Reopening the economy; cost benefit analysis

Elsewhere I provide a back-of-the-envelope cost-benefit statistic to guide reopening the economy from the covid shutdowns; here I want to extend the model a bit.  In that post I work in terms of the sort of model in which we use a reproduction number — in particular, a fairly homogeneous model.  There I look at assessing the cost of creating opportunities for the disease to spread, and here I want to allow at least some variation in that.

For at least the past month I have been largely thinking about the epidemiology of the disease in terms of multiple populations; these could be counties or states, and it was county and state data that moved me toward this framework, but I've occasionally thought in terms of multiple populations in the same area, one of which engages in much more "social distancing" than the other.  I think the following crude model will be sufficient for my purposes: suppose time is discrete, and consider a vector v at each time, with each component indicating infectious cases in a particular area, and v_t+1=Rv_t, where the reproduction number R is now a matrix instead of a number.  I don't suppose that it's constant with time, but I am going to consider changing one element R_ij at a single moment in time with R unchanged at all other times; to be clear, it may be changing over time, but the counterfactual follows the same path as the baseline scenario except for a single element of the matrix at a single time.

For T>t+1 let M_T be the product of the transition matrices from time t+1 to time T-1, such that v_T=M_TRv_t. The change in v_T,l due to a change in R_ij is the amount of that change times M_T,li v_t,j. If you can place a cost on each exposure — a cost that may be different at different times and different for different populations — and multiply each row of each M_T by the relevant cost and then add up the M, you get a matrix C; the associated marginal cost of an increase in R_ij is C_liv_t,j, i.e. the relevant cost from the C matrix times the current prevalence in the population from which we're increasing the spread.[1]  The real question, now, is how to get any kind of bead on C.  In parallel with the earlier post, I'll note that the sum over rows of CRv is the total cost of all future infections; this will allow us to make contact with other attempts to do a cost benefit analysis on the entire crisis.

I'm not sure there's benefit in producing more formulas by imposing more structure on C; I could note that, if R were constant (and all of its eigenvalues below 1) and the costs were constant or exponentially declining, then we would basically have C=(1-R)^-1, and even with varying R we can maybe read something off of that. It's worse to transmit infections to places that tend to transmit more infections; it's worse to transmit infections to places that do so, and so on.

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[1] This may seem on some level obvious; I would kind of hope it does.  Note an implication, though: there is substantial benefit in cutting transmission from a hotspot, and that benefit is largely independent of whether it's to a hotspot or seeding a new location.  In conversation I sometimes get the impression that people think that, well, if we're trading people between hotspots, that doesn't matter, but if each infected person in each place transmits the disease, on average, to 0.8 people in their hotspot and 0.4 in the other, you will get exponential growth that could be eliminated by stopping the interchange.