Bayesian inference of antigenic and non-antigenic variables from haemagglutination inhibition assays for influenza surveillance

Haemagglutination inhibition (HI) assays are typically used for comparing and characterizing influenza viruses. Data obtained from the assays (titres) are used quantitatively to determine antigenic differences between influenza strains. However, the use of these titres has been criticized as they sometimes fail to capture accurate antigenic differences between strains. Our previous analytical work revealed how antigenic and non-antigenic variables contribute to the titres. Building on this previous work, we have developed a Bayesian method for decoupling antigenic and non-antigenic contributions to the titres in this paper. We apply this method to a compendium of HI titres of influenza A (H3N2) viruses curated from 1968 to 2016. Remarkably, the results of this fit indicate that the non-antigenic variable, which is inversely correlated with viral avidity for the red blood cells used in HI assays, oscillates during the course of influenza virus evolution, with a period that corresponds roughly to the timescale on which antigenic variants replace each other. Together, the results suggest that the new Bayesian method is applicable to the analysis of long-term dynamics of both antigenic and non-antigenic properties of influenza virus.

Also, if a random variable X follows a gamma distribution, then the probability distribution function of X is given by: where 0 > x , the shape 0 > α and the rate 0 > β .
For an entire collection of HI titres, H, it follows from applying the Bayes' Theorem that the full posterior density of all the parameters can be expressed as: and H is the collection of the entire HI titres. Note that is the prior.
Therefore, expanding the expression to the right hand side of S6 results in the following:

S7
From S7, it can be readily shown that the full conditional distribution of all the parameters, θ , are summarized as follows: .
The superscripts and subscripts of variables denote value of a variable for a particular virus. For instance, A Y denotes the concentration of antibodies derived against virus Y. Additional details of Bayesian inferences are discussed in the literature [3,4].

Sampling the values of Parameters
Since the full conditionals have been determined, we proceed to construct Markov chain samples with the Gipps sampler [5,6]. We expect the samples to be acceptable since they are produced from the entire domain of the posterior distribution [7]. In addition, autocorrelation function plots of the samples of each parameter further supported the independence of the samples of estimates ( Figures   S1-S3). Given independent samples, large samples of size 100,000 were selected with 2% as burnin [8]. Moreover, no significant differences between estimates of different simulations with the chosen sample size were observed which motivates the choice of the sample size. We obtained the mean of the samples as the estimate of the HI titre parameters.
In order to guide the sampling within acceptable range of the HI titres, initial values for the priors were searched from the literature. In particular, Ndifon and coworkers [9] estimated that high concentrations of antibodies in antisera from influenza virus-infected animals are of the order of 1.67nM. Furthermore, the average affinities of such antibodies were also found to range from 6M per M to 1G per M for influenza A (H3N2) subtypes [1]. On the other hand, initial values of non antigenic variables were estimated from the viral avidities (up to 50ug/mL RDE) as reported in [10].
This is because the avidity of virus for red cells mainly composes the non-antigenic variables [10].
More specifically, utilizing the prior knowledge about the HI titre parameters in the literature, initial values of the parameters within the expected ranges were randomly generated from the log-normal distribution such that the means of the samples were similar to the estimates reported in the earlier studies. This log-normal distribution was selected since log-transforming such samples will result in a normal distribution required for the sampling and the inference as indicated in Equation S2.
Furthermore, the estimates obtained in this manner compared favourably well with the literature when the HI titre data were compared with the titre from the aggregation of the estimates obtained by our methods described herein. In particular, the Mann-Whitney test (with 0.05 level of significance) showed that there were no statistically significant differences between the two titres (p = 0.91). R codes were written to perform all simulations.