A reproduction rate which perfectly fits Covid-19
aa r X i v : . [ q - b i o . P E ] M a y A reproduction rate which perfectly fits Covid-19
Christoph Bandt ∗ June 4, 2020
Abstract
We present a simple technique to compare the development of the Covid-19 epidemic indifferent regions, based only on the time series of confirmed cases. Weekly new infections,taken for every day, are interpreted as infection potential of Covid-19. We derive a robusttime-varying reproduction rate for the infection potential, including asymptomatic cases,which does not depend on death rate or testing intensity. It requires few assumptions andshows a more plausible time course than official reproduction rates in several countries.
Daily and weekly Corona numbers.
The Covid-19 pandemic is commonly described bycumulated numbers of confirmed cases, deaths and recoveries, for each day t and each countryor region. We use only the series C t of confirmed cases from the database [16] of Johns HopkinsUniversity in its version from 25 May 2020. The letter t denotes a date. If t is today, then t − t − C t is a function of time with a meaning in the first stageof an epidemic, but not for lockdown conditions. The numbers of new infections N t = C t − C t − are the original data collected each day. They show tremendous variation, including a periodiccomponent reflecting the weekly rhythm of health administrations [16, 3]. The best way torepresent the course of the epidemic is by weekly new infections W t = C t − C t − = N t + N t − + ... + N t − (1)for each day t [15, 3], called Covid-19 activity in the maps of [8]. In Figure 1 they are shown forfive countries as incidences, to adjust for population size. Three stages of the epidemic, and the delay principle.
In a first stage, the epidemicin a country grows exponentially. Weak countermeasures reduce the rate of growth, but will notstop the growth [5]. Brazil is still in this phase. The other countries have implemented stronglockdown measures and managed to stop the growth.When dealing with Covid-19, we have to distinguish infection time, symptom time, andobservation time.
From the time of infection up to the observation of a case in the statistics,there is a delay of 2 up to 3 weeks [7, 8]. The time between infection and onset of symptomsis the incubation period which man cannot control. It is between 2 and 14 days, with a meanof about 5 days [17]. The other part of the delay includes patient’s hesitation and visit to the ∗ Institute of Mathematics, University of Greifswald, 17487 Greifswald, Germany, [email protected] weekly confirmed cases per 10000 people BrazilIranRussiaTurkeyUS
25 Mar 1 Apr 8 Apr 15 Apr 22 Apr 29 Apr 6 May 13 May20 May0.511.52
Observed reproduction rate of Covid-19
BrazilIranRussiaTurkeyUS
Figure 1: Weekly cases per capita (left) and reproduction numbers (right) for the United States,Russia, Brazil, Iran and Turkey from March 23 up to May 24, 2020.doctor, testing, laboratory work, and report of the case from local officials up to the government.This time usually includes one or two weekends.At the lockdown day the delay principle implies a bad prediction: we remain in stage 1 withincreasing case numbers for 2 or 3 weeks. Even if the lockdown immediately stops all infections,it cannot avoid the infections which took place before. That is a painful long waiting time forpeople, politicians and the media who expect immediate relief. Stronger measures are oftenimplemented to demonstrate activity against the pandemic.After successful containment, and 2-3 weeks waiting time, the apex of infections is reachedand a second stage of decline of infections starts. For Iran and Turkey in Figure 1, the initialdecline was almost as fast as the previous increase, for the USA it started slowly. Small regionshave a good chance to get straight to a low level. Large countries often slow down on intermediatelevel so that lockdown measures are extended. Due to the delay principle, lifting of containmentmeasures can influence the statistics only 2-3 weeks later. So there is a good prediction for atleast 2 weeks.In a third stage, the goal is to keep the level of infections low and avoid new waves. In Figure1, Iran is in the third phase, opposing a second wave. Asian countries like Vietnam, Thailand,South Korea and Taiwan are on an extremely low level of infections. European countries andthe USA can hardly reach that level without severe economic losses. Some Eastern Europeancountries, Norway and Austria presently have the best prospects.
The difference between infections and confirmed cases.
Most infections by Covid-19 are caused by people who have little or no symptoms, or no symptoms yet. The number N t of observed new positive cases at day t is smaller than the unknown number I t of new infectedpersons for day t : I t = c · N t for some constant c > . (2)The parameter c is unknown and hard to estimate. According to [10] it can be anywherebetween 3 and 300. Note that c differs from country to country. In the USA a lot of testing wasperformed, so N t includes many mild cases. Certainly c is much higher for Brazil where only2evere cases were tested. Thus the level of infection in Figure 1, taken as weekly incidence ofconfirmed cases, is not comparable between countries without further adjustment. The shapeof curves is comparable, however, and indicates the stage of the epidemic.On the right of Figure 1 there is a function which does not depend on the intensity of testingand allows comparison of different regions. The concept of reproduction rate.
The reproduction rate R is a central concept inepidemiology [4, 6, 2, 12, 11]. It is ‘the average number’ of people who are infected by one sickperson. If R > , the number of infections will increase exponentially. If R < , the number ofinfections will decline. In Figure 1, Brazil and Iran have reproduction numbers above 1. Theircase numbers are continuing to rise in the coming days. Russia just managed to get below1, which means that case numbers are going to decrease further. Turkey and the USA havesubcritical reproduction numbers near to one, so it is not clear how long the decline of casenumbers will continue.Case numbers and reproduction rates in Figure 1 must be studied together. Supercriticalreproduction on high level of infections will lead to disaster while on low level of infections itcan be tolerated for some time. Subcritical reproduction on high infection level can lead toimmediate relief while on low level it will not be felt so much. On the whole, reproduction rateis a good concept for large infection numbers. When new infections can be counted with fingers, R becomes obsolete.Note that reproduction rate, like life expectancy, is an abstract notion and not a statisticalaverage. It is calculated from a variety of models [1, 4, 12, 14], often with differing results. Mostmodels require estimation of model parameters. The value of R then comes with a confidenceinterval indicating its accuracy. The basic model assumptions are taken for granted, however.Our approach is non-parametric, with few assumptions explained below. We claim that theassumptions fit the properties of Covid-19 and of the given data extremely well. The infection potential.
The following two sections are theoretic. We derive a formulafor R t , where t now denotes symptom time - the day of onset of symptoms. Now N t will denotethe number of sick persons with symptoms day t which will later be confirmed by a test andenter the statistics. Our main assumption is that a sick person remains infectious for 7 days.
More specifically,these are the two days before noting symptoms and four days after. For asymptomatic cases thereshould also be a ‘symptom day’ which separates the infectious period into two and four days.Let I t be the unknown number of all symptomatic and asymptomatic people with ‘symptomday’ t. We assume that equation (2) holds for this definition of N t and I t . We assume that c canbe considered as a constant for a period of 10 days.Now we consider the infection potential at time t, that is, all asymptomatic or symptomaticpersons which are infectious at day t. This includes I t and the infectious people with symptomday 1 or 2 days after t or up to 4 days before t. So the infection potential at time t is I t − + I t − + ... + I t +2 = c · ( N t − + N t − + ... + N t +2 ) = c · W t +2 . (3) The infection potential at day t is just the weekly number of new infections W t +2 two days later,multiplied by the unknown constant c which changes slowly with time. This means that the3hape of curves in Figure 1 reflects the course of the epidemic, including asymptomatic cases.The level must be corrected by a factor for each country.We did neglect variations in the period and degree of infectivity of a sick person. Our aimwas to get a simple formula. Moreover, weekly sums are already in use and have the advantageof reducing weekly periods in the numbers of daily new infections [15, 8, 3].
Daily and total reproduction number.
Once we defined the infection potential, wecan study its performance. We assume that the incubation time of Covid-19 is exactly 5 days[17]. That is, the infection potential for infections at symptom day t must be taken at t − . Using equations (2) and (3), the daily reproduction number for infections at symptoms day t is r t = infections at day t infection potential at day t − I t cW t − = N t W t − . (4)The essential point is that the unknown constant c cancels out. Moreover, since the infectionpotential is a weekly sum of infection numbers I t , the total reproduction rate of the infectionpotential at time t is a sum of seven successive daily reproduction rates: R t = r t + r t − + ... + r t − . (5)This is written as a backward sum since it will be calculated daily for the most recent timepoint t. This is the ‘observed reproduction rate’ at observation time t. It refers to the infectionswhich took place 2-3 weeks ago. For Germany, we would date it back by 17 days - 14 daysdelay between infection and observation plus 3 days because r t − is the middle term of (5).However, an exact estimate of this delay is only necessary when we investigate the effect ofcertain lockdown measures. The main purpose of a reproduction rate is the study of the presentand the prediction of observations of the next few days. The smooth reproduction rate R ∗ . Note that (5) is again a moving sum with a smooth-ing effect. With length 7, it will further reduce weekly periods in the data. Nevertheless, thegraph of R t often has plenty of little corners in case of irregularly collected data N t . For thisreason, we use a smooth version of daily reproduction rate. In the formula r t = N t /W t − wereplace the numerator by ( N t + N t − ) and the denominator by ( W t − + W t − + W t − ) . Thisaccounts for the fact that the incubation period is not constant, and not all cases have beentested on their symptoms day. The resulting daily and total reproduction rates are r ∗ t = 3( N t + N t − )2( W t − + W t − + W t − ) and R ∗ t = r ∗ t + r ∗ t − + ... + r ∗ t − . (6)This smooth version was used for Figure 1. As we see below, there is little difference between R and R ∗ . Since the most recent value N t is used only with factor , we lose half a day in actuality,but we gain a lot of smoothness. We tried a number of similar smooth versions, and found that R ∗ is a good choice although random variations in incubation period, period of being infectious,day of testing minus symptoms day etc. are larger than expressed by (6). A longer moving sumin numerator or denominator would mean further loss of actuality, however.4 omparison with established methods. Let us compare R and R ∗ with the reproduc-tion numbers used by health authorities in Austria, Germany, and Sweden. Figure 2 shows theweekly incidences for the three countries. The level of Germany and Austria is comparable.Sweden did test only severe cases and consequently has a higher incidence. Sweden has been inthe center of public interest since the pandemic is managed without hard lockdown. Because ofthe multiplicative nature of reproduction, incidences are presented on logarithmic scale.
25 Mar 1 Apr 8 Apr 15 Apr 22 Apr 29 Apr 6 May 13 May 20 May0.30.5124 weekly confirmed cases per 10000 people
AustriaGermanySweden
Observed reproduction rate of Covid-19 in Austria
AGES R R * Observed reproduction rate of Covid-19 in Germany
RKI R R * Observed reproduction rate of Covid-19 in Sweden
FHMH R R * Figure 2: Weekly incidences for Austria, Germany, and Sweden, March 25 to May 24, 2020, andofficial reproduction functions for each of the countries, compared with our R and R ∗ . The health authority AGES in Austria calculates reproduction rates by a method of Coriet al. [4] which beside case numbers requires the distribution of serial intervals - time distancesbetween a positive case and a person infected by that case. They estimated the distribution usingtheir contact tracing work. Details are found in [14], reproduction numbers can be downloadedfrom the website. The method works for small regions, even for very few cases. All cases aredated to the time of the first positive laboratory test. The reproduction function is slowlyvarying. The functions R t and R ∗ t show more variation, indicate the changes between R >
R < R is calcu-lated as a quotient of two weekly sums with difference of four days, which is assumed to be the5erial interval. This method is a bit unstable and resulted in values R ≥ R and R ∗ . In Sweden, the health authority uses the method of [4] as in Austria, but with a serial intervalestimation of Nishiura et al. [13] who found a mean value of 4.8. Their reproduction functionwent below 1 before April 10 and stayed below one. Compared with the weekly averages in Figure2 this seems too optimistic. Our R and R ∗ are clearly larger in March, and then fluctuate aroundone. We think this describes the development of Swedish case numbers more accurately.Our simple reproduction rates perform well also for American data, where a Bayesianmethodology was used to compute R on rt.live , and for Brazil, where Figure 1 confirmsthe predictions of [12]. There is always little difference between R and R ∗ . The smooth versionlooks nicer and seems more convenient for prediction.
References [1] Matthias an der Heiden and Osamah Hamouda. Sch¨atzung der aktuellen Entwicklung derSARS-CoV-2- Epidemie in Deutschland? Nowcasting.
Epidemiologisches Bulletin , 17, 2020.[2] Jeffrey K Aronson, Jon Brassey, and Kamal R Mahtani. “When will it be over?”: Anintroduction to viral reproduction numbers, R and R e . ,retrieved 25 May 2020.[3] Christoph Bandt. Transparent Covid-19 prediction. http://arxiv.org/abs/2004.04732 .[4] A. Cori, N.M. Ferguson, C. Fraser, and S. Cauchemez. A new framework and soft-ware to estimate time-varying reproduction numbers during epidemics. American Jour-nal of Epidemiology , 17(19):1505–1512, 2013. https://doi.org/10.1093/aje/kwt133 , https://CRAN.R-project.org/package=EpiEstim .[5] Neil M Ferguson, Daniel Laydon, and Gemma Nedjati-Gilani et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare de-mand. Imperial College London (16-03-2020) .[6] Luca Ferretti, Chris Wymant, and Michelle Kendall et al. Quantifying SARS-CoV-2 trans-mission suggests epidemic control with digital contact tracing.
Science , 368:eabb6936, 2020.[7] European Center for Disease Prevention and Control. Covid-19. , retrieved 25 May 2020.[8] Robert Koch Institut. Daily situation reports. ,2020.[9] Robert Koch Institut. Nowcasting und R-Sch¨atzung. , retrieved 25 May 2020.610] John P.A. Ioannidis. A fiasco in the making? as the coronavirus pandemic takes hold, weare making decisions without reliable data. .[11] Ying Liu, Albert A Gayle, Annelies Wilder-Smith, and Joacim Rockl¨ov. The reproductivenumber of COVID-19 is higher compared to SARS coronavirus.
J. Travel Medicine , 27(2),2020. https://academic.oup.com/jtm/article/27/2/taaa021/5735319 .[12] Thomas A Mellan, Henrique H Hoeltgebaum, and Swapnil Mishra et al. Estimating thenumber of infections and the impact of non-pharmaceutical interventions on COVID-19 in11 european countries.
Imperial College London (30-03-2020) .[13] H. Nishiura, N.M. Linton, and A.R. Akhmetzhanov. Serial interval for novel coronavirus(COVID-19) infections.
International Journal of Infectious Diseases , 93:284–286, 2020.[14] Lukas Richter, Daniela Schmid, and Ernst Stadlober. Methodenbeschreibung frdie Schtzung von epidemiologischen Parametern des COVID19 Ausbruchs, ¨osterreich. .[15] Max Roser, Hannah Ritchie, Esteban Ortiz-Ospina, and Joe Hasell. Coronavirus pandemic(covid-19). https://ourworldindata.org/coronavirus , retrieved 25 May 2020.[16] Johns Hopkins University. Coronavirus Covid-19 globalcases by the center for systems science and engineering. https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html , data on https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series ,update of 25 May 2020.[17] Worldometer. Coronavirus incubation period.