They also pointed us to this article from December 2019, Contemporary statistical inference for infectious disease models using Stan, by Anastasia Chatzilena et al. I guess this particular paper will be useful for people getting started in this area, or for epidemiologists who’ve been hearing about Stan and would like to know how to use it for differential equation models in epidemiology. I have not read the article in detail.
We’re also doing some research on how to do inference for differential equations more efficiently in Stan. Nothing ready to report here, but new things will come soon, I hope. One idea is to run the differential equation solver on a coarser time scale in the NUTS updating and, use importance sampling to correct the errors, and then run the solver on the finer time scale in the generated quantities block.