Pandemics and Societal Choices (Part I)
Thanks to all who participated in our first Eco 100 online meeting today. I'll post an edited recording soon. I hope to be a bit more adept with the zoom software next time (e.g. to allow more questions and participation).In this blog post, I'll summarize some of the main points. The short deck of slides I used is here. Links to the videos and sources I relied on heavily, below.
We want to get to the economics of this situation but it's hard to do that without a first basic understanding of how epidemics spread and how they can end.
Contagion and Exponential growth
This is by no means the first global pandemic. There have been many throughout history. The last large 1918 flu pandemic killed possibly as many as 50 million people. Fortunately for us, modern medicine and communications allow us to take some more effective measures to prevent and treat people with disease, but the fast-moving nature of the epidemic caught many countries unprepared.
Grant Sanderson who makes great math videos at the 3blue1brown youtube channel has a nice explainer on epidemics dynamics (Dario in our class pointed to a possibly more accessible video making similar points here). There's a bit of math involved but high school level stuff.
During the early outbreak in China the daily growth rate $E\cdot p$ was about 20 percent per day. If you plot out the actual number of cases $N_d$ over time in China you get a curve like this:
The exponential relationship above can be rewritten as
$$
N_d = (1+E\cdot p)^d \cdot N_0$$
where $N_0$ is the initial number of infected people at some start day 0.
If you take the logarithm of both sides you get
$$
log(N_d) = d \cdot log(1+E \cdot p) + log(N_0)
$$
Hence the log of the number of cases rises linearly with the number of days $d$, with a slope given by the growth rate $E \cdot p$. So if you display the log of the number of cases in China you from the chart above you get:
which seems well described by a linear fit. Each vertical grid line step up represents a ten-fold growth in the number of infected. From the diagram above we can deduce that it takes just 16 days for the number of infected to be multiplied tenfold.
The scary thing about exponentials is how quickly the numbers can grow. An old trick science question makes the point and runs something like this:
One way to get a better feel for how fast things are moving is to calculate how long it will take for the number of cases to double in size. If daily cases $N_d$ are growing at g percent per day, it's easy enough to solve for an exact answer, but you'll need a calculator (just solve for $d$ in the equation $2 \cdot N_0 = (1+g)^d \cdot N_0$). But there's a useful trick called The Rule of 72 that will give a you very good approximation (so long as growth rates are not crazy high or low). The rule states that if the rate of growth is $g$ then the population will double in size in $d^*=\frac{72}{g}$ periods. For example, if the rate of growth of confirmed Covid19 cases in the USA is growing at 33 percent per day (as it has been in Italy, the USA and many other countries as shown below; source: FT), then it will take just $\frac{72}{0.33}=2.18$ days for the number of infected to double.
If science could develop an effective vaccine to provide immunity to large portions of the population we would get to an inflection point quickly but many observers say that that scenario may still be a year or a year and a half away. Before then at least part of the herd immunity will come from people getting infected and then recovering -- many people have reported having quite mild cases. The problem with waiting for that, of course, is that people in higher-risk groups may get seriously ill or die, and hospitals may be overwhelmed. Although it remains a controversial position, some observers have argued that, compared to the very costly social distancing measures we are now taking which are tanking the economy, it might be wiser to concentrate resources on protecting and caring for those most at risk and accept that others will get infected and recover.
The Washington Post has some great simulations to illustrate how different policies might work. Visit the site yourself to see how they simulated each of the following four scenarios (clockwise from top left) to see how the curve would be flattened (the colors represent uninfected, infected, and recovered): (1) no policy; (2) border restrictions; (3) 'moderate social distancing' and (4) 'extensive social distancing'
So where are we now? The situation changes day by day. As I post this on 3/21 there are more than 10K confirmed cases in NY State, accounting for nearly half of the cases nationally (even though NY State is only 6 percent of the population. The Federal Emergency Management Agency (FEMA) has declared NYC a 'Major Disaster' area.
What about the economic and social fallout? As I mentioned in my last post Societies have been forced to face a truly unpleasant tradeoff: the need to apply ever-stricter social distancing or else let the rate of progression of the disease rise to levels that might overwhelm the health sector. But social distancing measures heavily punish some sectors of the economy, in particular, the labor-intensive service sector where most work, particularly those on lower incomes. Society could lessen the immediate economic costs by using less social distancing, but this risks faster infections and more deaths, particularly among the more vulnerable. From what is being said now, we must, unfortunately, continue to make such choices not just for the next several weeks, but possibly for many months or more than a year.
More on actual and anticipated impacts on the economy and debates surrounding the implmeneted or proposed extrordinary measures by the Fed and the President and Congress. in the next classes.
In a simple model the number of cases a disease tomorrow (day $d+1$) is given by the number of cases today $N_d$ times one plus the growth rate of the disease. This last term can be thought of as given by the product of the average number of people someone infected each day $E$ times the probability $p$ that any of those exposures becoming an infection. Hence:
$$ N_{d+1} = (1+E \cdot p) N_d $$During the early outbreak in China the daily growth rate $E\cdot p$ was about 20 percent per day. If you plot out the actual number of cases $N_d$ over time in China you get a curve like this:
$$
N_d = (1+E\cdot p)^d \cdot N_0$$
where $N_0$ is the initial number of infected people at some start day 0.
If you take the logarithm of both sides you get
$$
log(N_d) = d \cdot log(1+E \cdot p) + log(N_0)
$$
Hence the log of the number of cases rises linearly with the number of days $d$, with a slope given by the growth rate $E \cdot p$. So if you display the log of the number of cases in China you from the chart above you get:
The scary thing about exponentials is how quickly the numbers can grow. An old trick science question makes the point and runs something like this:
A lily pad sits on a pond. It doubles in size every day. It has taken 600 days for the lily pad to grow large enough cover half of a very large pond. How many more days will it take for the lily pad to cover the entire pond?The scary answer, of course, is that it will take just one more day! Humans have a hard time comprehending exponential growth which may explain why politicians and policymakers were slow to understand the need for faster policy responses.
One way to get a better feel for how fast things are moving is to calculate how long it will take for the number of cases to double in size. If daily cases $N_d$ are growing at g percent per day, it's easy enough to solve for an exact answer, but you'll need a calculator (just solve for $d$ in the equation $2 \cdot N_0 = (1+g)^d \cdot N_0$). But there's a useful trick called The Rule of 72 that will give a you very good approximation (so long as growth rates are not crazy high or low). The rule states that if the rate of growth is $g$ then the population will double in size in $d^*=\frac{72}{g}$ periods. For example, if the rate of growth of confirmed Covid19 cases in the USA is growing at 33 percent per day (as it has been in Italy, the USA and many other countries as shown below; source: FT), then it will take just $\frac{72}{0.33}=2.18$ days for the number of infected to double.
This chart shows, on the vertical, the logarithm of the number of cases $log(N_d)$ in each country (grid lines a). On the horizontal time-axis, they are counting the number of days since each country reached its 100th case to give a sense of how fast each country is growing from the same base. The USA is a few days behind (the first cases started later) but on the same growth trajectory as countries such as Italy and Spain whose health care systems have become overwhelmed by an influx of severe cases (caveat: these numbers show confirmed cases which depends in part on how well things are being measured; the USA has more hospital beds than Italy). Here is another piece explaining the use of log charts that compares USA to Italy.
Social Distancing, Quarantines and Herd immunity
The growth rate $g=E \cdot p$ was determined by how many other people each infected person encounters per day $E$ times the probability that any such contact results in a transmission $p$. These numbers might be altered via policy. Social distancing measures such as New York and many other states have now impoesed involving the closure of schools, bars, restaurants and event gatherings lowers the number of encounters $E$. Information campaigns to get people to keep a distance, clean their hands frequently, not touch their face, etc, can lower $p$. These measures aim to flatten the curve: slowing the rate of growth in order to lower the daily count to a level hopefully below the healthcare system's capacity to cope:
Now obviously the number of cases keep growing forever since, before long, the entire planet would be infected with nobody left to infect. Another factor is that although some may become gravely ill and even die, the vast majority of those infected will recover and gain immunity. That then means that in future contacts with infected people they will not become infected and will no propagate the virus. Once this is considered the exponential curve becomes a logistic curve like the one shown below.
The Washington Post has some great simulations to illustrate how different policies might work. Visit the site yourself to see how they simulated each of the following four scenarios (clockwise from top left) to see how the curve would be flattened (the colors represent uninfected, infected, and recovered): (1) no policy; (2) border restrictions; (3) 'moderate social distancing' and (4) 'extensive social distancing'
So where are we now? The situation changes day by day. As I post this on 3/21 there are more than 10K confirmed cases in NY State, accounting for nearly half of the cases nationally (even though NY State is only 6 percent of the population. The Federal Emergency Management Agency (FEMA) has declared NYC a 'Major Disaster' area.
What about the economic and social fallout? As I mentioned in my last post Societies have been forced to face a truly unpleasant tradeoff: the need to apply ever-stricter social distancing or else let the rate of progression of the disease rise to levels that might overwhelm the health sector. But social distancing measures heavily punish some sectors of the economy, in particular, the labor-intensive service sector where most work, particularly those on lower incomes. Society could lessen the immediate economic costs by using less social distancing, but this risks faster infections and more deaths, particularly among the more vulnerable. From what is being said now, we must, unfortunately, continue to make such choices not just for the next several weeks, but possibly for many months or more than a year.
More on actual and anticipated impacts on the economy and debates surrounding the implmeneted or proposed extrordinary measures by the Fed and the President and Congress. in the next classes.
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