model_details.md

Model details

These are the mathematical details of the models used to capture and forecast vaccine uptake. There are currently just one model: a mixture of a logistic and linear function. This model proposes a latent true uptake curve, which is subject to observation error. A hierarchy accounts for the unique effects of grouping factors (e.g. season, geography, age) on model parameters.

Logistic Plus Linear (LPL) Model

Notation

The following notation will be used for the LPL model:

• $t$ = time since the start of the season, expressed as the fraction of a year elapsed
• $V_t^{obs}$ = number of people surveyed at time $t$ who are vaccinated
• $N_t^{obs}$ = total number of people surveyed at time $t$
• $c_t$ = latent true cumulative uptake on day $t$
• $G$ = grouping factors (e.g. season, geographic area, age group, race/ethnicity), indexed by $i$ with $I$ total factors

Summary

At a high level, the LPL model is structured as follows:

$$\begin{align*} &V_{t,G}^{obs} \sim \text{Pr}(V_{t,G}^{obs}~|~c_{t,G},~N_{t,G}^{obs}) \\\ &c_{t,G} := f_{\text{Logistic + Linear}}(t,~\phi_G) \\\ &\phi_G \sim \text{Pr}(\phi_G~|~\xi) \\\ &\xi \sim \text{Pr}(\xi) \\\ \end{align*}$$

Here, $t$ is rescaled by dividing by 365, so that $t$ represents the proportion of a season elapsed. Additionally, $V_{t,G}^{obs}$ and $N_{t,G}^{obs}$ are inferred from $c_{t,G}^{obs}$ and its reported 95% confidence interval, by assuming the latter is a Wald interval representing $1.96$ standard errors of the mean in each direction from $c_{t,G}^{obs}$. As a result, the standard error of the mean $\sigma_{t,G}^{SEM}$ is considered known for each data point, and $V_{t,G}^{obs}$ and $N_{t,G}^{obs}$ are as follows:

$$\begin{align*} &N_{t,G}^{obs} = \frac{c_{t,G}^{obs} \cdot (1-c_{t,G}^{obs})}{{\sigma_{t,G}^{SEM}}^2} \\\ &V_{t,G}^{obs} = N_{t,G}^{obs} \cdot c_{t,G}^{obs} \\\ \end{align*}$$

Observation Layer

The observed uptake is considered a draw from the beta-binomial distribution, governed in part by the true latent uptake in the population.

$$\begin{align*} &V_{t,G_1,...,G_I}^{obs} \sim \text{BetaBinomial}(\text{shape1 = }\alpha_{t,G_1,...,G_I}, \text{ shape2 = }\beta_{t,G_1,...,G_I}, \text{ N = }N_{t,G_1,...,G_I}^{obs}) \\\ \end{align*}$$

Note that the shape parameters $\alpha$ and $\beta$ are not declared explicitly. Rather they are implied by an alternate mean and concentration parametrization, described below.

Functional Structure

The model's functional structure describes the latent true uptake curve:

$$\begin{align*} &c_{t,G_1,...,G_I} = \frac{A_{G_1,...,G_I}}{1 + e^{-n \cdot (t - H)}} + M_{G_1,...,G_I} \cdot t \\\ \end{align*}$$

$c_{t,G_1,...,G_I}$ serves as the mean of the beta distribution in the beta-binomial likelihood in the observation-layer. A fixed concentration parameter $d$ is also required. From the mean and concentration, the two shape parameters of the beta distribution are as follows:

$$\begin{align*} &\alpha_{t,G_1,...,G_I} = c_{t,G_1,...,G_I} \cdot d \\\ &\beta_{t,G_1,...,G_I} = (1 - c_{t,G_1,...,G_I}) \cdot d \\\ \end{align*}$$

Hierarchical Structure

Certain parameters of the latent true uptake curve have group-specific deviations, determined as follows:

$$\begin{align*} &A_{G_1,...,G_I} = A + A_{G_1} + ... + A_{G_I} \\\ &\frac{A_{G_i}}{\sigma_{A_{G_i}}} \sim \text{Normal}(\text{location = }0, \text{ scale = }1) ~\forall~i~\text{ in } 1, ..., I \\\ \end{align*}$$

and similarly for $M$.

Priors

$$\begin{align*} &A \sim \text{Beta}(\text{shape1 = }100.0, \text{ shape2 = }180.0) \\\ &\sigma_{A_{G_i}} \sim \text{Exponential}(\text{rate = }40.0) ~\forall~i~\text{ in } 1, ..., I \\\ &H \sim \text{Beta}(\text{shape1 = }100.0, \text{ shape2 = }225.0) \\\ &n \sim \text{Gamma}(\text{shape = }25.0, \text{ rate = }1.0) \\\ &M \sim \text{Gamma}(\text{shape = }1.0, \text{ rate = }10.0) \\\ &\sigma_{M_{G_i}} \sim \text{Exponential}(\text{rate = }40.0) ~\forall~i~\text{ in } 1, ..., I \\\ &d \sim \text{Gamma}(\text{shape = }350.0, \text{ rate = }1.0) \\\ \end{align*}$$