Everyone in Ecuador is closely watching the curve of the number of Ecuadorian coronavirus cases over time. As dead bodies begin to accumulate in the streets of Ecuador’s largest city, Guayaquil (not all due to coronavirus, to be sure), all of us anxiously look for signs that the curve is beginning to flatten, that the curfew, transit restrictions, and economic shutdown are having some effect. Last week the number of new cases per day finally did begin to drop from one day to the next. Were our countermeasures having some effect? Many of us, including me, thought so. Here is what the graph of new confirmed (tested) cases per day looked like as of March 30 (which is Day 18 on this graph):
However, looking more closely, there is something odd about this graph. There is too much variability for it to be accurately describing a disease which has a long and variable incubation time. Symptoms of Covid-19 usually don’t show up for three to eight days after infection; the median waiting time until symptoms show up is five days after exposure, with most people showing symptoms after between three and eight days. People don’t get tested and aren’t registered in this graph until after they show symptoms, so the people who test positive on any given day were infected between three and eight days earlier. So the positive test counts for each day are influenced by all of the infection rates of the previous two to eight days. This makes it mathematically impossible for the graph to change much from one day to the next, if the graph were really measuring what it is supposed to be measuring.
Let’s play with some hypothetical data to see how much the graph could vary from day to day. Let’s first look at a very extreme case: let’s say the probability of infection doubles every day, and then suddenly one day everyone is put into complete isolation and the infection rate instantly drops to zero. To make this graph I have to know the proportion of infected people who show symptoms after three days, the proportion who show symptoms after four days, etc. A good guess (given that the median for covid-19 is known to be 5 days, and that in one sample of 45 infections on the same day, some people did report symptoms on Day 3 and on Day 8) is that 10% get symptoms after three days, 20% get symptoms after four days, 20% get symptoms after five days, 20% after 6 days, 20% after seven days, and 10% after eight days. (The exact values of these numbers aren’t too important for the point I am making.)
So, under these conditions, how fast can the graph of new positive test cases per day drop when the actual infection rate suddenly drops to zero? Here is the answer (the infection rate drops to zero on Day 1):
Note that even after the infections are completely stopped from Day 1 onward, the graph continues to rise steeply for a few days as people infected earlier begin to show symptoms and get positive tests. The rise begins to slow down three days after the infections stop (Day 4 on the graph), and then it gently but steadily declines.
That was what happens when the disease is spreading exponentially until it suddenly disappears. Let’s look at the other extreme– what happens if the infection rate is holding steady until it drops to zero on Day 1? Symptoms continue to show up, and people continue to register new positive cases, for several more days after people cease getting infected. Then the curve starts to go gently down.
In real life, of course, the infection rate could never go from a high number to zero overnight. So any real curve will be even smoother than these. Likewise the actual range of delays between infection and test result is wider than the numbers I used, and this should also make the real curve even smoother than these.
Now look again at the Ecuadorian graph at the top of this post. It shows much more dramatic drops than any of these hypothetical graphs in which the disease is completely cured overnight. The Ecuadorian graph is just not possible under any sort of natural situation.
Indeed, when I add in the data from March 31 (which is Day 19 on the graph below) to April 5, the Ecuadorian graph reveals that the apparent dip in cases on March 29 and March 30 meant absolutely nothing:
Could this be blamed on random statistical variation from day to day? Probably not. The number of infections per day has what statisticians call a “Poisson distribution”, and its range of random variation (the “standard deviation”) is known to equal the square root of the mean value. If we had many days with the same infection probability, the number of infections per day would almost always be within two standard deviations of the mean. The highest daily value in graph above, 456, could be expected to vary from 413 to 498. The value just two days earlier was 42 new positive cases, far beyond the expected range of random variation given the value of 456 two days later.
So what can explain the Ecuadorian graph? At first I thought that test kits might be so scarce that the number of new verified coronavirus cases per day is almost entirely dependent on the availability of test kits. So this would really be a graph of test kit availability, nothing more. However, Javier Robayo points out that the real limitation is the lack of people able to run the tests, each of which takes about six hours. The biggest dips happen on weekends, suggesting that this kind of human factor is indeed shaping the graph. Sadly, this all means that patterns on this graph can offer no guidance to Ecuador’s citizens or its decision-makers.
Further evidence for this pessimism can be found by comparing the death rate per day with the new cases per day. Daily coronavirus death totals are also not reliable in Ecuador, as the government only lists deaths confirmed by testing for the virus. For example the health ministry reports a total to date of 180 coronavirus deaths, and mentions but does not tabulate an additional 159 deaths that were “probably” coronavirus (https://www.elcomercio.com/actualidad/contagios-muertes-covid19-balance-ecuador.html accessed April 5). But just to be conservative, let’s accept the official figure of 180 deaths as of April 5. Those people must have become infected in the preceding several weeks. The mortality of this virus is about one in a hundred or a bit less:
If there are 180 people dead, there must have been 180*100= 18000 infected people a week or so before April 5. But the official number of infected people for March 30 is only about 1900 people. This suggests that there are really almost ten times more people infected than the official numbers show. And if, as seems likely, the real death toll is twice the reported rate, then the real number of infected people is about twenty times greater than the official figures show. As of today, that would be 67000, which is more than 4000 cases per million inhabitants. This would be about twice the reported per capita infection rate of New York City.
This estimate of 34000-67000 infected depends on the death rate for Covid-19 really being 1% in Ecuador. If medical care in Ecuador were much worse than elsewhere in the world, perhaps the death rate is actually much higher here. If the death rate were 20% instead of 1%, the numbers of deaths and infections would match better. I am not sure which result is more frightening, that the rate of infection is one of the highest reported in the world, or that the death rate is one of the highest in the world. The truth is probably somewhere in between. I don’t think any of the possibilities are good.
The President of Ecuador himself agrees with this frightening conclusion: “Sabemos que tanto en número de contagios, como de fallecimientos, los registros oficiales se quedan cortos. La realidad siempre supera el número de pruebas y la velocidad con la que se presta la atención”.
[“We know that in both the number infected, and in the number of deaths, the official counts fall short. The reality is always exceeding the number of tests and the pace of attention.”- Lenin Moreno, President of Ecuador.]
And now Banos is running low on food…..We hope the internet and electricity don’t fail.
Lou Jost, Fundacion EcoMinga