I think I now understand the problem identified in my previous post, at least partially. Looking at the section "Make the susceptible neighbors of infected agents meet their neighbors" in the efficient method, and in particular the line Flatten[Lookup[neighbors,Normal[stateAgents["I"]]],1], this picks out all agents that are connected to an infected agent. However, in cases where an agent is connected to more than one infected agent they are then double-counted, effectively being given more than one opportunity to be infected in a single step (note that the fact that they are more likely to be infected if they are connected to more than one infected agent is already taken account of elsewhere).

A simple fix is to change the line to DeleteDuplicates[Flatten[Lookup[neighbors,Normal[stateAgents["I"]]],1]]. This significantly improves the agreement with the slower implementation, although there is still some discrepancy, with the efficient method now slightly favoring a later and lower peak.

I made that change and recomputed everything. None of the results look significantly affected, but thanks for finding that bug. The main post has been updated to show results from the new version.

Thanks for this very interesting post.

I did some experiments with your two different implementations and found that they behave differently in a statistically significant sense. To verify this, I ran your simple example a large number of times (1000 simulations) using both implementations and extracted the time and magnitude of the infection peak. The attached histograms show the results. Orange is the "Efficient implementation" and blue is the other implementation. You can see that the infection peak is earlier and higher when run with the efficient implementation, suggesting that it has a higher effective transmission rate.

I thought I understood both implementations, but there must be something I am missing as I can't see why they should behave differently. Can anyone explain why I am finding differences?

Nice post, Christopher, thanks. It shows the power of Mathematica to generate and manipulate efficiently highly structured systems. We're working on similar structured-contact covid model for local settings (e.g. hospitals, city districts et al). Would be glad to share it, when ready

Dear Christopher, Thank you for your prompt reply! I believe I did not run out of memory. I am trying again, using ParallelTable.

Impressive project! Congratulations! I wonder what is the minimum configuration of a Windows 10 Pro workstation to evaluate the notebook, as I was not able to evaluate superGraphSimsByConnectivity, using a workstation with an Intel Core i9-9900K @ 3.60 GHz CPU, 32.0 GB RAM, and Mathematica 12.1.

Excellent post @Christopher Wolfram , I was working along the same lines. I'll switch the focus to cover other interesting aspects of using this type of models.

Outstanding post, Christopher!

It's intriguing that the shape of the infected curve in the Simple Example section is fairly symmetrical and does not have the long exponential tail.

Generating the phase plane analysis from the simulations is amazing. I've seen (and done) these analyses using the traditional calculus-based approach, but it's wonderful that you can do the same with this model and show that many of the same principles apply.