---
abstract: 'The number of corona virus (COVID-19) infections grows worldwide. In order to create short term predictions to prepare for the extent of the global pandemic we analyze infection data from the top 25 affected countries. It is shown that all country-specific infection rates follow a power law growth behavior and the scaling exponents per country are calculated. We find two different growth patterns: either steady power law growth from the very beginning with moderate scaling exponents of 3-5 or explosive power law growth with dramatic scaling exponents of 8-11. In the case of the USA we even find an exponent of 16.59. By means of data analysis we confirm that instituting strict measures of lock-downs combined with a strong adherence by the population are effective means to bring the growth rates down. While many countries have instituted measures there are only three countries (Denmark, Norway, and South Korea) so far where such lock-downs led to a significant reduction of the growth rate. In the case of Denmark we calculate the reduction of the scaling exponents to move from 6.82 to 1.47.'
author:
- 'H.M. Singer'
bibliography:
- 'covid01.bib'
date: '2020-03-25'
title: 'Short-term predictions of country-specific Covid-19 infection rates based on power law scaling exponents'
---

Introduction {#introduction .unnumbered}
============

Since the identification of a novel novel corona virus (COVID-19) in Wuhan, China in December 2019 the virus kept spreading rapidly throughout the world resulting in a global pandemic. As of March 25th 10:30 CET 424,048 people in over 170 countries and regions have been infected [@arcgis]. Due to its high infectivity no slowing down of the spread is currently in sight. A medical cure or vaccine is not available yet. This pandemic brings health care systems worldwide to their limits. The mortality rate is calculated to be 2.5% [@LAI2020105924] (comparison: SARS 9.6% [@SMITH20063113] and H1N1 influence 0.6%[@VaillantEtAl2009]). As the number of infections rises many governments around the world have instituted drastic lock-downs and curfews and called for social distancing and work from home to reduce the rate at which the virus spreads.

The severity of the infections with the COVID-19 is not evenly distributed with respect to age. While children do not seem to be affected much the risk of complications and negative progression seems to increase with increasing age the elderly population (70+) being affected most (5-11% mortality rate) [@jcm9020523]. Preexisting health conditions such as cardiovascular diseases, diabetes, respiratory diseases or cancer [@DRIGGIN2020; @jcm9020523] and smoking [@Vardavas2020] seem to increase the risk of complications as well.

In order for health care systems to be able to predict the short term and longer term number of infected patients different models have been used. Epidemiological analysis models assumes an exponential growth and a fixed reproduction number $n$, i.e. the number of people infected by a sick person. For $n>1$ and an incubation time $\tau$ the total number of infections is assumed to grow exponentially $$n^{\frac{t}{\tau}}=e^{at}$$

This approach is the basis for a number of different predictive studies such as [@ZHAO2020214; @schmitt2020simplified].

Other approaches to estimate and predict the number of infections are for example based on ’mechanistic-statistical’ approaches coupling a SIR ODE model describing the unobserved epidemiological dynamics combined with a probabilistic model describing the data acquisition process and a statistical inference method [@roques2020mechanisticstatistical], time series modeling [@deb2020time], iterative maps [@botha2020simple], or agent based simulations [@chang2020modelling] as well as logistic maps[@Hermanowicz2020.02.04.20020461; @ROOSA2020256].

The problem with exponential approaches is that it assumes that any infected person will infect the same number $n$ more people not taking into account the human tendency to live and work in relatively closed groups. Therefore such models tend to overestimate the number of infected people as time progresses, simply because there are no more new people to infect in the individuals particular group. On the other hand if the disease were to spread only at the periphery of a compact region of infections then a parabolic growth $t^{2}$ would be expected [@br2020quadratic]. It is thus to be assumed that after an initial short exponential growth a slow down should be observed where the growth of newly infected people increases less than exponentially but probably faster than $t^{2}$ as humans are not generally limited to the confines of a compact region. Given what is known about human interactions and structures mostly showing small world phenomena and scale free networks [@Watts1998; @Watts2004] the assumption of a power law behavior is reasonable. Early analysis in February 2020 [@Ziff2020.02.16.20023820] showed this indeed to be true. After a short initial exponential growth period the number of infections, as well as the number of recovered and deceased patients follows a power law: $$n(t)=Bt^{\gamma}$$

In the case of the infection spreading in China the exponent $\gamma$ was determined to be $2.27$ [@Ziff2020.02.16.20023820] and $2.48$ [@li2020scaling] (with more data). In an update to [@Ziff2020.02.16.20023820] it was suggested to take the saturation and slow down in the rate of new infections into account by tapering off the growth with an exponential slow down as suggested in [@Vazquez_2006] $$n(t)=Ct^{\gamma_{cutoff}}e^{-\frac{t}{t_{0}}}$$

with $\gamma_{cutoff}=3.09$ and $t_{0}=8.90$ days. As a second parameter $t_{0}$ is introduced the original scaling exponent $\gamma$ changes slightly to $\gamma_{cutoff}>\gamma$ as a result of the fitting procedure.

Method {#method .unnumbered}
======

All data related to COVID-19 was downloaded from the publicly available JHU-CSSE (2020) data source provided continuously by the Johns Hopkins University Center for Systems Science and Engineering (JHUCSSE) [@csse_github]. Aggregate data calculation for countries with different regions or states were performed after collecting by a simple summation over all associated regions or provinces (for the countries USA, China, and Canada). The data for Australia was omitted because the aggregated data in [@csse_github] did not correspond to the value given in [@arcgis].

We have selected the data sets with the 25 most infections as presented in [@arcgis] on 2020-03-24 11:00 CET. The countries and their infection counts are given in Table I.

      Country       \#infections (2020-03-24)
  ---------------- ---------------------------
       China                  81588
       Italy                  63927
        USA                   46481
       Spain                  35212
      Germany                 30081
        Iran                  24811
       France                 20149
    South Korea               9037
    Switzerland               8795
   United Kingdom             6733
    Netherlands               4767
      Austria                 4742
      Belgium                 4269
       Norway                 2647
       Canada                 2088
      Portugal                2060
       Sweden                 2059
       Brazil                 1960
      Denmark                 1703
       Israel                 1656
      Malaysia                1624
       Turkey                 1529
       Japan                  1140
       India                   511
       Russia                  444

  : Countries with the most COVID-19 infections (2020-03-24)

We have plotted the data for each country in a log-log plot and determined the range of power-law behavior. From there the scaling exponent for each country was extracted. As an example the data for Germany is shown in a log-log plot in Fig. 1. The range used to extract the scaling exponent is denoted with two vertical lines. The best fit line (determined by linear regression) is superimposed on the data points. In Fig. 2 the short term prediction of the total number of infections is superimposed onto the data points for the the next 7 days.

![Log-log plot of the total number of infections for Germany as a function of time. The range of scale free behavior is denoted with two vertical lines. The best fit line (linear regression) is superimposed on the data points.](covid19_germany_loglog)

![Short term prediction for the next 7 days of the total number of infections for Germany superimposed on the available data points.](covid19_germany)

The scaling exponent for Germany was found to be $\gamma_{Germany}=11.59$.

In order to cross-check the validity of the determined scaling exponent we have recalculated the scaling exponent for China in the power law range and found $\gamma_{China}=2.20$ in excellent agreement with [@Ziff2020.02.16.20023820]. Fig. 3 shows the log-log plot for China.

![Log-log plot of the total number of infections for China as a function of time. The range of scale free behavior is denoted with two vertical lines. The best fit line (linear regression) is superimposed on the data points. A second fit line for the saturation range is superimposed as well.](covid19_china_loglog)

![Available data for the total number of infections in China. Superimposed are the scale free power law growth ($\gamma_{China,1}=2.20)$ and the the saturation with $\gamma_{China,2}=0.024$.](covid19_china)

Since no country apart from China exhibits signs of a saturation we have used the simple power law fit instead of the cutoff power law fit [@Vazquez_2006] to determine all scaling exponents.

Results {#results .unnumbered}
=======

We have extracted the scaling exponents for all the countries in Table I. As of 2020-03-24 there was only China which showed a saturation of the total number of infections. All the other countries have not reached yet such a level. We could distinguish three different stages:

1.  **Growth stage:** the total number of infections is in the power law range and a single scaling exponent could be extracted. Currently most countries fall into this category.

2.  **Slow down stage:** due to government imposed strict measures to slow down the spread of the virus the initial power law range split into a second one with a smaller scaling exponent. The total number is still increasing as a power law but more slowly than before. We have found only three countries where this was the case: Denmark, Norway, and South Korea.

3.  **Saturation stage:** the power law regime has ended and the total number of infections slows down to small numbers close to 0. This was only the case for China (see Fig. 3 and Fig. 4).

To demonstrate the slow down stage we present the data from Denmark in Fig. 5 as a log-log plot and Fig. 6 as short term predictions for the total number of infections. We find $\gamma_{Denmark,1}=6.82$ pre-intervention and $\gamma_{Denmark,2}=1.47$ post-intervention. It is, however, important to note that the data so far clearly does not show a saturation merely a slow down into a different power law regime with a different scaling exponent.

![Log-log plot of the total number of infections for Denmark as a function of time. The two ranges of scale free behaviors are shown between two vertical lines. The best fit lines (linear regressions) are superimposed on the data points. We find $\gamma_{Denmark,1}=6.82$ pre-intervention and $\gamma_{Denmark,2}=1.47$ post-intervention.](covid19_denmark_loglog)

![Short term prediction for the next 30 days of the total number of infections for Denmark. Superimposed are the scale free power law growth $\gamma_{Denmark,1}=6.82$ and the post-intervention slow down stage with $\gamma_{Denmark,2}=1.47$.](covid19_denmark)

We have plotted all country data in a log-log plot starting from the respective day of the first infection in Fig. 7. If more than one person was recorded to be infected on the first day we have subtracted the difference to 1 from the data set to plot all the data from the same origin. Since we are only interested in the power law exponent in the range of large total numbers of infections this subtraction did not affect the outcome of the plot and the result. Since, however, the number of recorded infections on the first day in China was 548 the data points from China do not follow the other countries’ general development.

As can be seen in Fig. 7 the growth rate of the total number of infections can be roughly divided into two distinct growth patterns:

1.  steady power law growth from the beginning in the blue (online) regime where we have found a scaling exponent of $\gamma_{steady}\text{\ensuremath{\approx}}4.8\,(2-6)$.

2.  explosive power law growth in the red (online) regime: after a long incubation time with hardly any infections the total number of infections increases rapidly with a scaling exponent of $\gamma_{explosive}\text{\ensuremath{\approx}}10\,(8-16)$.

![All 25 country data superimposed in a log-log plot starting from the day of the first infection. Since the first recorded data from China is 548 infections the China data does not show the same increase as the other curves that started with 1. Two distinct growth patterns can be distinguished: blue range (online) steady power law growth from the very beginning with a $\gamma_{steady}\text{\ensuremath{\approx}}4.8\,(2-6)$ and red range (online) with a long incubation time and an explosive power law growth with a $\gamma_{explosive}\text{\ensuremath{\approx}}10\,(8-11)$.](covid19_allcountries_loglog_prep)

The exact calculations of the power law scaling exponents are given in Table II. The scaling exponents are displayed graphically in Fig. 8.

      Country       $\gamma$ (pre)   $\gamma_{2}$(post)
  ---------------- ---------------- --------------------
       China             2.22        0.024 (saturation)
        Iran             2.26       
       Japan             3.38       
    Netherlands          3.63       
       Sweden            4.50       
    Switzerland          4.72       
      Portugal           4.89       
      Austria            4.90       
       Norway            5.25               2.08
       Turkey            5.91       
       Brazil            6.39       
       Israel            6.60       
       Italy             6.80       
      Denmark            6.82               1.48
      Belgium            8.88       
      Malaysia           9.05       
   United Kingdom        9.62       
       India             9.76       
       France           10.14       
       Russia           10.33       
       Spain            10.34       
    South Korea         10.88               0.83
      Germany           11.58       
       Canada           11.67       
        USA             16.59       

  : Scaling exponents $\gamma$ per country. Countries with a slow down stage or a saturation stage after government interventions and lock-downs have a second scaling exponent $\gamma_{2}$ attached. The countries are sorted according to increasing scaling exponent $\gamma$ from steady growth to explosive growth.

![Scaling exponents per country sorted in ascending order. Countries with slow down stages or saturation stages have the post-intervention lock-down (post-LD) scaling exponent $\gamma_{2}$ plotted next to the inital $\gamma$.](scaling_2020_03_23)

Conclusions {#conclusions .unnumbered}
===========

Our initial assumption was that cultural habits of proximity and cleanliness such as the habit of washing hands as well as the geographical location would influence the speed at which the total number of infection increases but at least on a per country comparison such a distinction was not discernible. Neither cultural habits nor social distance and personal space [@social_dist] seem to have a noticeable influence. Also the latitude and the according temperature differences do not seem to play any role.

A puzzling fact is that neighboring countries do not seem to exhibit similar scaling exponents such as Belgium (8.88) to the Netherlands (3.63) or France (10.14) to Switzerland (4.72) or Germany (11.58) to Austria (4.9).

We have confirmed that the implementation of lock-downs had an impact on the spread however, not the governmental sanctions *per se* but obviously the adherence to it by the people. Notably there was hardly any noticeable impact on the lock-down in Italy whereas the measures in Denmark and Norway were very effective. The complete lock-down of Wuhan in China seemed to have been so far the most effective measure, however, China is two months ahead of all the other countries so no direct comparison can be drawn at the moment. South Korea deserves a special mention since the wide range testing of a large part of the population and consequent isolation of confirmed cases seems to have prevented any further growth (dropping from 10.88 to 0.83).

A puzzling data point is Iran which has the second lowest scaling exponent (2.26) almost identical to China (2.20). As Iran to our knowledge did not introduce any severe and strict measures and lock-downs we are tempted to believe that the reported numbers do not correspond to the actual numbers of infected people in that country. Perhaps many people in Iran are not able to seek medical attention or prefer not to?

It is worth noting that the currently highest spread is found in the USA with a scaling exponent of 16.59. We do not know why the outbreak in the US was so explosive.

We have found that the stricter the lock-down measures have been instituted and the higher the adherence of the population to those measures the faster the exponent dropped and the more noticeable pre- and post-measure regimes were separated.

Countries where there were no strict measures (or not yet) and no adherence to those measures if implemented or if they were introduced very late in the process have scaling exponents of $\thicksim8-11$ (USA 16.59). Countries with earlier lock-down measures and stricter adherence have exponents of $\thicksim3-5$.

In conclusion the data supports evidence that the spread is not exponential but can be described as a power law which indicates that all investigated countries and societies seem to be organized as small world networks and show scale-free behavior in the total number of infections. Different governmental measures and adherence of the population to those measures leads to strikingly different growth rates and scaling exponents. The power law approach fits the recorded data very well and allows accurate short term predictions of the total number of infections which allows health care systems to prepare and to effectively plan necessary staff, infrastructure and in particular intensive care units and to triage new patients more efficiently.

The author reports no funding related to this research and has no conflicting financial interests.
