The following is condensed and abridged. For full articles, please see Physical Measure of a Pandemic; April 15, 2020 and Physical Modelling of a Pandemic; March 30, 2020
These are strange times, the likes of which we probably see once in a generation, and it is amazing that it is happening across the world simultaneously, and in the age of internet. As physicists, as much as one would like to carry on , it is hard to ignore what is going on around you, and try to understand it, the same way we understand any other physical system. So here it goes:
As we approach any other physics problem, we start from first principles that describe the system of interest. What bothers me about the standard treatments is that, by reducing the system to ordinary differential equations, it ignores this geographical diversity and thus might miss important phenomenological features of the evolution.
Instead, I will use an inhomogeneous linear growth+diffusion equation to describe the early phase of an epidemic, prior to large-scale immunity or social intervention, where we can decompose the solution into modes with exponential growth (or decay). This results in two localized results we may consider as different communities, with different rates of growth of the epidemic. Incorporating probability distribution, we predict a super-exponential growth at early times, which is in contrast to the exponential expectation from simple uniform models. The reason for this is clearly that we are dealing with a distribution of growth rates, and the later times will be dominated by populations with faster growth rates, no matter how small they start.
Logarithmic derivative of the total fatality for US (blue), Canada (red), Italy (green), and South Korea (black), as of March 29th. For US and Canada, t=1 is at the report of first death, while the data for Italy and South Korea are shifted forward by 25 and 40 days, respectively. The curve shows the best fit to US data.
Fitting the model to the mortality data raises more questions - is it possible that the true onset of the Italian (South Korean) epidemic was 25 (40) days before the first death was attributed to the Covid-19 epidemic? What if there is no universality, and the different behaviors are dictated by environmental and social factors?
The best-fit model (to US, Canada, and Italy) predicts that Canadian mortality will pass 1000 around April 9th, while the US mortality will pass 10,000 around April 5th. While the latter milestone did indeed come to pass, the Canadian rate has fortunately slowed down since the end of March.
In fact, exactly how the spread of a contagion slows down was beyond the scope of my first stage of analysis, but clearly it is what we care about the most as a society. So, given the wealth of data that is available on the spread of COVID-19 across the world, I decided to figure out how the data can tell us when a slow-down might be happening.
There are different ways that different outlets decide to visualize this. Probably one of the better ones is the number of daily confirmed fatalities as a function of time (right). While this is a good measure, it still has a few caveats. One obvious caveat is that there could be many deaths that were missed, especially at peaks of outbreaks (e.g., due to death at home), or misidentification of the cause of death. Another caveat is that it is hard to compare different countries, due to their different populations, or different sizes of outbreaks.
To solve the latter problem, the Relative Daily Increase in Reported Death is introduced, and a rolling average over 14 days is implemented to reduce the scatter.
What we really need is this number for the infected individuals, but given the arbitrariness and shortcomings in testings across municipalities, and the prevalence of asymptomatic individuals, the mortality numbers (even with all its shortcomings) may give a more reliable measure of the epidemic. However, there is clearly a time-delay between the exposure and death, which needs to be taken into account if one wants to relate the mortality statistics to the epidemic dynamics. According to current clinical studies, the incubation period (the period from exposure to onset of symptoms) is 4 to 7 days, while the time from onset of symptoms to death (on average) is 17 to 19 days (both at 95% confidence level). Given my lack of knowledge of how these uncertainties may be correlated, I take the conservative range of 21-26 days, as the expected time delay from exposure to death. The 21-26 day delay implies that it takes at least 21 days for any effect of, e.g., social distancing to show up, but it may take up to 40 days to see the full effect.
Comparing the COVID-19 mortality growth rates of Canada and the U.S., we see that both countries show an early rise in death growth rates, with a precipitous drop in mortality growth rate following the peak. But what is responsible for these precipitous drops in mortality growth rates? The natural response might be that preventative measures, such as lockdowns and social distancing have started to work. Using Google Mobility Reports, we notice that the activities in both countries start to drop around March 15th, and take almost a week to reach their plateau. If this drop can mitigate the spread of COVID-19, given the 21-day delay that we discussed above, it should only start to affect the mortality after April 5th, which could well fit the rapid drop in Canadian mortality growth on April 9th. But what about the US, whose mortality growth peaked on March 25th? For that, the spread of the virus must have slowed down well before March 4th. Even looking at the mobility reports for Washington and New York States, which are respectively the earliest and the biggest US epicenters of the pandemic, show little change in social activities prior to March 8th. More curiously, even though US and Canada started their social distancing around March 15th, the precipitous drop in Canadian mortality rate around April 9th, has no counterpart in US’s data.
To understand how unique the situation with the US might (or might not) be, I decided to compare it to mortality growth rate for other countries. In the plot at left, I am comparing this measure for the 9 countries with the highest reported total death. I have shifted all the curves, so that their peak is at t=0.
It appears that the mortality growth rates fall precipitously, in essentially an identical fashion to the US, past the peak: all countries appear to peak, and linearly decrease, with similar slopes that cross zero within 21-27 days. The two exceptions are the UK and France that appear to undergo secondary and tertiary outbreaks.
Similar to the US, there appears to be no correlation between the dates of the lockdown and those of the peak mortality rate, no matter how effective the lockdown has been.
There is an interesting coincidence between the numbers that we discussed above. The 21-27 days from the peak to when it crosses zero (i.e. max. daily fatality), which is common amongst the 9 countries with biggest total death is identical to the 21-26 days exposure-to-death period, inferred for COVID-19. Therefore, if we are near the peak of the epidemic and you suddenly stop the infections today, then most fatalities will happen 21-26 days from today. This would explain the universality of the universal precipitous drop mortality growth rate.
But what would suddenly stop the spread at the peak? As the last plot showed, there doesn’t seem to be an obvious causal relationship between the lockdown and the peak in mortality growth. In fact, one may speculate that the lockdown might be more of a sociological response to large mortality growth.
To shed light on this situation, let’s make a comparison between four countries, Canada and Austria (with more effective lockdown), USA (with less effective lockdown), and Sweden (with no effective lockdown), pictured right.
We see that, for Austria and Canada, as we argued above, the rapid drop and small peak is a likely (and timely) response to social distancing measures. However, countries with late or no lockdown will reach a universal maximum, until they presumably reach some version of “herd immunity”, at which point the death rate drops in a universal fashion (case in point: US and Sweden curves are indistinguishable beyond the peak, despite vastly different populations). The problem with “herd immunity”, however, is that it may need to happen over and over again, within different communities in a country, as it seems to be happening in France and the UK.
I guess one lesson might be that you may always learn new things by plotting different statistical measures of a physical phenomenon, including a pandemic. I also think the universal behavior of mortality growth rate, as I have defined here, at least with countries with large number of deaths was a surprise to me, and something that I hadn’t seen anywhere else. I also believe the existence of a dark onset of epidemic for most countries in Europe and China (having missed the early rise), might be another lesson from these plots. As to the most important question, i.e. the role of “herd immunity” vs. “lockdown”, it appears hard to distinguish the two possibilities, as they both appear to happen around the peaks of a pandemic in a community. However, countries with poorer and/or later lockdowns appear to see higher peaks and/or slower decays of the mortality growth rate (translating to a larger overall fatality per capita). Ultimately though, this is intimately related to the prevalence of asymptomatic and undetected spreaders, the more of them around, the less fatal is COVID-19 and we reach “herd immunity” faster, while an effective lockdown and contact tracing will become harder.
