Why social distancing does not work that well, part 2

In part 1 I explain, using some simple math and intuition, why the hype about social distancing is unfounded and unlikely to do much to prevent he spread of Covid19 in the US.

The problems with social distancing are, as summarized in part 1:

1. Damage to businesses and local and national economy by having to close. Any possible lives saved due to social distancing has to be weighed against the cost of lost economic activity and loss of business.

2. People do a lot of interacting in non-social settings such as at the food store, shopping, on the street (such as being in a public area where there are people nearby), when taking public transportation, at work, or domicile (household or apartment).

3. Social events tend to be brief and infrequent wheres non-social interactions are more commonplace.

The last factor especially explains why it is not that effective. The math shows, using some assumptions about the frequency of social events, that the reduction of the rate of infection is too small to flatten the curve much, or at lest not nearly as drastically depicted in that viral picture shown in part 1. There is some flattening and delaying of the peak, but not enough to possibly justify shutting down businesses.

People sometime get mad or defensive when you tell them it does not work that well or by questioning the efficacy. “It does work, do you want more people to get sick?” Yes, of course, it works, but the question is, how much. Also, the benefit of preventable cases has to be weighed against economic loss due to distancing. Is preventing a few percent of possible deaths worth tens of billions of economic loss and damage?

To model this, we first need to determine the daily infection rate. This can be estimated with an exponential function (technically, we should use a logistic function, but this is close enough). For the first 14 day of March, the total number of cases rose from 68 to 3000.

So solving 3000=68e^(14r) gives r=.27

This means the total number of cases is growing by 27%/day.

A 5-day asymptomatic period in which the virus spreads gives an r-naught of 2.35. This is because r=naught is approximately equal to 1+asymptomatic period times daily growth rate.

So how do we get the so called flattening of the curve and delay? That requires using a logistic curve for modeling the spread of infection within a finite population.

So we have r=.27. The pop size of the US is approx 3.3*10^8. The initial case count is 65. Taking the derivative of the logic curve with respect to time gives the distinctive bell curves that people have been sharing on social media. One can also see that without delay, it takes about 100 days for almost everyone in the US to be infected; with delay, about 80.

The doublebell curve is shown below:

The purple plot shows flattening and delaying of the peak by letting r=.20, just as depicted by the viral Vox image. But the vox image shows the flattering to be even more exaggerated.

So how much does social distancing help? A problem is that we need to estimate how much socializing people do. I am going to restrict the definition of ‘distancing’ to large events and gatherings, such as sporting events, concerts, and Broadway. My estimate based on some research is that approximately 150 million Americans attend such events per year, some multiple times. However, there is also a fraction of Americans who don’t attend such events. The current rate, r=.27, assumes no distancing. Distancing means this value should be reduced, but by how much.
A reason why the reduction is not that much is because the incubation period, in which people are infectious but don’t know it, is around 5 days. So this means for distancing to work, asymptomatic carriers who would have attended such an event in this small 5-day window are unable to do so. This is not many people relative to the total number of infected. So consider there are 6,000 infected Americans who are asymptomatic carriers (extrapolation of the Princess cruise data in part 1). If half of such Americas will attend such an event (the other half maybe don’t like crowded events, or prefer to watch sports at home, or don’t like going to plays, etc.), that means 3000. But the window is just 5 days. So the odds of at least one of these people attending such an event in this window is 5/365*1/2 ( around .6%). So out of 6,000 carriers, closing such events prevents 36 interactions. What does this mean in terms of r? It means we need to derive a new value for r that takes into account a fraction high risk (distancing) and low risk carriers.

The solution to the logistic differential equation with two rates and a weighting is trivial: the rates are added as follows into a new value for r: r_1*w_1+r_2*(1-w_1)

The .27 value can be broken into two parts: a social group and non-social group:

.006*(r_2+r_1)+.994*(r_2)=.27

r_1 is the additional rate of infection due to large social events. r_2 is the baseline rate for non-social interaction such as food shopping or public transportation.

When distancing is implemented, we have r_2 as the new rate. But the problem is we don’t know what r_1 is. But we have some clues. The infection radius is generally accepted to be 6 feet, so if we consider a large group gathering, an affected individual may infect some people in his or her vicinity, but not dozens or hundreds, unlike a much more contagious virus such as measles. This further wakens the case for distancing.

To be continued