Yesterday, Tuesday, March 18 at 10 AM, the JHU web site said that there were 6519 cases in the U.S. Today, Wednesday, March 19, at 4 AM, it was showing 9415 cases. That is an increase of roughly 50 percent. That increase in known cases is a combination of two factors: increased testing (an artificial factor), which raises the number of known cases to the number of actual cases; and spreading of actual cases. I don’t know how much is due to each, but if you are looking for evidence that the virus is not spreading exponentially, an increase of 50 percent per day is not a good sign.

Now for some grim math. Let C be the number of known cases, H be the ratio of hospitalizations to known cases, and D be the ratio of deaths to hospitalizations. Then we have:

(1) total deaths = DxHxC

For example, if there are 1000 known cases (C=1000), 5 percent of these are hospitalized, and 20 percent of those who are hospitalized die, then deaths = 1000x.05x.20 = 10. Note that in this particular example, I assumed that no one dies who is not hospitalized. In reality some people will die without being hospitalized, and they will count in D.

Note that in this equation, HxC is the case mortality rate. In the numerical example, it is .05x.20 = .01, or one percent.

Next, we can do a logarithmic derivative approximation to write

(2) g = d + h + c

where g is the growth rate of deaths, d is the growth rate of D, h is the growth rate of H, and c is the growth rate of C. Note that this approximation only works for SMALL values of d, h, and c, not for big numbers like 50.

Suppose that cases grow at a rate of 4 percent (c = .04). Then if the hospitalization rate falls by 4 percent (h = -.04), that would offset the growth rate in cases.

Assume that soon the growth rate of cases will reflect true spreading, and the bump from increased testing will be behind us. Then going forward, there is reason for optimism in all three components of (2). The rate of death of hospitalized patients should fall as we get better treatment protocols and find useful drugs. The rate of hospitalization should fall as we get better at triage and we also find more effective treatment protocols that reduce time in hospital. It also could fall if we get better at protecting high-risk populations, so that more of the people who get the virus do not experience severe symptoms. Finally, the rate of growth of cases should fall as the effects of social distancing kick in.

If the rate of hospitalization does not fall fast enough (h turns sufficiently negative), then as long as c, the growth rate of cases, remains positive, we may at some point run out of facilities to treat seriously ill patients. The limiting factor in facilities might not be space and equipment–it could be the supply of health care workers. In any case, once we exceed capacity, that would cause a spike in d, the growth rate of deaths relative to hospitalizations. The growth rate in deaths would be high in such a scenario.

There are web sites that track total cases, C, and total deaths. What would help in this framework is to have H, the proportion of known cases that are hospitalized. As I searched for that data, at first I found what appears to be misinformation:

Up to 1 in 5 younger adults in the U.S. infected with coronavirus wind up in the hospital, according to a new analysis by the Centers for Disease Control and Prevention.

Baloney sandwich. What the report says is

Among 508 (12%) patients known to have been hospitalized, 9% were aged ≥85 years, 26% were aged 65–84 years, 17% were aged 55–64 years, 18% were 45–54 years, and 20% were aged 20–44 years. Less than 1% of hospitalizations were among persons aged ≤19 years

That is, 20 percent of those hospitalized were in the 20-44 year age group, not that 20 percent of the cases in that age group are hospitalized. Since 508 were hospitalized, that means that about 102 in the 20-44 age group were hospitalized.

As I understand it, at the time the report was run, there were 4226 cases, and 29 percent of these were in the 20-44 age group. That means that there were about 845 cases in that age group. So the rate of hospitalization within that age group was 102/845, or a bit under 12 percent. Still a big number, and an indication that letting this “low-risk” population all get infected soon may not be a good strategy. But see my final note.

Overall, dividing 508/4226 gives a value for H of just over 12 percent. With cases having more than doubled since the report was run, in order to hold steady we would need H to have fallen below 6 percent.

Final note: the value of H in the report is greatly overstated to the extent that people without severe symptoms did not get tested, and hence did not show up as cases. That could be a lot of 20-44 year-olds, which would make their H much lower.

I wish we had a dashboard that provided reliable numbers for H. I wish we were testing a random sample of the population so that we could estimate key numbers with more confidence.

There are a lot of people who are calling the social-distancing movement a “panic” that is needlessly wrecking the economy. But I thin we can all agree that it buys us some time by slowing the spread of the virus. Here is a list of advantages that I see from this.

1. We can produce more ventilators before the demand peaks.

2. We can keep our health care workers healthier longer.

3. We can evaluate treatment protocols.

4. We can test many more people, and we can analyze the data we obtain from doing so, before making further course corrections.

5. It is possible that the course correction that we need is even stronger quarantines. But we could never do that without first finding out that the milder social-distancing measures have been tried first and failed.

I listened to Russ Roberts and Tyler Cowen discuss the coronavirus. If you missed it, maybe you can find it at the Mercatus video archive at some point. Here is rough paraphrase (caricature?) of part of the dialogue.

Russ: Think of this as a three-week vacation. Why can’t the economy recover from a 3-week vacation?

Tyler: A lot of organizational capital will be lost.

Russ: Wha???

Tyler: Employers and employees know how to work together. When those relationships need to change, it takes a lot of time for matching and training to work out.

Russ: You seem to think that the virus will be a factor for a long time. Why?

Tyler: Suppose that in the very short run we get it under control through social distancing. That means that a lot of people will not have had it yet. What is likely to happen is that there will be a series of outbreaks, and that means a series of shutdowns. It means that it will be a long time before people feel comfortable going to locations where they will encounter crowds.

Think of this as the problem of herd non-immunity. It means that we could go a long time during which people change their behavior to avoid catching/spreading the virus.

To me, this makes two things important. One is the process of testing and approving a treatment. If a treatment can be shown to work, then we can be much more relaxed about allowing people to get the virus.

The other is having a random testing program. John Iaoannidis is getting a lot of flak, some of it deserved, for what he wrote yesterday. But I very much agree with this:

The most valuable piece of information for answering those questions would be to know the current prevalence of the infection in a random sample of a population and to repeat this exercise at regular time intervals to estimate the incidence of new infections. Sadly, that’s information we don’t have.

If the government won’t do this (and I have little confidence that they will), then I hope some corporation or non-profit will take it on.

He writes,

I bet that by the end of the calendar year, the number of deaths that can clearly be attributed to the disease will be greater than 100,000.

Note that he hopes he loses his bet, as do we all.

Think of 100,000 as 10 million cases times a 1 percent death rate.

Why might we get more than 10 million cases? First, it is possible that the number of cases that have not been officially detected is 100x the number of official cases. Because some infected people do not have symptoms, some who have symptoms are not going to get tested, and some who want to be tested have been, until recently, turned away.

Even if the true number of cases in the U.S. is only 10,000 today, if that were to double ten times we would be at 10 million. If the doubling time were a week, it would take only ten weeks to be over 10 million. And as of now the doubling rate is faster than that in many European countries and in the U.S.

But given the sharp reduction in travel and large gatherings that has taken place, I expect that the doubling rate will slow down. Suppose that, after these changes have been in place for a few weeks, we find that it is taking a month or more to double the number of cases. That would make it less likely that we hit the 10 million total by the end of the year. (Although, again, it is hard to know where we are starting from.)

We might also find that we have a lower death rate. Compared with other countries, we have less smoking and more capacity in our health care system. And the steps that we take to protect at-risk populations from the virus may prove effective.

This might be a time to update the status of two hypothetical bets of mine. First, I hypothetically bet that no centrist candidate would arrive at the Democratic convention with more than 40 percent of the delegates. That now looks like a bad bet.

More recently, I hypothetically bet that the number of Covid-19 cases in the U.S. would be more than 12,000 by the end of this week. As of Tuesday evening, the total was over 6000, and it appeared to be doubling every two or three days. Unfortunately, it looks like I will turn out to be correct on that one.

A reader writes,

I’m seeing millions of people, primarily at the middle to lower end of the economic spectrum, who will be financially ruined by the response.

While remaining humane and just, is there a more cost effective approach to the problem, particularly since it seems concentrated in a rather narrow subset of the population (those with compromised immune systems and the 60+ cohort)? E.g. instead of isolating the entire population, why not just isolate those most at risk and compensate them accordingly? How do we weigh the ethical concerns of one group as against another? Why do such questions seem so out of bounds and antisocial?

1. I think that even the “low-risk” population, if they move about freely, are likely to overwhelm the health care system. I doubt that encouraging them to go about their business is the first-best strategy. But it might work out better than other approaches.

2. What we are doing now, which is lockdown-lite, is also not the first-best strategy. It may exacerbate economic pain while failing to do enough to stop the virus. It may be an instance of my two-weeks-behind hypothesis. Still, I think we ought to support the lockdown-lite approach–as you know, my wife and I adopted self-quarantining last Thursday. We ought to give it a chance.

3. I suspect that the first-best strategy is a total, nationwide lockdown for two weeks, enforced militarily. The intent would be to deprive the virus of hosts. Even then, in order to contain subsequent outbreaks, we would have to continue to encourage social distancing, require people to keep track of contacts, and do a lot of random testing of asymptomatic people.

Think of this as comparable to the Arab oil embargo. Probably the harder the adjustment we make sooner, the better it will be. The macroeconomic policies of that era made things worse, and I expect the same to happen today.

I cringe at all the talk of monetary and fiscal “stimulus.” What this means is that governments will use the crisis to enlarge their share of the economy, which will hurt the adjustment process, not help it. It will also “solve” the problem of debt collapse by piling on more debt.

Robin Hanson wrote,

it isn’t crazy to consider cutting pandemic deaths via more infection inequality, including via deliberate exposure.

Pointer from Bryan Caplan.

Consider the following strategy:

1. Separate the population into low-risk and high-risk groups, based on their conditional probability of death if they get the virus. For example, young people with healthy immune systems vs. older people and/or those with compromised immune systems. Separate them not only conceptually, but physically–don’t let anyone from one group get near someone from the other group.

2. Then, allow the low-risk group to become infected, while keeping them away from the high-risk group.

3. Once the low-risk group have recovered, let the two groups mix.

Two reasons to hesitate about doing this. One is that it is not certain that people who have had the virus are immune. There are anecdotes about people re-acquiring the disease. Perhaps there are multiple strains, rather than “the” virus.

A second reason to hesitate is the high rate of death among health care providers, many of whom are young with healthy immune systems. This suggests that there are some other factors that affect risk, and you want to know more about those other factors before you try this approach.

My own “out of the box” suggestion is a program to test a random sample of people who are asymptomatic. That would give us a better idea of the dynamics of virus spreading.