# On R0

Epistemic Status: As with all of my Coronavirus posts, I am not any kind of expert. I am a person thinking out loud, who will doubtless make many mistakes. Treat accordingly. However, the concrete policy proposal contained herein seems right and I endorse it strongly.

Partly a response to (Overcoming Bias): [Beware R0 Variance](http://www.overcomingbias.com/2020/04/beware-r0-variance.html)

Previously (not required): [Taking Initial Viral Load Seriously](https://thezvi.wordpress.com/2020/04/01/taking-initial-viral-load-seriously/)

Also related to Covid-19: [Coronavirus is Here](https://thezvi.wordpress.com/2020/03/02/coronavirus-is-here/), [An Open Letter To The Congregation Regarding The Upcoming Holiday](https://thezvi.wordpress.com/2020/03/06/an-open-letter-to-the-congregation-regarding-the-upcoming-holiday/), [Let My People Stay Home](https://thezvi.wordpress.com/2020/03/17/let-my-people-stay-home/),

Ultimately, it’s always all been about R0.

If you get and keep R0 substantially below one, infections fall off. Covid-19 is squashed.

If you can keep R0 below one and let normal life happen, life can return to normal. If you can’t, life can’t.

If R0 remains above one, infections continue to rise until something changes.

What will it take to do that?

Interventions that reduce contact and exposure, or reduce the danger of each contact or exposure, or change their dynamics in useful ways, reduce R0. Every person already infected is almost certainly immune (at least for now) which also reduces R0.

You need some combination of interventions and immunity from previous infection that sufficiently reduces the initial R0.

This is not an [o-ring production function](https://en.wikipedia.org/wiki/O-ring_theory_of_economic_development). You are not as vulnerable as your weakest link. Not everything you do has to be correct. This isn’t a designed puzzle where there are *exactly enough* interventions available to solve the problem and you need to do them all. Instead, we have a variety of possible actions and need to pick the cheapest basket that reliably accomplishes the mission.

### **What Was Initial R0?**

As Robin notes, R0 does not start out and never is one number. It depends on the surrounding disease environment. In some places it will start out very high. I’ve seen plausible estimates for New York City as high as 8. In other places R0 might be lower than one to begin with, because people barely interact with each other in normal times. That hermit with fifty years experience social distancing? His local region’s R0 is about zilch.

To get intuition pumping one still needs a good ‘default’ R0 before any substantial interventions take place, so we can adjust after that for interventions and anything about a particular place that changes R0 substantially versus the default.

[This document](https://docs.google.com/document/d/1QdoCEXmZqj_4ptywXOKFe8I8rS6ijveqvdYXWT1quQI/edit#) collects, among other useful things, a bunch of estimates of R0. If you average out all numbers that don’t involve an intervention it comes to about 3.36.

We need to reconcile that with serial interval and doubling times. Looking at the same document for serial interval, we get a minimum of 4 days and an average of 5 days, with an upper bound around a week.

It now seems clear that most of even those sounding early warnings predicted doubling times far longer than those we later observed. Community spread is fast. For a while a lot of places seemed to have doubling times of 2.5 days (and New York City may plausibly have been as low as 1.3 days). If the doubling time is 2.5 days and the serial interval is 5 days then R0 should be about 4, so each serial interval can let us double twice. I have been using that as a reasonable baseline overall guess.

It does seem Bucky and LessWrong [got it right on March 9](https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks), which *would have materially impacted my life plans* even that late in the game*,* but I didn’t see that in time and only updated at least a week or so later. My usual sources all had 4-5 days or longer, causing me to move much too slowly.

### **Interaction Interactions**

Exposures to Covid-19 outside the household and outside the medical system all seem to have sufficiently low individual probability of infection that we can add their infection probabilities together to get the expected infection count. This is a tiny error, but not enough to matter much.

If we presume that a typical place starts with R0=4, then we need roughly a 75% reduction in total quantity of exposures to get R0=1. It’s more complicated than that, in a bit we will beware and also embrace R0 variance, but I want to start with the simple version first.

How much do various interventions cut exposures? What types of exposure are cut how much?

Here is a practical list of ways people might be exposed:

1.  Inside of a household or living facility.
2.  Social interactions.
3.  Medical system.
4.  Prison, jail, other detention.
5.  Courts.
6.  At work indoors.
7.  At work outdoors.
8.  Recreational indoors out of household.
9.  Recreational outdoors.
10.  Errands and tasks not otherwise included in the list.
11.  Commuting, especially mass transit.
12.  Travel other than commuting, especially air travel, trains and so on.
13.  Crowd events now gone: Stadiums, theaters, concerts, rallies, etc.
14.  Crowd events mostly gone: Religious services, holidays.
15.  Grocery stores from other shoppers.
16.  Restaurants, including takeout, excluding from the food.
17.  Packages, excluding the food itself.
18.  Food, including both groceries and restaurants.
19.  Delivery contact points.
20.  Schools.

What am I missing here?

It is easy to see that *most* of these have been squashed as vectors far more than 75%, but some of them have not been and may even have risen.

In particular, #1 (Inside of Household), #3 (Medical), #4 (Imprisonment), #9 (Recreational Outdoors) and #15 (Grocery Stores) are plausibly as bad or worse than their normal life baselines. I believe we have cut back on #5 (Courts) in most places but those that do remain still seem quite bad.

Inside of household exposure gets worse when you always stay home, which plausibly overrides any extra precautions taken. The good news is that it then gets very hard for many secondary in-household infections to spread further, as we’ll discuss later.

The medical system frequently lacks proper protective equipment and is not giving health care workers proper time to quarantine. These are both very bad. The horrible consolation here is that each health care worker only gets sick once, they know to be careful when not on the job, and in heavily infected areas anyone who does not think they have Covid-19 or a true emergency situation is avoiding all health care workers entirely. There might not be that many others left to infect here after a few weeks.

The prison systems make even ordinary-life levels of social distancing or hygiene physically impossible, so basically everyone we don’t release is probably going to get exposed. We should be releasing a lot of people but mostly aren’t, although new arrests are slowing dramatically. Prisons are presumably headed quickly to herd immunity levels, and the guards will also mostly get infected at some point. Our prison system is pretty terrible even in normal times and this is worse. The ‘good news’ again is that this vector dies out relatively quickly due to herd immunity.

Grocery stores are often a madhouse now, and people need more stuff from them. Delivery services are maxed out and have failed to expand that much. Here in Warwick the local ShopRite is gigantic but impossible to safely use. The website is permanently overloaded and there are never available slots even for pick-up let alone delivery. Even the aisles are more dangerous than anything else we might do, and checking out is a disaster. Long lines in places others can’t avoid. The counterargument is that before a lot of people went very often and would end up very, very close to each other, whereas now at least people are trying to not do that. Grocery delivery might both be the worst thing left for most people, and still better than it was before interventions.

Recreational outdoors could have gotten better or worse, since people take precautions but are also far more desperate to get outside and are crowding some areas.

#2 (Social Interactions), #6 (At Work Indoors), #7 (At Work Outdoors), #8 (Recreational Indoors), #10 (Errands), #12 (Travel), #13 (Stadiums), #14 (Worship), #16 (Restaurants), #19 (Delivery Contact Points) and #20 (Schools) all seem squashed reasonably well, at least 75% in most places. Some are effectively down almost 100%.

#11 (Commuting) is down a lot but perhaps not 75%. That could be a problem in places where commuting often involves mass transit, as riding in one’s car is not a big concern. One of New York City’s biggest problems has been that the subway, even with ridership (and in a show of true bureaucratic insanity, number of trains) down by half or more, this vector is still a gigantic problem that most other places don’t have.

#17 (Packages) and #18 (Food) depend on how many people are actually taking precautions on these, which is unclear. Presumably they haven’t gotten net worse overall, but it is possible. I do not think they are large contributors to the initial R0. Those being careful are 90%+ safer. Those not being careful depend on the precautions on the other end, and are getting more packages.

Or another list, which would be infection vectors more generally:

1.  Direct physical contact.
2.  Droplets.
3.  Surfaces.
4.  Fecal/oral.
5.  Miasma (to extent this is a thing).

Direct physical contact is *way* down. Even a simple ‘do the things you’d normally do, other than those that involve a lot of direct physical contact, and try not to touch anyone while doing them’ seems like it should be good for 75%+. We’re going much farther than that.

Droplets are a function of social distancing and presumably follow roughly an inverse square law for each interaction because physics. People are being told to think of a Boolean at 6 feet, which is obviously wrong, but given people’s defaults their actual reactions should result in large cuts. It’s hard to think they aren’t down 75%+ even before masks. Masks then help a lot as well, and the tide is turning on wearing them. If all of that isn’t enough, it’s presumably a large underestimation of the rise in grocery store exposures, which we can and should limit with extreme prejudice as I suggest in the next section.

Surfaces are something people previously mostly ignored and are now trying to dodge, plus we are mostly not going outside, plus a lot of wearing gloves and washing and not touching faces, so 75%+ reduction seems clear.

Miasma is a function of hanging out in packed places, which has to also be down 75%+ (and may or may not be a vector at all, or one worth worrying about).

The only vector that isn’t obviously down at least 75% would be fecal/oral. Increased hand washing and reduced face touching seem like they’d be big games here, as would gloves worn during food preparation and doing most food preparation inside the household. Many are taking additional strong food precautions.

A third method is to look at interventions, and estimate how much reduction each one accomplishes.

How much do we get from hand washing and face avoiding? How much do we get from social distancing? Wearing masks? Working from home? Closing schools? Closing restaurants? Cancelling events? Closing stadiums and hoses of worship? And so on.

### **Low Hanging Fruit: Safer Grocery Delivery**

Doing that analysis made buying groceries stick out as the biggest remaining easily avoidable obstacle, and the one place things have gotten much worse for average people.

For most people able to work from home, getting groceries safely is the main barrier to maintaining an effective quarantine.

Delivery and pick-up services exist, but they are not scaling up fast enough and are at maximum capacity. Stores are packed. It is difficult to retain the work force, let alone expand it, as things get more dangerous and stressful. When we last investigated the local ShopRite, there were only three open checkout aisles. Getting even a pickup time has become impossible. Most people cannot afford Instacart’s ~30% markup, which is the reason Instacart is still available at all, but that means it does not provide a general solution.

A single trip to many grocery stores, even while taking all realistic precautions and moving quickly, is likely *most* of the exposure for *most* people working from home or not working, even if you only count exposure to other shoppers. Store workers are at high risk from everyone coming in and out.

This is also the only major exposure that most people have that hasn’t already been cut by 75% or more. If we could knock it out, it would be a huge blow to R0. It would also be a huge safety boost to many of our most vulnerable.

The good news is that this problem is so big we can afford to throw massive amounts of money at it. I propose we do exactly that. There is no need to be subtle or careful, here.

I propose a straightforward $20/hour direct wage subsidy to all grocery store and restaurant workers whose focus is a combination of check-out, pick-up and delivery, up to 20% of the revenue from goods sold (you need *some* cap if you’re letting people hire at a good wage for free, these numbers don’t seem obviously wrong, and going for the elegance of calling this “a 20/20 vision” seemed nice).

In exchange, we ask only that all take-out and delivery charges must be waved – delivery has to be same cost as in-store purchase. This encourages the use of the pick-up and delivery services without concern about price. As much as I love allocation by price in most situations, we want to force massive scaling here rather than efficient allocation before scaling is finished.

We include existing workers to not punish anyone who scaled up early, to encourage employee retention, and because they are heroes who deserve the hazard pay. As Henry Ford and many others have shown us, if you want extraordinary effort, pay well above the ‘market wage’ and you will get rewarded. Given what all this is costing us, the cost here is chump change.

And also because we already wanted to throw money at people, our existing methods are taking months to work, and this seems like as good a way as any to get things going.

The resulting lowering (or at least, not raising) of food prices across the board then acts as a progressive subsidy to everyone, taking the same role as sending out universal checks. Competition is a thing and everyone’s gotta eat. We also create jobs.

That’s version one. I’m sure it can be improved, but it seems like an obviously vastly better deal than anything else on the table, and seems shovel ready.

(A parallel wage subsidy to health care workers on the front lines likely also makes sense for many of the same reasons, especially as hospitals depend on elective procedures to keep the lights on and are in many places cutting doctor pay, and would presumably have broad support.)

With that out of the way, we now return to examining R0.

### **In-Household versus Out-of-Household**

In versus out of household was a key distinction [when considering initial viral load](https://thezvi.wordpress.com/2020/04/01/taking-initial-viral-load-seriously/). It’s also a big consideration when modeling the spread of infections. This is especially true directly after instituting a lock down.

In-household transmissions go up rather than down when a lock down is instituted, since everyone in-household is now staying home more often, and thus interacting more often.

However, once you are locked down, are the resulting infections going to spread further?

The members of a household now have most of their interactions with each other, even more than they might have had before the lock down. Many, including most kids, will have a tiny amount of overall household exposure and an even smaller percentage of overall exposing of others to the household. If you stay home, you essentially cannot infect anyone outside the household, at all.

Thus, the bulk of in-household infections basically stop counting for the effective R0, because the resulting R0 for such infections is close to zero. We would be better off thinking about households as either ‘infected’ or ‘not infected’ the way we would previously have thought about individual people.

In turn, this means we will be doubly too pessimistic about lock down effects early on, since the ratio of in-household to out-of-household infections will spike before returning to normal levels and that is not only atypical but relatively harmless in terms of further infections.

### **Beware R0 Variance?**

R0 as one number is a big simplification. R0 variance in some times and places is bad and we need to be beware. In other times and places it is helpful. Details matter a lot, as does the big picture.

R0 variance raises overall R0. If we have two groups are mostly distinct and only interact rarely, one with R0=0.5 and one with R0=1.5, we get most of our infections in the R0=1.5 group, and R0 ends up well above 1. Thus, if we say that R0=4 in general, we’re saying that the average across groups or regions is less than that. We are not doing good modeling if we say that R0=4 in general but with variance so R0>4, although there will be scenarios where it temporarily goes over 4.

If our goal is to squash entirely and then reopen everything, R0 variance across places and groups naively looks quite bad. We need to squash in every group, or the high-R0 groups will reinfect the low-R0 groups where we squashed successfully. The nightmare is you get R0=0.5 in general via strong suppression measures, but with a lot of variance, and in some places it still isn’t enough or the measures weren’t adapted, and you need to stay shut down or it all comes back.

In other situations, the variance works in our favor.

If we can properly target our interventions to particular groups, places and events, R0 variance is great. R0 is gigantic in sports stadiums so we shut down the sports stadiums. If R0 was huge in New York City but under 1 in the rest of the country, we could close the bridges except to haul in supplies, and let 97% of our people carry on largely as normal. Wuhan being compact was great for China. National borders are a thing and can be enforced, and so on.

If we are willing and able to let things burn out to herd immunity in some places or among some groups, similar things happen. As an alternative to shutting New York City down indefinitely, we could simply declare defeat there, and allow New York City to lock down its most vulnerable as completely as possible while rapidly getting to herd immunity, while not following that strategy elsewhere. Then New York City is no longer a threat.

What is happening inside New York City likely looks a lot like that strategy, except for a subgroup rather than the whole area.

### **A Very Simple Model of New York City**

Some New Yorkers were able and willing to self-isolate, or even to flee the city entirely. Others were not.

There is a continuum between my parents, who are literally only opening the door to take in packages that they then sanitize, and those who have no choice but to ride packed subways every day to make ends meet, or are so young and selfish as to be indifferent to what happens.

Let’s simplify that a ton, and say that some people ride the subway and some don’t.

The people in the subway are packed tight, because the MTA cut the number of trains in proportion to the cut in ridership. Those still in the subway have several times the exposure to people that others have, and those exposures are to other strangers who ride the subway.

Among those who ride the subway a lot, infection rates get out of control very quickly. We should expect by now that those still riding the subways daily have essentially all been exposed enough at least a week ago to get infected if they aren’t somehow naturally immune – look at the admitted-to-be-undercounted death rates, assume those people were infected three weeks prior, look at the doubling rates during those times, and snowball the effects. There’s no need to do any math here.

By the end of April and quite possibly the end of March, the subway group is mostly immune, and has blown past steady state herd immunity levels even with its high levels of exposure. The city then has R0 levels that are even lower than if the subway was taken entirely out of play, since those people also interact elsewhere.

It’s definitely not the solution we would want! It’s a ton of infections. Also deaths. Still a lot less infections and deaths than doing the same to the whole city, which will now have a much easier time squashing than if everyone in the population was identical. It also means we can now use the subways to get essential workers where they need to get, without making the forward-looking problem worse.

### **Actual Exposure Inequality**

I consider the above scenario a *relatively equal* distribution of exposures, compared to the real situation.

Even in normal times, different people have radically different levels of exposure to crowds, to being physically close to others, to interacting with different as opposed to the same people repeatedly. Different people have radically different levels of caution and hygiene.

People have speculated about ‘super spreaders’ who are much more infectious than others, and who might account for the bulk of out-of-household infections. That might or might not be a thing. What is definitely a thing is that some people are in position to create additional orders of magnitude more exposure in others, especially others that are socially disjoint from them, than most people are in position to create.

The people who are in position to infect lots of others are, mostly, also the people in position to be infected by many others in the first place. If you are the pastor and shake hands with your whole congregation each week under normal circumstances, you might both have 100 times as much exposure to catching the virus, and 100 times as many places to spread the virus.

Both directions follow a power law, and both directions are highly correlated.

Thus, the idea that to get R0 from 4 to 1 via herd immunity requires 75% immunity rates *doesn’t make even a little bit of sense* to me. People won’t be infected at random. And yet this is the standard calculation used everywhere I look.

One follow-up question is, are those who have more exposure more commonly linked up with each other than with others?

In some cases, clearly yes. The subway is a clean example of those at risk mostly exposing each other. So are all the young men who continued not social distancing after they were ordered to distance. They mostly hung around each other. The reason Florida’s spring break was likely to bad was because those there were engaging in *unusual-for-them* behavior, and would later go back to other places and act normally. Under normal circumstances, if the beach or park is packed, then the group of people willing to go there will put each other at great risk but may not put others at such high risk. This gets us a lot of effective immunity at low cost.

Consider how we got to R0=4. If one third of the people interact twice as much as everyone else, in both directions, and all interactions are random, than at the start they constitute half of all infections. Thus, we start out at R0=2.67 for infections in the bigger group. If the third who interact more was immune, that’s half of possible infections, so that goes to R0=1.33, so you only need a quarter of the less interactive group immune to get R0=1 with a 50% infection rate rather than 75%. You can’t get that whole win since you wouldn’t actually infect the whole more-interaction group that fast, but you could certainly get half this effect and get down around 62%.

The real ratios are *much bigger* than that. It doesn’t take that much of this effect to get R0=1 in the bulk of the population even before anyone is immune or does any social distancing.

If I had to guess what percentage of people we would need to *naturally* infect (rather than selecting them on purpose) to get herd immunity with no behavioral alterations? I would guess something like 35%, rather than 75%. And I’d expect that the first 3% infected, which I am guessing is about where the United States is right now, would get us *far more than 10%* of the effect of 35%. 

That’s without us doing deliberate infection or antibody testing in any intelligent way. We can do even better if we can create and/or identify immune citizens and put them in positions of high two-way exposure. Unless we completely drop the ball (which would fit our existing patterns so far) we should start to see substantial gains from antibody testing by the end of April.

Thus, I’ve become more of an optimist going forward. Variance makes the problem anti-inductive in the sense that the virus will thrive wherever conditions are best, but friendly longer term once containment fails anyway, because it separates out problems that solve themselves relatively quickly at lower cost.

I expect that the way we get out of this is a combination of herd immunity coming on stronger and faster than we expect, combined with the cumulative effects of various interventions, many of which can be maintained in a “Phase II” style world without that high a cost, mostly via voluntary action. We can continue to not shake hands and wear masks and avoid large gatherings and have a third of people work from home and so on at a relatively tiny cost. Then we combine that with test and trace, which helps even if it isn’t ubiquitous, and we can have an acceptable situation while we await a vaccine.

### **Postscript on Recent Data**

Since I started writing this several days ago, the official numbers have improved considerably in America and abroad. New infections are leveling off, test positive rates are down, and deaths are no longer growing exponentially. This is faster than even my at-the-time prediction that new infections (rather than new detected infections) peaked specifically in New York City on April 1, and the stock market has rallied accordingly. 

I do think we should be deeply skeptical of these numbers. New York’s numbers especially seem to mostly reflect hitting maximum hospital capacity and maximum testing capacity, both of which are mostly static. We only count deaths if they involve positive tests and in practice in New York that means hospitalization. So when we see deaths leveling off faster than would make sense given how deaths lag, we likely shouldn’t interpret even deaths as meaning much more than ‘the numbers are higher than this but we have no idea how much higher.’ With even deaths clearly undercounted, calculations that start with official death counts, without adjustments, are going to be underestimates. 

The percentage of tests that are positive is probably the best data we have, but even that seems to be strangely noisy. American positive test rates were in the 23%-26% range for four days, then dropped to 15% before increasing back to 19% for 4/5 and 4/6. This is even more stark if you exclude the epicenter in New York and New Jersey (where positive rates have been between 46% and 51% with 20k-31k daily tests each day since 3/30), with the rest of the country having four days from 15.7%-18.3% positive rates, then three days from 9.5%-12.9% (all data from [the Covid tracking project](https://covidtracking.com/data)). That has to reflect strange distributions in reported tests and/or incorrect data. So it’s all full of noise, and probably stays that way until we get antibody tests. Hopefully we get better data soon.