Choice of alpha value in the HI protection curve

[mhoehle]

morevac/src/infect_cpp_2.cpp

Line 61 in 93b9c49

inf_comp = (1/(1 + exp(1.299*((log(200)-delta_x(i,j)-log(2.844))))));

The Ainslie and Riley (2022) paper writes on p. 2942 that $\alpha=1.045$ was selected. The Coudeville et al. (2010) is a little mysterious about the prior of $\alpha$ (for some not so clear reason they write \alpha \sim LogN(\mu_{alpha}, \sigma_{\alpha}^2) with a point estimate for $\mu_{\alpha}$ in Tab. 3 reported as 2.844.
You seem to choose to log-transform this value, however, I think the prior really is $\alpha \sim N(\mu_{\alpha}, \sigma_{\alpha}^2)$, because then $\exp(2.844)$ would match a 50% protection titer of around 17, which they report to be around 1:17 for the ALL model.

Also, I need $\alpha=2.844$ to reproduce their figure 4.

#' Infection protection curve according to Coudeville et al. (2010)
coudeville <- function(T, alpha=2.844, beta=1.299) {
  plogis( beta*(log(T) - alpha))
}
#' Plot for the two alpha values
df <- data.frame(T=c(seq(0.1,1,length=100),seq(1, 200, length=1000))) %>% rowwise %>% mutate(pi_2.844 = coudeville(T), pi_1.045=coudeville(T, alpha=log(2.844)))
ggplot(df, aes(x=T, y=pi_2.844, alpha="2.844")) + 
  geom_line() + 
  geom_line(aes(y=pi_1.045, alpha="1.045")) + 
  theme_minimal() + 
  #xlab("log(T)") +
  scale_color_brewer(pal="Set1") +
   scale_y_continuous(breaks=seq(0,1,by=0.2)) +
  scale_x_continuous(breaks=seq(0,200,by=20)) + #trans='log2') +
  ylab(expression(pi(T~"|"~alpha,beta)))

My question is just, if I'm missing some important thoughts in your paper or if $\alpha=2.844$ would be the more appropriate choice?