Chad Jones and Jesús Fernández-Villaverde have updated their SIR model with social distancing. A part I find very intriguing is that they impute the infection rate and the reproduction rate from death rate data. The infection rate \(I_t\) is given by \[I_t = \frac{1}{\delta \gamma} \left( \frac{d_{t+2}-d_{t+1}}{\theta} - d_{t+1} \right)\] where the greek letters are parameters they estimate by fitting the path of deaths over time, and \(d_t\) is the daily death rate. Though deaths only happen a few weeks after infection, you can reverse the model dynamics to figure out how many are infected today from how many are dying today. (Well, tomorrow and the day after). They similarly infer today's reproduction rate \(R_0\) from the next three days death rates.
Now, there is clearly some inaccuracy here, and I've been pestering them to provide standard errors. There is some noise in daily deaths and once you start double and triple differencing them, the noise is larger.
But as I think about behavioral and policy responses, these are the numbers we need. How many people in this state, city, zip code, grocery store, bar, are infectious right now? 1 in 10? 1 in 100? 1 in 1000? 1 in 10,000? Is the virus spreading or slowly decaying, with reproduction rate below one? Just how careful do we need to be? Is wiping down, surfaces or spraying luggage with disinfectant remotely cost-effective? Where are hot spots?
If we had spent 1/1,000,000 of the $5 trillion the government is spending on random testing, we would know the answer to this question. We don't. We do have death data. So a measurement with error of the thing we need the most is potentially quite valuable.
Reproduction rates seem to stabilize around one, as my little behavioral model suggested.
The fraction currently infectious is tiny. Still, half a percent is half a percent. If you run in to 100 people a day you're going to get it in two days. (A commenter corrects my sloppiness here -- "run into" has to have enough close interaction to transfer the virus.)
Go look up your location in Table 1 (too big to include) The SF Bay Area only has 0.04% infected! That Whole Foods is pretty safe. But the reproduction rate is still above one.
Their dashboard has up to date results for lots of places.
