First Law of the peak: Peak is not immediate after the lockdown but comes many weeks later.
Continuing from the previous post on the Zeroth Law:
Well, by now it is obvious to everyone that has endured weeks of lockdown that the peak comes several weeks later. How many weeks later is of course the question that everyone is curious about now. Let us take the example of the number of cases in Austria, which was reasonably early in its response. It was probably one of the first European countries to close borders with other member states of the European Union. Similarly social distancing, lockdown were strictly implemented since the 16th of March. The result?
In the picture above, the daily new cases in Austria were around 200. But by the time the peak was reached, these cases were over a 1200 a day. Now what we notice is that: 1. The number of cases continued to increase despite the lockdown 2. The actual turn around happened many weeks later. Let me try to address both of these:
Cases increasing despite lockdown: Does this imply the lockdown is a success or a failure? Interestingly in this time just after lockdown, several interesting things happened in many countries. Mass migrations to hometowns was the simplest. Then many new cases with a traceable contact come out, which happened in South Korea, which happened in India. These cases which can be traced back to the same origin, a religious congregation, a Chinese or Persian New Year celebration, or a carnival celebration before Easter, or a football match take up the center stage of attention. This is a phase all countries went through. Since a majority can be traced to these super spreading events, it looked as if without these cases the numbers of new cases would have dropped immediately dropped. Unfortunately it is not as simple. True without such super spreader events, the overall infections will be lesser, may be even by 50%? But what is hidden behind this story is that another 50% of the cases which can not be easily traced to one single source continue to increase in number despite the lockdown. The reality is that all the countries before they were hit by COVID19 thought they were different. China thought it was Wuhan’s problem, rest of the world thought it was China’s problem. Soon Europe that it was Italy’s problem, and in a blink everyone had COVID19 in their backyard. Same with the cases after lockdown. The public perception everywhere has been that without some spurious sources, there will be no new cases. Certainly there will be fewer cases, but not an immediate decline in cases as some mathematical models would predict.
If the cases continue to increase, how do we know whether lockdown is working? Or why bother to even continue the lockdown? The purpose the lockdown is that the contact between the infected individuals and the susceptible individuals should be avoided. Mathematically when the new infections depend on the existing infections, the growth is exponential (dI/dt α I) where as if the people who are already infected are separated, there are many other relational dependences, but certainly the above proportionality is not valid. Which makes the growth reduce from an exponential to a sub-exponential behavior as observed in China (https://science.sciencemag.org/content/early/2020/04/07/science.abb4557) to a mostly linear behavior as observed in most other countries (https://www.medrxiv.org/content/10.1101/2020.04.07.20055772v1). The less acceptable growth rate post-lockdown which is still seen in some countries is an exponential but with a different rate constant. Not bad, but not great either.
Linear, Quadratic, and all other Sub-exponential growth rates show that the lockdown is in fact working, with some efficiency. As poor as the efficiency may appear to be it is still much better than the continuation of the pre-lockdown scenario. Human ears, eyes are good at dealing with exponential data by performing biological logarithms on them. We hear sounds of different decibels, which is the logarithm of the intensity of the sounds. Same with light we can both navigate our way in a very poorly lit room and at least for a fraction of a second gaze at the sun, because the light intensity is processed logarithmically. However the brain works fairly linearly, for most. If the number of cases today are 100, tomorrow are 200, it is natural to expect 100,200,300,400,.. However in an exponential growth phase it is not the difference but the ratio that matters and hence they will grow more like 100,200,400,800,..The more one waits the more disastrous they numbers become. So, one is desperate to buy themselves time with lockdowns, however poor the efficiency may be.
If the lockdown is partly or fully functional, how long should one wait for the peak? The simplest definition of the peak is a trend that the cases to be expected in the coming days are fewer than the cases that were seen in the past. One basic reason to see the cases even after lockdown is the “incubation time”. This is the time from the infection to the appearance of the symptoms. For COVID19, the incubation time has been constructed from the patient contact histories, by carefully reconstructing the stories of when they were exposed to infected people. It is not always so easy. It can be done if the person 1 that infected person 2 showed symptoms and found out that (s)he was COVID19 positive. In COVID19, up to 86% of the infections have remained asymptomatic which means one does not even know they are transmitting the infection (https://science.sciencemag.org/content/early/2020/04/24/science.abb3221).
Ok. Coming back to the incubation story, let us believe that the exposure history can be reconstructed for some, can we use it for obtaining the distribution of the incubation times? In fact, it has been done. It gives a Weibull/log-normal distribution with a median of 5.2 days, and lasting up to 15 days easily. Which means a person who was infected just on the day of the lockdown may show symptoms 15 days later. What happens in this period? If it is a perfect lockdown, the person is likely to transmit the infection to the family (s)he shares the house with, who may again thankfully remain asymptomatic with a great chance or show symptoms after some more days. In an imperfect lockdown it spills over beyond the 4-walls. The incubation time explains why even in a perfect lockdown you will hear of newer cases.
There is a little contradiction here, as you may have realized. The incubation can last up to 15 days. But the graphic of Austria above was showing cases up to 30 days (https://www.medrxiv.org/content/10.1101/2020.04.15.20067058v1). It is easy to brush off saying the lockdown may not have been perfect. True. But that explanation is like the famous Chinese idiom about cutting the feet to fit the shoe, takes away all hope of predictability by sounding like there is no room for quantifying this. This is something we will take up in the coming posts.
In simple terms, if one imagines driving a vehicle, lockdown is the removal of the foot from the accelerator. It should not be confused with applying a brake. Then the system moves with its own inertia, incubation before it slows down and then halts. The three stages are thus: acceleration, deceleration, and the stage in between which is the peak. Governments, Media, Public, all over the world have been anxious about it. Every few hours or every day they check to see if they crossed over the peak. Does it make sense to check every few hours? or even every single day? We discuss in the next one —
Next Post: Second Law of the Peak: The peak is buried in fluctuations
