infection with a log titre of 0. As discussed above, some of the inferred infection histories had runs of infections in consecutive time periods (e.g., Z_i = [0, 0, 0, 1, 1, 1, 0, 0]). We removed these consecutive infections from the probability of infection statistics (e.g., [0, 0, 0, 1, 1, 1, 0, 0] becomes [0, 0, 0, 1, 0, 0, 0, 0]). We repeated this process for 1,000 posterior samples to generate median and 95% CrI estimates on the relationship between titre and relative risk of infection.

Validation using self-reported vaccination

We validated our inference approach by comparing the model-predicted probability of infection to self-reported influenza vaccination. Vaccination coverage was low in this cohort (S1 Table), and self-reported vaccination status is not a perfect reflection of true vaccination status. Furthermore, vaccination status was reported only within a window of time rather than on a specific date: individuals were asked if they had ever received an influenza vaccination and could give one of 6 answers: never vaccinated; in the same calendar year; in the preceding calendar year; 2 to 5 years ago; 5+ years ago; or unsure. Individuals were also asked if they had been vaccinated since their last study visit. Where an individual reported vaccination in one of these time periods, we define the covered time period as the “vaccination window.”

To assess the accuracy of our inference approach, we defined sensitivity as the proportion of vaccination windows +/− 3 months which contained at least 1 inferred infection with more than 25% posterior probability (i.e., we expect vaccination windows to contain inferred infections if our inference is accurate). However, a complication is that these windows are wide—an individual reporting vaccination 2 to 5 years ago has a 3-year wide vaccination window, and thus the individual is likely to have been infected during that time regardless of vaccination. Therefore to provide a comparable null simulation, we took each of the windows of time with reported vaccination and reassigned them to randomly chosen individuals at random times (e.g., if individual 1 reported vaccination between Q1-2013 and Q1-2015; the null simulation reassignment might move the window to individual 2 in Q2-1980 to Q2-1982, regardless of individual 2’s vaccination status). We repeated this process 100 times to generate 95% uncertainty intervals for the null simulation. We then compared the proportion of vaccination windows which contained inferred infections based on reported vaccination to the null simulations. If the upper 95% uncertainty interval of the null simulation was lower than the proportion of vaccination windows with inferred infections using the real data, we took this as evidence that our model was more likely to infer antibody boosting events that coincided with self-reported vaccination than in randomly selected time periods.