We will collect and combine data from previously published articles and reports for this aim.

Preliminary results

We have created a simple DAG (Figure 1) describing what we believe are the most important variables on the causal pathway between vaccine/circulating strain antigenic distance (AD) and vaccine effectiveness (VE). In this framework, we believe that the circulating strain (CS) and vaccine strain (VS) will each have direct effects on population VE, as well as leading to the antigenic distance result. We also assume that pre-season immunity (PT), as indicated by HAI titer, will impact VE, although we will not have this information as part of our data set. The vaccine strain in the previous season’s vaccine (PVS) may also impact VE, as described in the antigenic distance hypothesis (Skowronski et al. 2017). Vaccine uptake (VU) may also impact VE through changes in direct and indirect vaccination effects (Eichner et al. 2017). Demographic factors (D) such as age, race, etc. influence VE both directly and indirectly through (VU). We plan to collect data on CS, VS, PVS, and VE in all cases, and will also collect and examine impacts of VU and D as data are available. AD will be calculated using VS and CS.

Figure 1: Relevant exposures, outcomes, and covariates on the causal pathway between antigenic distance (AD) and vaccine effectiveness (VE). Exposures (yellow) include AD, circulating influenza strain (CS), vaccine strain (VS), pre-season HAI titers (PT), previous year vaccine strain (PVS), and population vaccine uptake (VU). Demographics (D) is included as a covariate (orange). Our outcome (gray) of interest is VE.

Proposed studies

We will calculate VE using multiple methods, including p_epitope distance (Gupta, Earl, and Deem 2006; pan2011?; Pan and Deem 2016), Grantham distance (Grantham 1974), and temporal distance (Auladell et al. 2022; Boyoglu-Barnum et al. 2021-03-24, 2021-03; Hinojosa et al. 2020-10-22, 2020-10; Li et al. 2021-02-23, 2021-02; Yang et al. 2020; Carlock and Ross 2023). We will then use Bayesian mixed-effects meta-regression (Equation 1 and Table 1) to examine the effect of our predictors (Table 2) on the VE outcome.

\[ \begin{aligned} \hat\theta_k &\sim \mathcal{N}(\theta_k,\sigma^2_k)\\ \theta_k &\sim \mathcal{N}(\mu_k, \tau^2), \quad \mu_k = \beta_0 + \sum_{j=1}^p\beta_jx_{j,k} + b^{(s)}_{s[k]} + b^{(x_1t)}_{t[k]} \cdot x_{1,k} \\ \tau^2 &\sim \text{Student's} \space t^+(3,0,1)\\ \begin{bmatrix} b_s^{(s)} \\ b_s^{(x_1s)}\\ \end{bmatrix} &\sim \text{MVN}(\vec0, \Sigma)\\ \Sigma_s &= (\text{diag}(\sigma_s)\text{L}_s) \times(\text{diag}(\sigma_s)\text{L}_s)^\text{T}\\ \beta_j &\sim \mathcal{N}(0, \sigma^2_j)\\ \sigma^2_{(\cdot)} &\sim \text{Student's} \space t^+(3,0,1)\\ \text{L}_t &\sim \text{LJT}(2)\\ \end{aligned} \tag{1}\]

Table 1: Descriptions of terms used in Equation 1

Term	Interpretation
\(\hat\theta_k\)	Observed estimate from study \(k\)
\(\theta_k\)	True underlying effect for study \(k\)
\(\sigma^2_k\)	Known sampling variance for \(\hat\theta_k\)
\(\beta_0\)	Fixed intercept
\(\beta_j\)	Fixed slopes for predictors \(x_j, j \ge 1\) as described in Table 2
\(\sigma^2_j\)	Prior variance for fixed effect \(\beta_j\)
\(\tau^2\)	Between-study variance for true effects \(\theta_k\)
\(\Sigma\)	Covariance matrix governing the joint variability and correlation of subtype-varying random effects
\(\begin{bmatrix} b_s^{(s)} \newline b_s^{(x_1s)} \end{bmatrix}\)	Jointly modeled random intercept (\(b_s^{(s)}\)) and random slope for \(x_1\) (\(b_s^{(x_1s)}\)), varying by influenza subtype \(s\)
\(b^{(s)}_{s[k]}\)	Random intercept/effect for influenza subtype/lineage \(s\) for study \(k\)
\(b^{(x_1s)}_{s[k]} \cdot x_{1,k}\)	Random slope for antigenic distance \(x_1\), varying by subtype \(s\) for study \(k\)
\(\sigma^2_{(\cdot)}\)	Prior variance terms for hierarchical parameters, all governed by a half-Student’s \(t\) distribution
L\(_t\)	Cholesky factor of a correlation matrix for time-varying random effects, with a Lewandowski–Kurowicka–Joe (LKJ) prior

Table 2: Descriptions of fixed effects \(\beta_j\)

Predictor	Interpretation
\(d\)	Antigenic distance between vaccine and circulating strain
\(t\)	Season
\(a\)	Age group
\(p_s\)	Proportion of infections caused by subtype/lineage \(s\)
\(r\)	Number of consecutive years relevant vaccine strain has been included in vaccine as of season \(t\)
\(c\)	Antigenic distance between this season’s and last season’s relevant vaccine strain

Expected outcomes

Figure 2: Theorized potential relationships between antigenic distance and VE. Created with BioRender.com.

References

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