Hierarchical Bayesian models
In serosv: Model Infectious Disease Parameters from Serosurveys

knitr::opts_chunk$set(
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library(serosv)

Parametric Bayesian framework

Currently, serosv only has models under parametric Bayesian framework

Proposed approach

Prevalence has a parametric form $\pi(a_i, \alpha)$ where $\alpha$ is a parameter vector

One can constraint the parameter space of the prior distribution $P(\alpha)$ in order to achieve the desired monotonicity of the posterior distribution $P(\pi_1, \pi_2, ..., \pi_m|y,n)$

Where:

$n = (n_1, n_2, ..., n_m)$ and $n_i$ is the sample size at age $a_i$

$y = (y_1, y_2, ..., y_m)$ and $y_i$ is the number of infected individual from the $n_i$ sampled subjects

Farrington

Refer to Chapter 10.3.1

Proposed model

The model for prevalence is as followed

$$ \pi (a) = 1 - exp{ \frac{\alpha_1}{\alpha_2}ae^{-\alpha_2 a} + \frac{1}{\alpha_2}(\frac{\alpha_1}{\alpha_2} - \alpha_3)(e^{-\alpha_2 a} - 1) -\alpha_3 a } $$

For likelihood model, independent binomial distribution are assumed for the number of infected individuals at age $a_i$

$$ y_i \sim Bin(n_i, \pi_i), \text{ for } i = 1,2,3,...m $$

The constraint on the parameter space can be incorporated by assuming truncated normal distribution for the components of $\alpha$, $\alpha = (\alpha_1, \alpha_2, \alpha_3)$ in $\pi_i = \pi(a_i,\alpha)$

$$ \alpha_j \sim \text{truncated } \mathcal{N}(\mu_j, \tau_j), \text{ } j = 1,2,3 $$

The joint posterior distribution for $\alpha$ can be derived by combining the likelihood and prior as followed

$$ P(\alpha|y) \propto \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha)) \prod^3_{i=1}-\frac{1}{\tau_j}\text{exp}(\frac{1}{2\tau^2_j} (\alpha_j - \mu_j)^2) $$

Where the flat hyperprior distribution is defined as followed:
- $\mu_j \sim \mathcal{N}(0, 10000)$
- $\tau^{-2}_j \sim \Gamma(100,100)$

The full conditional distribution of $\alpha_i$ is thus $$ P(\alpha_i|\alpha_j,\alpha_k, k, j \neq i) \propto -\frac{1}{\tau_i}\text{exp}(\frac{1}{2\tau^2_i} (\alpha_i - \mu_i)^2) \prod^m_{i=1} \text{Bin}(y_i|n_i, \pi(a_i, \alpha)) $$

Fitting data

To fit Farrington model, use hierarchical_bayesian_model() and define type = "far2" or type = "far3" where

type = "far2" refers to Farrington model with 2 parameters ($\alpha_3 = 0$)
type = "far3" refers to Farrington model with 3 parameters ($\alpha_3 > 0$)

df <- mumps_uk_1986_1987
model <- hierarchical_bayesian_model(age = df$age, pos = df$pos, tot = df$tot, type="far3")

model$info
plot(model)

Log-logistic

Proposed approach

The model for seroprevalence is as followed

$$ \pi(a) = \frac{\beta a^\alpha}{1 + \beta a^\alpha}, \text{ } \alpha, \beta > 0 $$

The likelihood is specified to be the same as Farrington model ($y_i \sim Bin(n_i, \pi_i)$) with

$$ \text{logit}(\pi(a)) = \alpha_2 + \alpha_1\log(a) $$

Where $\alpha_2 = \text{log}(\beta)$

The prior model of $\alpha_1$ is specified as $\alpha_1 \sim \text{truncated } \mathcal{N}(\mu_1, \tau_1)$ with flat hyperprior as in Farrington model

$\beta$ is constrained to be positive by specifying $\alpha_2 \sim \mathcal{N}(\mu_2, \tau_2)$

The full conditional distribution of $\alpha_1$ is thus

$$ P(\alpha_1|\alpha_2) \propto -\frac{1}{\tau_1} \text{exp} (\frac{1}{2 \tau_1^2} (\alpha_1 - \mu_1)^2) \prod_{i=1}^m \text{Bin}(y_i|n_i,\pi(a_i, \alpha_1, \alpha_2) ) $$

And $\alpha_2$ can be derived in the same way

Fitting data

To fit Log-logistic model, use hierarchical_bayesian_model() and define type = "log_logistic"

df <- rubella_uk_1986_1987
model <- hierarchical_bayesian_model(age = df$age, pos = df$pos, tot = df$tot, type="log_logistic")

model$type
plot(model)