When I was in 7th grade one day, Mr. Houseman, our science teacher, gave us a math problem.
He told us the story of the miser and the blacksmith.
A blacksmith is putting shoes on a miser’s horse. The blacksmith charges $10 for the job. The miser refuses to pay. “Ok,” the blacksmith says, “I will make a deal with you. There are 8 nails in each horseshoe. The horse has four shoes. That would be 32 nails in all. I will ask you to pay one penny for the first nail, two pennies for the next nail, four pennies for the next nail, eight pennies for the next nail, etc… until the 32nd nail.”
Our teacher asked us if we would take the deal if we were the miser. Most of us said, “yes.” I mean – it is just pennies. It won’t add up to very much.
He then had us start doing the math and told us to bring our answers to class the next day. We were told that we had to do the math by hand (no calculator) and that we had to show our work.
If any of you have ever done this problem, you know the secret.
Later that night at home I had a huge piece of paper filled with numbers. With each additional nail the numbers got bigger and bigger. It was hard to keep it straight and to do the math by hand.
The total cost was $42,949,672.95 (almost 43 million dollars).[1]
What is the trick? It has to do with exponential growth. If you want to see more interesting illustrations, search for exponential growth on YouTube.[2][3] What you will see will be a variety of similar illustrations. They are really quite wild.
It doesn’t make sense. It defies common sense.
But it is real.
Why does this matter and what does it have to do with COVID 19?
COVID 19 in a society without suppression has an Ro or rate of spread of 2. This means that it follows the same math as the miser and the blacksmith problem. That means that how it spreads will not make sense from a gut level or common-sense approach. That math however can be alarming. If it were to spread unchecked, it can seem like nothing is happening and then seemingly out of nowhere – a simple $10 job becomes $43 million. Or a few cases become tens of thousands and hospitals and health systems become overwhelmed.
But if suppression measures are done and are effective, the Ro can go down and the exponential growth goes away or is reduced.
In early to mid-March, our health system and the experts they relied on started using epidemiologic mathematical methods to predict what might happen. The graphs and predictions were terrifying. We were looking at needing thousands and thousands of hospital beds, intensive care rooms and ventilators. The potential was far greater than any health system could ever handle.
The next models looked at what would happen if we implemented social distancing. They started out estimating based on a 30% social distancing model. This assumes that we can never completely 100% isolate everyone from everyone else. If you have to pick a number what number is realistic? Experts suggested that the reasonable degree of social distancing that we could expect in our population would be around 30%.
The models started giving us data that we could potentially work with. The demand still looked enormous, but it was at least in the ballpark. The predictions included a peak of 2500+ patients in the hospital with 1000 patients in the ICU and around 700 on ventilators at a time. It meant that we would exceed all possible capacity of the hospital system. Teams went to work implementing plans to build out more and more capacity depending on what happened.
Something different happened in West Michigan, however. Cell phone data has suggested that the degree of social distancing has in reality been around 58%. This has had a dramatic difference on what has happened. Instead of the huge surge in numbers of patients with COVID 19, it has been a steady stream of patients with numbers in the range of 50-80 patients per day in the hospital.
Does this mean that the models were wrong?
The answer lies with the miser and the blacksmith.
It has to do with exponential growth. If we achieve a high degree of social distancing the exponent drops. Depending on the exponent, wide variations in the numbers of cases at a time can occur. The math shows that what happens can go beyond what common sense will tell us.
The projections depend very much on the percent of social distancing that is achieved. If it loosens up a small amount (which seems likely based on the trends in society) then the numbers look like there will be a higher peak (perhaps around 500 patients per day in the hospital).
Is this all theoretical? Nope.
It suddenly makes sense of why we can have had such dramatically different outcomes in places like New York City or Detroit compared to Grand Rapids. Those places that had time to implement social distancing before the exponential growth took off have a lot lower numbers of cases. The experiences in the US (and the world) match the math.
So – we might only see about 50-100 patients in our hospital system at a time (the current state), or we might peak at 500 (if social distancing loosens a little bit), or we might have 2000 (if restrictions are loosened a lot.)
But then where do we go from here?
As I stated in a prior blog[4], the problem is not with the math or the effectiveness of social distancing. It has all worked.
The problem now has to do with sustainability. The tail end of the curves are now extending far into 2021. We are all going to be forced to figure out different solutions on how we can live through this. We need to somehow maintain some degree of safe social distancing – but we have to be able to do it for a lot longer than any of us would like.
In some ways it is like trying to open the valve on a fire hose just a tiny amount so that you can get a drink. We are very thirsty. Not opening the valve is not really an option. Maybe we will be good enough to open it up a little bit. Maybe we won’t and we will get blasted in the face. Maybe the number of cases will go up a modest and manageable amount. Maybe the number of cases will surge and we will see repeats of New York City and Detroit.
A Wall Street Journal headline yesterday described May 2020 as being one of the greatest social experiments in history.[5] The huge experiments are going on in states that have loosened up social distancing (like Georgia and Texas.) I am really hoping and praying that these go well. If they do perhaps they can provide lessons and a guide for all of the rest of us.
I don’t have the answers.
But I did find it very interesting to look at YouTube posts from 2019, 2016, and 2013 that described potential pandemics and the exponential math that has happened in our world.
The miser and the blacksmith and what I thought was an odd assignment from Mr. Houseman comes back to me so many years later. It makes so much sense to me for what is happening in our world now. Who would have thought it?
[1] https://www.pedagonet.com/brain/brain43.htm
[2] https://www.youtube.com/watch?v=Tm38xA3KcQY
[3] https://www.youtube.com/watch?v=0BSaMH4hINY
[4] https://manmedicineandmike.com/the-path-forward/
[5] https://www.wsj.com/articles/america-wants-to-reopen-from-coronavirus-but-disagrees-about-how-11588350721
4 replies on “The Story of the Miser and the Blacksmith and COVID 19”
A story that all of us need to read. You were a very sharp 7th grade science student. I didn’t know what you would become as an adult, but I knew that it would be something big and important. Lewis Houseman
Thanks so much! You were an excellent teacher. Amazing to see how what you taught continues to influence us even now!
But in time everyone at some point ‘meets’ the virus, right? We’re protecting the health care system which is necessary. I’m finding it hard to believe that we can keep it away from the elderly forever. Am I understanding it correctly?
Good question. The reason for all the social distancing is absolutely to keep the number of very ill at a manageable level at any given time. It is not necessarily to keep us all from getting COVID 19. However – there is a hope that not everyone has to get COVID. That would be if we get to “herd immunity” = about 70% have had it such that it is no longer being spread in the community. Theoretically also you could stop transmission in your community and have it go away – but our societies are so mobile – that would only be a temporary fix. The last hope would be a vaccine. In terms of the elderly – the most realistic hopes are keeping them from getting it until there is either a vaccine or herd immunity.