Website https://valeriupredoi.github.io/

Table of Contents

• Model
• Results
• Conclusions
• Resources
• Software
• Data

Model

Linear Fit

We assume the following model for the cumulative number of reported cases and deaths: at any given time t (measured in days), the cumulative number of cases will be:

$$N(t) = N0exp(bt)$$

where N0 is an initial number of cases, rate b is the growth rate (in units 1/day or day-1), and exp is the exponential number = 2.72. Similarily, deaths have the distribution:

D(t) = D0exp(mt)

where m is the growth rate for deaths and D0 is an initial number.

Doubling times and daily increments

Line-fitting ln(N) = f(t) and ln(D) = f(t) will give us rates b and m, and will allow us to estimate the doubling times for reported cases and deaths:

double_time_cases = ln(2)/b

and

double_time_deaths = ln(2)/m

With these, we can rewrite the evolution of both reported number of cases and reported deaths:

N(days) = N0 x 2^(days/double_time_cases)

and

D(days) = D0 x 2^(days/double_time_deaths)

where days is the number of days the evolution is measured on; the larger double_time_cases and double_time_deaths the smaller N(t) and D(t) are and for large enough double_time_cases >> days and double_time_deaths >> days, the exponential evolution can be approximated to:

N(days) = N0 x (1 + days/double_time_cases)

and

D(days) = D0 x (1 + days/double_time_deaths)

that is a linear evolution with time, which much slower than the exponential one.

We can also estimate a local, daily basic reproductive number: for a daily evolution, starting from N cases the previous day, the total number of cases would be

dN + N = Nexp(b)

where dN is the daily increase in cases, so the relative variation in number of cases in one day will be:

dN/N = exp(b) - 1

this is a rough estimate of a daily basic reproductive number, which is < 1, but over an infectious period of X days (I assume 10 in this case) will be > 1. It somewhat resembles the definition of R0 as a basic reproductive number since it measures how fast the number of cases changes over the infectious period, but it is not a rigorous computation of R0. We can rewrite the formula for the daily increase as

b = ln(1 + P)

where P = dN/N represents the daily measured relative increase in reported number of cases; this is in fact a function of time b(t) = ln(1 + P(t)) since both b and P will decrease over time (see next section).

We can now express the basic reproductive number R in terms of reported cases doubling time:

R = exp(ln2/double_time_cases) - 1

Time evolution of rates

Growth rates b and m do not stay constant over longer (>10-12 days) periods of time, it is noticed they are gradually decreasing with time (observed decrease by 50% every 10 days). Therefore, the linear fits are not performed on all data points from the start of the epidemic, but are, in fact, performed on subsets of data points to maximize the quality of fit. This generates a set of rates b and m over time.

We perform two types of fits: one for all available data and another for the last 5 days. If the fit for the last 5 days is better than the overall data fit (compare coefficient of determination R) and if the coefficient of determination for the last 5 days fit is > 0.98 then we chose the linear parameters for the last 5 days fir (most of the cases show that the last 5 days fit is generally better; 0.98 means a very high quality fit).

Estimating the actual number of cases

The reported number of cases is unreliable data, so it is desired to estimate the actual number of existing cases at any given day (the analysis performed daily, so the actual number of cases is given at the today day).

For this purpose we use the deaths data D(t) and its growth rate m and a set of plausible mortality fractions of the virus M = [0.5, 1, 2, 3 and 4]%: we construct the plausible actual number of cases

C(t) = D(t) x 1/M

and shift C(t) in time by a fixed delay of 14 days (assumed as average duration between time of infection and time of death) and compute the number of cases on the day when the deaths reported today were cases right after infection C(t - 14). To get the number of actual cases of today, C(t - 14) needs to be extrapolated 14 days (14: as of April 9, previous results with 20 days have been redone) later (today): for that we use the rate of growth for deaths, m, and construct a function f(m) to best represent the evolution of m = m(t) for the next 14 days. Note that we are not using the rate of growth for reported cases b since we consider it to be unreliable. The reason why we need to construct f(m) is to best represent the growth rate of the actual cases, which we assume to be close to the evolution over the next 14 or so days of m (case evolution is well mirrored by the deaths evolution after 14+ days).

A lot of the deaths rates m have been noticed to drop to half after ~10 days, so we chose f(m) = m/2 and thresholding m at 0.05 and apply the 0.5 factor only for m > 0.05 since those rates < 0.05 are very stable and have not been observed to change over longer periods of time. This way we can estimate the actual number of cases today as:

C(today) = 1/M x D(today) x exp(m x delay=14)

for m < 0.05 and

C(today) = 1/M x D(today) x exp(0.5 x m x delay=14)

for m >= 0.05.

C(today) is a function of observables D(today) and two free parameters M and delay and may have the same value for two different combinations of (M, delay) but for the realistic set of M = [0.5, 1, 2, 3 and 4]% only delays of >= 14 days are a realistic parameter value, that excludes unrealistic delays of < 2 days.

Further we compute country current cases population percentages as C(today)/POP and testing percentages as Reported cases/C(today).

Results

Visualization

Maps are courtesy of datawrapper and use data from this daily updated tables. Map source can be found here

• Reported cases doubling time Interactive Map
• Reported deaths doubling time Interactive Map
• Actual infections percentages of country populations Interactive Map
• Estimated underreporting: Interactive Map

Rolling averages

We plot the 7-day rolling averages for absolute numbers of daily increases for deaths and the same but country population weighted (in units of 1000 inhabitants).

7-day rolling window averages for daily increases in numbers of deaths (absolute numbers)

7-day rolling window averages for daily increases in numbers of deaths (weighted by country population in units of 1000 inhabitants; note that Belgium's numbers are very high due to their reporting all deaths, see article)

Upper bound on mortality: Israel

Israel is the only country that has solid testing results (16,000+ TpM as of April 10) and has sufficiently good data for us to place an upper limit on the mortality mortality there: M=2%

Actual and simulated cases in Israel	Doubling times in Israel

Representative countries

We compare results obtained from simulating the actual number of cases for different countries giving percentages of infect out of total country population for different mortality fraction scenarios (0.5, 1 and 2% respecitively). Comparison is made with the Imperial College London report from 30/03/2020, different in-country studies and a study from Georg-August University, Germany.

Comparison labels:

• Numbers labelled IC for 30-03-2020 from the Imperial College London report.
• Numbers labelled GA for 31-03-2020 from the Georg August University, Germany study.

There is very good overlap with these results suggesting a consistent mortality fraction per country of M=0.5-1%.

Tables header:

• Date: date for data
• C: number of reported cases
• D: number of reported deaths
• DayR C %: daily increase rate for reported cases in percentage
• DayR D %: daily increase rate for reported deaths in percentage
• DoubT C (d): doubling time for entire reported cases population in days
• DoubT D (d): doubling time for entire reported deaths population in days
• %pop M=0.5%: percentage of country population infected for mortality M=0.5%
• %pop M=1%: percentage of country population infected for mortality M=1%
• %pop M=2%: percentage of country population infected for mortality M=2%
• Test% M=0.5%: country testing efficiency in percent for mortality M=0.5%
• Test% M=1%: country testing efficiency in percent for mortality M=1%
• Test% M=2%: country testing efficiency in percent for mortality M=2%
• Square brackets for 31 March data: values obtained for a 10-day and 20-day delay (equivalent to a 95% confidence interval) for each mortality M.

Key parameters:

• Parameter space for simulations: 14 day delay between deaths and simulated cases and projected rate 0.5 x current death rate or current death rate if current death rate < 0.05 (or 5%)
• For 31 March data we give in [ ] the values obtained for a 10-day and 20-day delay

Spain

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
12/03	2277	54	40	50	1.8	1.4	0.79	0.39	0.20	0.6	1.2	2.5
21/02	25374	1375	19	24	3.6	2.9	3.18	1.59	0.80	1.7	3.4	6.8
31/03	95923	8464	9	13	7.4	5.5	8.71[6.8-12.7]	4.36[3.4-6.4]	2.18[1.7-3.2]	2.4	4.7	9.4
11/04	158273	16081	4	5	18.7	14.8	13.28	6.64	3.32	2.6	5.1	10.2

• Population percentage %pop IC: 15% ci=[3.7%-41%] implies underestimation vs IC, probably M=0.5[-1]% best overlap.
• Population percentage %pop GA: 12.24% implies slight underestimation vs GA, probably M=0.5% for longer delays 14-20 days.

Israel

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
31/03	5358	20	23	34	3.0	2.0	0.53	0.27	0.13	12.1	24.1	48.2
11/04	10408	95	6	13	12.5	5.3	0.56	0.28	0.14	22.0	44.0	88.1

Austria

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
31/03	10180	128	7	20	9.6	3.4	1.21[0.8-2.2]	0.60[0.4-1.1]	0.30[0.2-0.6]	9.5	19.0	38.0
11/04	13555	319	2	9	28.7	7.8	1.34	0.67	0.33	11.4	22.8	45.5

• Population percentage %pop IC: 1.1% ci=[0.36%-3.1%] implies M=0.5[-1]%.
• Population percentage %pop GA: 0.94% implies M=0.5% and longer delays ~16 days.
• Austrian study suggests 0.32% of total population infected around April 4-5, meaning exactly M=2%

France

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
12/03	2281	48	29	31	2.4	2.3	0.13	0.06	0.03	2.8	5.5	11.0
21/03	14282	562	16	27	4.4	2.6	1.16	0.58	0.29	1.9	3.8	7.6
31/03	52827	3532	11	14	6.4	4.9	2.91[2.2-4.5]	1.46[1.1-2.2]	0.73[0.6-1.1]	2.8	5.6	11.2
11/04	125931	13215	9	10	7.5	7.3	7.92	3.96	1.98	2.4	4.9	9.8

• Population percentage %pop IC: 3% ci=[1.1%-7.4%] implies M=0.5[-1]%.
• Population percentage %pop GA: 3.09% implies M=0.5% and longer delays ~16 days.

Germany

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
21/03	22213	84	26	33	2.7	2.1	0.21	0.10	0.05	12.9	25.8	51.6
31/03	71808	775	8	20	8.3	3.4	0.77[0.5-1.4]	0.39[0.3-0.7]	0.19[0.1-0.4]	11.2	22.3	44.6
11/04	122171	2767	4	12	16.2	5.7	1.56	0.78	0.39	9.4	18.8	37.6

• Population percentage %pop IC: 0.72% ci=[0.28%-1.8%] implies M=0.5[-1]%.
• Population percentage %pop GA: 0.55% implies M=0.5% for shorter dealys or M=1% for longer delays.
• Heinsberg study suggests a mortality M=0.37%, in line with my estimate and IC (if M=0.5%)

Denmark

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
21/03	1326	13	8	33	8.9	2.1	0.45	0.22	0.11	5.1	10.3	20.5
31/03	3039	90	8	26	8.7	2.7	1.90[1.1-4.1]	0.95[0.6-2]	0.48[0.3-1]	2.8	5.6	11.1
11/04	6014	247	7	7	10.1	9.7	1.41	0.71	0.35	7.4	14.8	29.6

• Population percentage %pop IC: 1.1% ci=[0.40%-3.1%] implies M=1[-2]%.
• Population percentage %pop GA: 0.41% implies M=2%.

Italy

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
12/03	12462	827	20	31	3.4	2.2	2.40	1.20	0.60	0.9	1.7	3.4
21/03	53578	4825	13	16	5.2	4.3	4.93	2.47	1.23	1.8	3.6	7.2
31/03	105792	12428	5	8	13.9	9.1	6.99[6-8.8]	3.49[3-4.4]	1.75[1.5-2.2]	2.5	5.0	10.0
11/04	147577	18849	3	3	25.4	21.1	9.85	4.92	2.46	2.5	4.9	9.9

• Population percentage %pop IC: 9.8% ci=[3.2%-26%] implies underestimation vs IC and M=0.5[-1]%.
• Population percentage %pop GA: 5% implies M=0.5% or understimated M=1%.

Switzerland

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
21/03	6575	75	23	29	3.0	2.4	1.30	0.65	0.33	5.9	11.8	23.7
31/03	16605	433	6	16	11.1	4.4	3.04[2.2-4.9]	1.52[1.1-2.4]	0.76[0.6-1.2]	6.4	12.8	25.7
11/04	24551	1002	3	7	19.9	10.1	3.79	1.90	0.95	7.6	15.2	30.4
17/04	26732	1281	1	4	55.0	18.3	5.11	2.55	1.28	6.1	12.3	24.6

• Population percentage %pop IC: 3.1% ci=[1.3%-7.6%] implies M=0.5[-1]%.
• Population percentage %pop GA: 2.72% implies M=0.5%.
• Wide antibody testing results published on April 25 quotes a total population infection of 5.5% which corresponds to M=0.5% from our model (5.1% population infected).

Norway

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
31/03	4641	39	5	18	13.2	3.9	0.50[0.4-0.9]	0.25[0.2-0.4]	0.13[0.1-0.2]	17.3	34.5	69.0
11/04	6314	113	3	11	24.9	6.5	0.90	0.45	0.22	13.2	26.3	52.7

• Population percentage %pop IC: 0.41% ci=[0.09%-1.2%] implies M=0.5[-1-2]%.
• Population percentage %pop GA: 0.23% implies M=1%.

Belgium

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
21/03	2815	67	21	48	3.4	1.5	3.30	1.65	0.83	0.7	1.5	3.0
31/03	12775	705	24	22	2.9	3.2	5.56[3.6-10.6]	2.78[1.8-5.3]	1.39[0.9-2.7]	2.0	4.0	8.0
11/04	26667	3019	6	14	11.3	5.0	13.90	6.95	3.48	1.7	3.3	6.7

• Population percentage %pop IC: 3.7%i ci=[1.3%-9.7%] implies M=0.5[-1-2]%.
• Population percentage %pop GA: 5.5% implies M=0.5%.

Sweden

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
21/03	1763	20	10	34	6.7	2.0	0.43	0.22	0.11	4.1	8.2	16.4
31/03	4435	180	9	27	7.8	2.5	2.45[1.4-5.6]	1.23[0.7-2.8]	0.61[0.4-1.4]	1.8	3.6	7.3
11/04	9685	870	8	14	9.1	5.0	4.64	2.32	1.16	2.1	4.2	8.4

• Population percentage %pop IC: 3.1% ci=[0.85%-8.4%] implies underestimation vs IC and M=0.5[-1]%.
• Population percentage %pop GA: 0.58% implies M=2%.

United Kingdom

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
31/03	25150	1789	14	20	5.1	3.4	2.22[1.5-4.1]	1.11[0.7-2.1]	0.55[0.4-1]	1.7	3.4	6.8
11/04	70272	8958	8	13	8.9	5.4	6.54	3.27	1.64	1.6	3.2	6.4

• Population percentage %pop IC: 2.7% ci=[1.2%-5.4%] implies M=0.5[-1]%.
• Population percentage %pop GA: 3.15% implies M=0.5% for longer delays ~17 days.

Romania

Date	C	D	DayR C %	DayR D %	DoubT C (d)	DoubT D (d)	%pop M=0.5%	%pop M=1%	%pop M=2%	Test% M=0.5%	Test% M=1%	Test% M=2%
31/03	2245	82	24	29	2.9	2.4	0.62[0.4-1.5]	0.31[0.2-0.7]	0.16[0.1-0.4]	1.8	3.7	7.4
11/04	5467	270	8	11	9.1	6.4	0.59	0.30	0.15	4.7	9.5	18.9

• Population percentage %pop GA: 0.48% implies M=0.5% for short ~10 days delays or M=1% for ~16day days delays.

Evolution of doubling time with time

Here we compute an R0 that is obtained from the doubling time and multiplied by 10 days (the average infectious period).

Date	Mean DoublT (days)	R0 (10 days)	Figure
16/03	3.1+/-1.9	3.3+/-1.4
27/03	3.4+/-2	2.9+/-1.4
31/03	4.2+/-2.4	2.6+/-1.5
05/04	8.9+/-4.3	1.0+/-0.4
12/04	12.6+/-7.3	0.7+/-0.3
Current	-	-

Conclusions

• Viral transmission is very high: R0 of up to 5-6 in mid-March (considering an infectious phase of 10 days, could be even higher if infectious phase is longer); definite decrease in transmission due to social distancing, of ~70% in the three weeks between 16/03/2020 and 05/04/2020
• Countries underreport cases in very high proportions; underreporting decreases with the increase of number of tests per million; the average reporting for the 13 representative countries we looked at is 13.6% (std.dev. 11.6%) for April 11, for a reference mortality M=1%; that means, on average, that 86% of actual infections may go unreported;
• Average mortality of 0.5-1% (possibly higher to 2% in certain countries, defintely not higher than 2%, considering reference countries we looked at).

Resources

Current repository resources:

• Country tables
• Current country plots
• March-2020 plots
• March-2020 situation report

Software

• Package
• Usage example: python cov_model/cov_lin_wrapper.py --countries COUNTRIES --regions REGIONS --month MONTH
• Command line args:
  • --countries: list of comma-sep strings or file (example: Italy,Germany)
  • --regions: list of comma-sep strings or file (example: California,Georgia)
  • --month: int (example: 3 (for March))
• Requirements:
• python2.7 or higher (ok with python3.x);
• Package xlrd available from PyPi via pip install xlrd;
• Package scipy: for an easy installation I recommend using miniconda/anaconda; for a pip installation on an older architecture and python2.7 you will have to install the lapack and blas libraries, a Fortran compiler and python-dev:
  • sudo apt-get install python-dev
  • sudo apt-get install gfortran
  • sudo apt-get install libblas3 liblapack3 liblapack-dev libblas-dev
  • sudo pip install scipy==0.16

Data

• UK Data: official data source
• Worldwide Data: Johns Hopkins University CSSE gitHub repository
• Testing efficiency numbers
• Heinsberg study
• Imperial College London, UK report
• Georg August University, Germany study
• Austrian large scale testing study