I don’t enough technical knowledge to understand your model, but I would first do what the error message suggests and set the init_r parameter lower then 2 and see what you get.
fit = sampling(
sm,
data=stan_data,
chains=3,
iter=600,
warmup=200,
thin=2,
init_r = .1
)
On the other hand, I would look for parameters that might need to have specified lower or upper boundaries and set them up on the pare meters block.
Finally, on my modeling experience, sometimes I have models where a parameter value cannot exceed values of the data. For these cases, I set the the initial values to a very small to avoid any issues. Might not be the case in your model, but there might be parameters that if initialized in a very extreme value might make the whole sum inside the poisson distribution negative. Which might be the source of your problem.