int<lower = 0> param_id_alpha; // parameter id of alpha (GP magnitude)
int<lower = 0> param_id_rho; // parameter id of rho (GP lengthscale)
int<lower = 0> param_id_R0; // parameter id of R0
int<lower = 0> param_id_fraction_observed; // parameter id of fraction_observed
int<lower = 0> param_id_reporting_overdispersion; // parameter id of reporting_overdispersion
int<lower = 0> param_id_pop; // parameter id of pop
