What to understand about the coronavirus exponent
I feel this may be useful to share in a longer post. The draconian measures that governments undertake around the world seem to a lot of people like overreaction. This is understandable. Exponents can fool the best of us.
This post is to illustrate some of the key issues at stake, as I understand them. I am not an epidemiologist, but I can put together a simple computer model. Below is what we we can learn from simple and easy-to-understand math.
Here is the most important figure from yesterday's article, which models the spread of coronavirus over many months:
The black curve illustrates a typical curve for a virus spreading through population. First, it rises exponentially. Then, after a significant fraction of the population acquires immunity, it falls off.
The total number of deaths in the US under this scenario is estimated at 2.2 million, not including any deaths that result from the hospital system being overwhelmed. The capacity of the medical system is the red line near the bottom of the graph. As you can see, it will stay completely overwhelmed for three months. So, even if you think the mortality statistics for this virus is overestimated, you should see that during that time a person of any age, who is in need of hospitalization for any reason, will be in danger of dying simply because they are unable to receive medical help. In effect, we as a society will face an ethical decision not to help people over a certain age, in order to save the lives of others. People over 70 who show symptoms have a 24.3% chance of requiring hospitalization, and 43.2% of those will require critical care. You may have to decide that between one in ten and one quarter of people in this age group will die.
If you do not want to face this choice, you will have to focus on the very small area at the lower left corner of the graph, to understand what it takes to keep the black line from ever rising above the red line. This means we would have to bring down the epidemic not by developing "herd immunity", but by making sure that the virus spreads slower.
This is where the exponents come to play.
Currently, the number of detected cases seems to double every 3 days. The doubling rate for real cases (i.e. including undetected) was early-on estimated between 4 and 6 days, but reanalysis of the Wuhan data also seems to yield 3 days. So we'll go with this number.
This is what doubling every 3 days means, starting with 100 cases on day 1. On day 30, the chart exceeds 100,000 cases. I put the "We are here" mark on day 11, which corresponds to a little over 1000 detected cases. This is where New York state stands today, March 17.
Doubling every 3 days means that in 30 days the number of cases increases by 2^10=1024, more than a thousandfold. Going from 100,000 to 1 million will take just another 10 days. Consider that 20% of the detected cases have required hospitalization, and also that the actual number of cases probably exceeds the number of detected cases by a factor between 25 and 50, and you will see why it is important to slow down this exponential growth.
Now for a less obvious part. The doubling time of 3 days means that each day the number of cases increases by 26%. Or, that each infected person infects 0.26 other people per day (if for the moment you neglect people who are no longer contagious). The way to slow down the exponent is to bring down this number, by limiting the number of contacts a potentially infected person has with other people. For this virus, a potentially contagious person may be anyone, even those not showing symptoms.
There are a number of parameters that determine the dynamics of active cases (i.e. the number of people who are sick at the moment). They include, for example, the number of days after which a person becomes contagious (3 to 5), and the number of days a person stays contagious (8 to 11). But the number of infections per sick person is still the key. Here is an example of how reducing this number can affect the dynamics.
The red line is the dynamics where cases continue to double every 3 days or so. The blue line is where the number of contacts is halved. The green line is where it is decreased 3 times. I put the vertical line at a point where the number of real cases is around 15,000. The middle graph shows the number of people in hospitals. For the first two scenarios, hospitals are overwhelmed within weeks to months.
Important: Treat this as an illustration rather than an actual prediction. We do not know the values of the key parameters well enough to predict the actual dynamics. Nobody knows. I think I was on the optimistic side here, but it's hard to tell. The range of the potential outcomes is wide. Reducing the number of contacts by a factor of 5 in my simulation may reverse the dynamics, but it's just hard to tell what number should be sufficient.
We also don't know by how much each measure affects the number of contacts. Is closing restaurants necessary? I don't have the answer. But given the uncertainty and the severity of consequences of getting it wrong, it makes sense to overreact rather than under-react. Then we can see the effect and adjust our measures as necessary.
A very important point: The speed of the action is key. China was able to suppress the epidemic because they acted both strongly and early. We are behind and unlikely to repeat the success. (For the range of unsatisfyingly poor options, take a look at the paper I referenced above.)
This graph illustrates the cost of delaying the action by only 2 days. 50% more cases! (Again, details may vary.)
I hope this helps explain the logic behind some of the events we are witnessing and the choices we face. I hope better estimates will come out in the coming days from people who can do this better than I can.
P.S. I am leaving out here the discussion of issues such as what to do after we get tired of quarantine well before the herd immunity is reached. Or whether the actual number of cases is so vastly underreported that a significant fraction of the population is already infected and the herd immunity will soon kick in. For the moment, the good news is that the measures we are currently taking are likely to work, in a sense of avoiding a major collapse of hospitals in the next two or three weeks.
P.P.S. I see a lot of discussion around the fact that both the fatality rate and the hospitalization rate are overestimated, because only the most severe cases end up being tested. These parameters are important for the eventual death toll and the exact timing of hospitals collapse. But they are not critically important if you are trying to figure out whether the hospitals collapse will happen at all. The reason why is that the number of people who end up in hospitals scales with the hospitalization rate linearly, whereas the exponential growth overpowers any linear function. It's not the question of 'if' but the question of 'when'. Here is a comparison of the number of people in hospitals for the hospitalization rate of 20% vs 5%.
