The relevant Covid-19 fatality rate

Much has been written in the past few days about whether the case fatality rate (CFR) for Covid-19 is actually much lower than the original estimate of about 3 to 4%. Globally, the CFR is highly variable ranging  from half a  percent in Germany to nearly 10% in Italy. The difference could be due to underlying differences in the populations or to the extent of testing. South Korea, which has done very wide scale testing, has a CFR of around 1.5%. However, whether the CFR is high or low is not the important parameter.  The number we must determine is the population fatality rate because even if most of the people who become infected with SARS-CoV-2 have mild or even no symptoms so the CFR is low, if most people are susceptible and the entire world population gets the virus then even a tenth of a percent of 7 billion is still a very large number.

What we don’t know yet is how much of the population is susceptible. Data from the cruise ship Diamond Princess showed that about 20% of the passengers and crew became infected but there were still some social distancing measures in place after the first case was detected so this does not necessarily imply that 80% of the world population is innately immune. A recent paper from Oxford argues that about half of the UK population may already have been infected and is no longer susceptible. However, I redid their analysis and find that widespread infection although possible is not very likely (details to follow — famous last words) but this can and should be verified by testing for anti-bodies in the population. The bottom line is that we need to test, test and test both for the virus and for anti-bodies before we will know how bad this will be.

How many Covid-19 cases are too many ?

The US death rate is approximately 900 per 100,000 people. Thus, for a medium sized city of a million there are on average 25 deaths per day. Not all of these deaths will be  preceded by hospital care of course but that gives an idea for the scale of the case load of the health care system. The doubling time for the number of cases of Covid-19 is about 5 days. At this moment, the US has over 25 thousand cases with 193 cases in Maryland, where I live, and over 11 thousand in New York. If the growth rate is unabated then in 5 days there will be almost 400 cases in Maryland and over 50 thousand in the US. The case-fatality rate for Covid-19 is still not fully known but let’s suppose it is 1% and let’s say 5% of those infected need hospital care. This means that 5 days from now there will be an extra 20 patients in Maryland and 2500 patients in the US. New York will be have an extra thousand patients. Many of these patients will need ventilators and most hospitals only have a few. It is easy to see that it will not take too long until every ventilator in the state and US will be in use. Also, with the shortage of protective gear,  some of the hospital staff will contract the virus and add to the problem. As conditions in hospitals deteriorate, the virus will spread to non-covid-19 patients. This is where northern Italy is now and the US is about 10 days behind them. This is the scenario that has been presented to the policy makers and they have chosen to take what may seem like extreme social distancing measures now. We may not be able to stop this virus but if we can slow the doubling time, which is related to how many people are infected by a person with the virus, then we can give the health care system a chance to catch up.

Probability of gun death

The tragedy in Oregon has reignited the gun debate. Gun control advocates argue that fewer guns mean fewer deaths while gun supporters argue that if citizens were armed then shooters could be stopped through vigilante action. These arguments can be quantified in a simple model of the probability of gun death, $p_d$:

$p_d = p_gp_u(1-p_gp_v) + p_gp_a$

where $p_g$ is the probability of having a gun, $p_u$ is the probability of being a criminal or  mentally unstable enough to become a shooter, $p_v$ is the probability of effective vigilante action, and $p_a$ is the probability of accidental death or suicide.  The probability of being killed by a gun is given by the probability of someone having a gun times the probability that they are unstable enough to use it. This is reduced by the probability of a potential victim having a gun times the probability of acting effectively to stop the shooter. Finally, there is also a probability of dying through an accident.

The first derivative of $p_d$ with respect to $p_g$ is $p_u - 2 p_u p_g p_v + p_a$ and the second derivative is negative. Thus, the minimum of $p_d$ cannot be in the interior $0 < p_g < 1$ and must be at the boundary. Given that $p_d = 0$ when $p_g=0$ and $p_d = p_u(1-p_v) + p_a$ when $p_g = 1$, the absolute minimum is found when no one has a gun. Even if vigilante action was 100% effective, there would still be gun deaths due to accidents. Now, some would argue that zero guns is not possible so we can examine if it is better to have fewer guns or more guns. $p_d$ is maximal at $p_g = (p_u + p_a)/(2p_u p_v)$. Thus, unless $p_v$ is greater than one half then even in the absence of accidents there is no situation where increasing the number of guns makes us safer. The bottom line is that if we want to reduce gun deaths we should either reduce the number of guns or make sure everyone is armed and has military training.

Information content of the brain revisited

My post – The gigabit machine, was reposted on the web aggregator site reddit.com recently.  Aside from increasing traffic to my blog by tenfold for a few days, the comments on reddit made me realize that I wasn’t completely clear in my post.  The original post was about a naive calculation of the information content in the brain and how it dwarfed the information content of the genome.  Here, I use the term information in the information theoretical sense, which is about how many bits must be specified to define a system.  So a single light switch that turns on and off has one bit of information while ten light switches have 10 bits.  If we suppose that the brain has about $10^{11}$ neurons, with about $10^4$ connections each, then there are $10^{15}$ total connections.  If we make the very gross assumption that each connection can be either “on” or “off”, then we arrive at $10^{15}$ bits.  This would be a lower bound on the amount of information required to specify the brain and it is already a really huge number.  The genome has 3 billion bases and each base can be one of four types or two bits, so this gives a total of 6 billion bits.  Hence, the information contained in the genome is just rounding noise compared to the potential information contained in the brain.  I then argued that education and training was insufficient to make up this shortfall and that most of the brain must be specified by uncontrolled events.

The criticism I received in the comments on reddit was that this doesn’t imply that the genome did not specify the brain. An example that was brought up was the Mandelbrot set where highly complex patterns can arise from a very simple dynamical system.  I thought this was a bad example because it takes a countably infinite amount of information to specify the Mandelbrot set but I understood the point which is that a dynamical system could easily generate complexity that appears to have higher information content.  I even used such an argument to dispel the notion that the brain must be simpler than the universe in this post.  However, the key point is that the high information content is only apparent; the actual information content of a given state is no larger than that contained in the original dynamical system and initial conditions.   What this would mean for the brain is that the genome alone could in principle set all the connections in the brain but these connections are not independent.  There would be correlations or other high order statistical relationships between them.  Another way to say this is that while in principle there are $2^{10^{15}}$ possible brains, the genome can only specify $2^{6\times10^{9}}$ of them, which is still a large number.  Hence, I believe that the conclusions of my original post still hold – the connections in the brain are either set mostly by random events or they are highly correlated (statistically related).

Evolution of overconfidence

A new paper on the evolution of overconfidence (arXiv:0909.4043v2) will appear shortly in Nature. (Hat tip to J.L. Delatre). It is well known in psychology that people generally overvalue themselves and it has always been a puzzle as to why.  This paper argues that under certain plausible conditions, it may have been evolutionarily advantageous to be overconfident.  One of the authors is James Fowler who has garnered recent fame for claiming with Nicholas Christakis that medically noninfectious phenomena such as obesity and divorce are socially contagious.  I have always been skeptical of these social network results and it seems like  there has been some recent push back.  Statistician and blogger Andrew Gelman has a summary of the critiques here.  The problem with these papers  fall in line with the same problems of many other clinical papers that I have posted on before (e.g. see here and here).  The evolution of overconfidence paper does not rely on statistics but on a simple evolutionary model.

The model  considers competition between two parties for some scarce resource.  Each party possess some heritable attribute and the one with the higher value of that attribute will win a contest and obtain the resource.   The model allows for three outcomes in any interaction: 1) winning a competition and obtaining the resource with value W-C (where C is the cost of competing), 2) claiming the resource without a fight with value W, and 3) losing a competition with a value -C.    The parties assess their own and their opponents attributes before deciding to compete.  If both parties had perfect information, participating in a contest would be unnecessary.  Both parties would realize who would win and the stronger of the two would claim the prize. However,  because of the error and biases in assessing attributes, resources will be contested. Overconfidence is represented as a positive bias in assessing oneself.  The authors chose a model that was simple enough to explicitly evaluate the outcomes of all possible situations and show that when the reward for winning is sufficiently large compared to the cost, then overconfidence is evolutionarily stable.

Here I will present a simpler toy model of why the result is plausible. Let P be the probability that a given party will win a competition on average and let Q be the probability that they will engage in a competition. Hence, Q is a measure of overconfidence.  Using these values, we can then compute the expectation value of an interaction:

$E = Q^2P (W-C) + Q(1-Q) W - Q^2(1-P) C$

(i.e. the probability of a competition and winning is $Q^2P$, the probability of  winning and not having to fight is $Q(1-Q)$, the probability of  losing a competition is $Q^2(1-P)$, and it doesn’t cost anything to not compete.)  The derivative of E with respect to Q is

$E' = 2 QP(W-C) + (1-2Q)W-2Q(1-P)C=2Q[(1-P)W-C]+W$

Hence, we see that if $(1-P)W > C$, i.e. the reward of winning sufficiently exceeds the cost of competing, then the expectation value is guaranteed to increase with increasing confidence. Of course this simple demonstration doesn’t prove that overconfidence is a stable strategy but it does affirm Woody Allen’s observation that “95% of life is just showing up.”

Productivity and ability

What makes some people more productive then others?  Is it innate ability, better training, hard work?  Although the meaning of productivity is subjective,  there are quantifiable differences between researchers in measures of productivity such as the  h-index.    Here I will argue that a small difference in ability or efficiency can lead to great differences in output.

Let’s consider a simple and admittedly flawed model of productivity.  Suppose we consider productivity to be the number of tasks you can complete and let $P$ represent the probability that you can accomplish a  task (i.e. efficiency).  A task could be anything from completing an integral, to writing a program, to sticking an electrode into a cell, or to finishing a paper.  The probability of completing $N$ independent tasks is $T=P^N$.  Conversely, the number of steps that can be completed with probability $T$ is $N = \log T/\log P$.  Now let $P = 1-\epsilon$, where $\epsilon$ is the failure probability.  Hence, for high efficiency (i.e. low failure rate),  we can expand the logarithm for small $\epsilon$ and obtain $N \propto \epsilon^{-1}$.  The number of tasks you can complete for a given probability  is inversely proportional to your failure rate.

The rate of change in productivity with respect to efficiency increases even faster with

$\frac{dN}{d P}\propto \epsilon^{-2}$

Hence, small differences in efficiency can lead to large differences in the number of tasks that can be completed and the gain is more dramatic if you have higher efficiency.  For example, if you go from being $90\%$ efficient (i.e. $\epsilon = .1$) to $95\%$ efficient (i.e. $\epsilon = .05$) then you will double the number of tasks you can complete. Going from $98\%$ to $99\%$ is also a doubling in productivity.  The model clearly disregards the fact that tasks are often correlated and have different probabilities for success.  I know  some people who have great trouble in revising and resubmitting papers to get published and thus they end up having low measured productivity even though they have accomplished a lot.   However, it seems to indicate that it is always worth improving your efficiency even by a small amount.

Some numbers for the BP leak

The Deepwater Horizon well is situated 1500 m below the surface of the Gulf of Mexico.  The hydrostatic pressure is approximately given by  the simple formula of $P_a+ g\rho h$ where $P_a = 100 \ kPa$ is the pressure of the atmosphere, $\rho = 1 \ g/ml = 1000 \ kg/m^3$   is the density of water, and $g = 10 \ m/s^2$ is the gravitational acceleration.  Putting the numbers together gives $1.5\times 10^7 \ kg/m s^2$, which is $15000 \ kPa$ or about 150 times atmospheric pressure.  Hence, the oil and natural gas must be under tremendous pressure to be able to leak out of the well at all.  It’s no wonder the Top Kill operation, where mud was pumped in at high pressure, did not work.

Currently, it is estimated that the leak rate is somewhere between 10,000 and 100,000 barrels of oil per day.  A barrel of oil is 159 litres or 0.159 cubic metres.  So basically 1600 to 16000 cubic metres of oil is leaking each day.  This amounts to a cube with sides of about 11 metres for the lower value and 25 metres for the upper one, which is about the length of a basketball court.  However, assuming that the oil forms a layer on the surface of the ocean that is 0.001 mm thick, this then corresponds to a slick with an area between 1,600 to 16,000 square kilometres.  Given that the leak has been going on for almost two months and the Gulf of Mexico is 160,000 square kilometres, this implies that the slick is either very thick, oil has started to wash up on shore, or a lot of the oil is still under the surface.