Estimation of Seasonal Influenza Attack Rates and Antibody Dynamics in Children Using Cross-Sectional Serological Data

Abstract Directly measuring evidence of influenza infections is difficult, especially in low-surveillance settings such as sub-Saharan Africa. Using a Bayesian model, we estimated unobserved infection times and underlying antibody responses to influenza A/H3N2, using cross-sectional serum antibody responses to 4 strains in children aged 24–60 months. Among the 242 individuals, we estimated a variable seasonal attack rate and found that most children had ≥1 infection before 2 years of age. Our results are consistent with previously published high attack rates in children. The modeling approach highlights how cross-sectional serological data can be used to estimate epidemiological dynamics.


Supplementary information Ethics Statement
This study was approved by The Gambia Government and UK Medical Research Council (MRC) joint ethics committee and the Medicines Control Agency of The Gambia.

Modelling antibody titre
We infer infection times by mathematically modelling the underlying antibody response individuals have to different test strains. We define the infection history for each individual i, Z i , as a vector of binary latent states for the presence or absence of infection at each time point.
The expected log-titre for an individual i against a strain circulating at time j and observed at time t was a linear combination of antibody responses from each prior infection, where µ l , µ s are the long-and short-term boosts of the antibody response. The short-term boost linearly wanes over time at a rate ω. The long-and short-term boosts are scaled down according to the long-and short-term level of cross reaction d l (), d s () between the infecting strain (k) and the test strain (j). Cross-reactivity was modelled as d l,s (j, k) = max(0, 1 − σ l,s δ k,j ) where δ k,j was the two-dimensional antigenic distance between strains k and j and σ l,s are the longand short-term cross-reaction parameters to be fitted. Here, the long-term boost is specified as a function of the short term boost as follows µ l = aµ s where a is bounded between 0 and 1.
Similarly, we specify the short term cross-reaction term as σ s = bσ l where b is bounded between 0 and 1.
We assume that the strains that circulated during our study period reflect those in the World Health Organisation (WHO) influenza vaccine recommendations for the Northern hemisphere (Table S1) and that within these circulation periods there was no detectable change in antigenic distance.
We used the inference framework described in [1] to infer the time of infection and June 26, 2020 1/6 The analysis was performed using the R package serosolver (https://github.com/seroanalytics/serosolver). The specific case study code can be found under the branch alternative parameterisation as well as a subset of anonymised data.

Priors
We assumed some prior knowledge of the distribution of antibody process parameters (Table S2).
The long term boost, waning and cross-reaction prior distributions were informed by independent model fits from data sets of HI titres from [2]. Table S2. Model parameters and prior distributions. Parameter Prior Distribution Long term boost (µ l ) Gamma(shape = 9, rate = 6). Mean = 1.5 Waning Estimation of childhood influenza attack rates process parameters using the Metropolis within Gibbs scheme presented by [1]. From these measures we can calculate epidemiological measures of interest.

Posteriors
Using the inferred time of infections, we calculated the age at first infection for individuals in the cohort. Figure S1 shows that age distribution for different estimated numbers of infections.
Those individuals with the highest number of infections also were the oldest at the time of sampling. The posterior distributions for the antibody process parameters for the titre data, where applicable the prior distribution density has been added in grey are shown in Figure S2.

Sensitivity analyses
For each sensitivity run, three chains were run and the Gelman-Rubin diagnostic was calculated for the antibody process parameters. In all cases, the upper limit of the confidence interval was less than 1.1, indicating that the chains had converged. For the main result, we ran three separate chains to confirm convergence ( Figure S3).

Antibody parameter prior information
In the main report and in Figure S2 we implemented prior distributions informed by previously published studies. With cross-sectional data, the waning rate of the antibody response cannot be estimated but it is of interest to understand how robust our attack rate estimates are to less prior information on the remaining antibody parameters. The estimated attack rates and likelihood chain plots in Figure S4 with prior information on the waning only, were similar to the estimated attack rates with all prior information incorporated.

Lower attack rate in 2013
Due to the low incidence of A/H3N2 observed in Senegal in 2013 [3], we imposed prior information that the 3rd quarter in 2013 had a low attack rate. In the absence of this additional prior information, we see that the estimated attack rates in 2013 has a low median but a much wider range ( Figure S5), and that the attack rates were similar in all other time points.
June 26, 2020 5/6 Estimation of childhood influenza attack rates Fig S5. Estimated attack rate for the Gambia cohorts with 95% quantile interval of three different MCMC chains using titre response when the lower prior in 2013 was not imposed.