key: cord-0770726-cum8woiz
authors: Krewer, K. L.; Bonn, M.
title: Time-Resolving the COVID-19 Outbreak using frequency domain analysis
date: 2020-05-11
journal: nan
DOI: 10.1101/2020.05.07.20094078
sha: 39c0fd5f05b4970fa1a561b7a2c5086d18ef382f
doc_id: 770726
cord_uid: cum8woiz

Difficulties assessing and predicting the current outbreak of the severe acute respiratory syndrome coronavirus 2 can be traced, in part, to the limitations of a static description of a dynamic system. Fourier transforming the time-domain data of infections and fatalities into the frequency domain makes the dynamics easily accessible. Defining a quantity like the case fatality as a spectral density allows a more sensible comparison between different countries and demographics during an ongoing outbreak. Such a case fatality informs not only how many of the confirmed cases end up as fatalities, but also when. For COVID-19, knowing this time and using the entire case fatality spectrum allows determining that an outbreak had entered a steady-state (most likely its end) about 14 days before this is obvious from time-domain data. The lag between confirmations and deaths also helps to estimate the effectiveness of contact management: The larger the lag, the less time the average confirmed person had to infect people before quarantine.

many of the confirmed infected appear to have died from COVID-19?" This is a bit simpler, but the 48 "appear" does leave room for interpretation. The problem with finding the infected is that many 49 present very mild symptoms that are indistinguishable from those associated with influenza and other 50 common respiratory diseases usually summed up as "the common cold". The severe cases, especially 51 those leading to deaths, are harder to miss. Therefore, Ward[8], and Flaxman et. al. [9] conclude that 52 the reported deaths are likely closer to the actual deaths than the confirmed are to the actual infected. 53

If one wants to fight a pandemic, not merely monitor it, another question becomes important: "In how 54 many cases do we know that infectious people stopped spreading the disease because they were put 55 into quarantine?" That number, too, is the number of confirmed cases, since confirmed infectious 56 persons generally will be quarantined. 57

So: How can we describe the time dependence of the observables of an epidemic? 58

The problem is that the confirmations C reported on the day t depend on all infections I on each day 59 before that day t and how long before they happened. Mathematically: is not a single number, but an entire spectrum of multiple frequencies. Further the case fatality ̃( ) 85 at each frequency is a complex number 1 composed of an amplitude |̃|( ) and a lag ( ): 86

The amplitude of the case fatality |̃|( ) answers the question: "What fraction of the confirmed 88 cases reported at frequency end up dying?" And the lag ( ): "How soon?" 89 1 We use the ̃ to indicate all complex numbers in this paper.

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The copyright holder for this preprint this version posted May 11, 2020 common; last but not least digital copies of this paper may be compressed using it. The compression 100

works because the most relevant information is contained in very few frequency steps. In the case of 101

China ex. Hubei, the amplitude spectra of confirmations and deaths in fig. 1 b) start fluctuation 102 randomly for frequencies larger than 0.06/day. This means data at these frequencies are most likely 103 dominated by statistical fluctuations and do not contain much useful information. We can therefore 104 limit our analysis to the 6 frequency steps below 0.06/day and still capture the relevant information 105 from two series of almost 100 time steps (days). Hence, we only plot the case fatality for these first 106 frequency steps in fig. 1 c) . We can see that the amplitude and the lag are quite constant, at almost 107 0.9 • 10 −2 fatalities/confirmation and ca. 11 days, respectively. This means the outbreak in China ex. 108

Hubei can be described in the simple terms that about 1/100 of confirmed cases died, and the death 109 was reported on average 11 days after the confirmation. This, however, could also have been inferred 110 by just overlapping the curves of confirmations and deaths, albeit less mathematically rigorous. We 111 aim to retrieve a good estimate of the case fatality in an ongoing outbreak, not just in one that is 112 essentially over. We stick with China ex. Hubei and ask the question: "What could we have known on 113 day 20 (February 11 th )?" Well, when we Fourier transform the rates reported up to day 20, we see in 114

fig. 1 c) that the 0 frequency value of the case fatality is 0.35 • 10 −2 fatalities/confirmation, much lower 115 than the final value. This is not surprising, because the 0-frequency value is just the accumulated 116 number of deaths divided by the accumulated number of confirmations. As explained above, this value 117 is quite useless as the outbreak is ongoing on day 20. 118

The second Fourier component, however, is already at 0.6 • 10 −2 fatalities/confirmation, much closer 119 to the final value. In general, we should average over the whole spectrum. We suppress the noise from 120 statistical fluctuations by first using a 7-day floating average over the daily reports and then weighting 121 the average by the spectral intensity of deaths, since the lower number of deaths, the larger their 122 relative statistical error. The 7-day floating average unfortunately delays the time traces by half a week, 123 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 11, 2020. . https://doi.org/10.1101/2020.05.07.20094078 doi: medRxiv preprint costing some valuable time and reducing time resolution. It is necessary for estimating amplitudes, 124 especially in countries which report significantly less on weekends. Even with the delay from the 7-day 125 average, the spectral average converges towards to final case fatality value around day 25, about two 126 weeks before the "static case fatality" does. 127

We also have a posteriori recognized that deaths lag the confirmations by about 11 days. When we 128 compare the deaths accumulated up to a certain day with the confirmations accumulated up to 11 129 days earlier, we also get an estimate of the case fatality amplitude that converges towards the final 130 value after day 25. While we only know this 11-day value a posteriori from our Amplitude spectra of the time traces from a). c) Case fatality amplitude (blue, left axis) and lag (red, right axis) for the frequencies above the noise floor, which is below 0.06/day. Full dots come from dividing the spectra from day 96, seen in b); empty dots from spectra derived from the first 20 days. d) Estimates of amplitude and lag of the case fatality by different methods as a function of time: Full lines are deaths reported up that time divided by confirmations up to the same time. Using the confirmations obtained 11 days earlier yields the dashed line. The dots are obtained from spectra taken up to that time, the amplitude is a weighted spectral average of based on 7-day averaged spectra, the lag is the value at first frequency above 0 from daily spectra. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 11, 2020. . https://doi.org/10.1101/2020.05.07.20094078 doi: medRxiv preprint days on February 7 th (day 16 ). The first symptoms set in shortly after a patient can be tested positive 134 and respiratory distress is how most severe acute respiratory syndrome corona virus 2 patients die, at 135 least those dying quickly. So those 8 days would have given reasonable initial guess for the lag. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 11, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 11, 2020. . https://doi.org/10.1101/2020.05.07.20094078 doi: medRxiv preprint reporting slow deaths, which may not have been caused by the infection, is the simplest explanation 187 we can find for the discrepancy. 188

We now turn to Germany and compare the numbers with the cases in China ex. Hubei. We can see 189 that the curve of confirmations in Germany has a very similar shape as the ex. Hubei standard would . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 11, 2020. . https://doi.org/10.1101/2020.05.07.20094078 doi: medRxiv preprint between onset of symptoms and quarantine for the cases confirmed today?" We count here the ability 216

to produce a positive test as a symptom. 5 days is a recent estimate of the average incubation time 217

[18]. If most infected have been quarantined before they became infectious, the average time between 218 symptoms and quarantined must be below 0 and cannot go lower than (minus) the average incubation 219 time. We urge to focus on reporting this time, rather than the precise numbers of confirmations. 220

Measuring timescales is more important and reliable than quantifying the time-dependent observables 221 in dynamic situations, since observables will change drastically over time, but timescales tend to be 222 determined or at least limited by underlying time constants, in this case the incubation time. This is 223 the underlying reasoning how we come up with the "ultimate question" and the answer. 224 Fig. 4 . Amplitude (blue, left axis, logarithmic) and lag (red, right axis) of the case fatality for territories with more than 100 deaths. The amplitude is estimated in two ways: fatalities divided by the confirmations 11 days prior are marked with x, averages of the amplitude spectrum with o. When those two estimates start matching, the outbreak was entering a steady state and much fewer infections happened than during its beginning, the latter happened one average infectionfatality lag prior. A large case fatality lag indicates a low infection quarantine lag, which means the confirmed cases had little time to infect other people before being quarantined.

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The copyright holder for this preprint this version posted May 11, 2020. to deaths by increasing testing and reporting. Lags may also be negative in very early stages of the 245 outbreak, when the 11-days-corrected estimate may massively overestimate the case fatality while 246 the spectral average massively underestimates it. This can be seen in the early stages of the Korean 247 timeline in fig. 2 b) . The data in fig. 4 indicates that for example Ecuador, Morocco, Bangladesh and 248

Saudi Arabia are currently in this early stage of their respective outbreaks. 249

South Korea was probably the only country that got its initial outbreak under control mainly by contact 250 management rather than social distancing. By now, however, many countries should have the test and 251 contact management infrastructure to do the same. Countries with a lag close to 10 days were already 252 within reach of this goal before. They can switch to this strategy now and monitor the situation by 253 reporting their answer to the ultimate question: "How much time did the infectious people have to 254 infect more people?" This can even be done in countries that do not have enough test resources, by 255 quarantining all even mildly symptomatic people and their contacts on suspicion and only test a small 256 3 Well, differences in treatment and especially reporting of late deaths may change it, as we discussed for Korea and China, but not by an order of magnitude.

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The copyright holder for this preprint this version posted May 11, 2020. . https://doi.org/10.1101/2020.05.07.20094078 doi: medRxiv preprint

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