I’m not convinced that this is the correct transformation for going from Ne(t)-> I(t).
The formula used here is based on the paper by Koelle et al and was derived under epidemic equilibrium. A different formula applies under exponential growth. See here:
Using that transformation would give you much bigger I(t), and FWIW I would also find that more realistic.
And in case it’s useful, I have also derived how it depends on the variance in transmission rates, which you can use as an alternative to the offspring distribution.
where m_i is the i’th moment of a distribution of transmission rates.
When the transmission rate is constant \beta that reduces to
which is the same as in Volz 2012. You could see what you get if you plug in a rate of about 108 transmissions/year which corresponds to R0=2.5 with a generation time of 8.4 days. It would be a lot bigger.
Another potential problem is related to the upper bound of Ne=10. I looked through the logs and it looks like the posterior bumps up against the boundary which would curtail the upper bound of case estimates.