Are you trying to infer the contact pattern from the infections? Or are you hoping to prescribe the contact pattern (eg via strong priors) and let this somehow “stabilize” the infection inference?
The second.
I think all of the other points can be handled “as is” by the model proposed here…
I have revisited the “simplex” model you have offered in that post, as it has potential. I’d like to get a better understanding of it but, after an online search, I cannot find a reference to such an approach.
I fail to understand how the transition rate between S and E_1, \beta \frac{I_1 + I_2}{N} S, is handled. See the Liverpool model description.
Let’s continue this discussion in the CoDatMo thread.