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.