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?
I think all of the other points can be handled “as is” by the model proposed here: CoDatMo Liverpool & UNINOVE models (slow ODE implementation and Trapezoidal Solver) - #45 by Funko_Unko
Though I don’t know what other developments have happened since then, but I guess maybe @s.maskell knows?