Confirmed and Deaths - cumulative and daily Cases (absolute date)

Case numbers are taken from (JHU 2020b). In order to compensate for the daily fluctuations, the mean number of cases over the past seven day (Mean_Daily_*) are added.

Confirmed and Deaths - cumulative and daily Cases per 100,000 Inhabitants (relative data)

Population numbers are taken from (UNO 2020). In order to compensate for the daily fluctuations, the mean number of cases over the past seven days (Mean_Daily_*) are added.

Confirmed and Deaths - cumulative and daily Cases per 100,000 Inhabitants (relative data)

Population numbers are taken from (UNO 2020). In order to compensate for the daily fluctuations, the mean number of cases over the past seven days (Mean_Daily_*) are added.

Case Fatality Rate - Proportion of deaths from confirmed cases

The number of confirmed cases is an early predictor of the number of deaths. The number of today’s deaths is already determined by the infections about by \(\sim19\) days ago or respectively by the confirmed cases about by \(\sim11\) days ago (see RWI 2020).

An average duration of Confirmed Infection to Death of \(12\) (lag-)days is assumed (country-independent for the sake of simplicity) for the calculations, since many tests are carried out in the meantime before symptoms appear.

However, this varies considerably depending on country-specific test rate and health system. In the worst health systems it may be only one day between infection confirmation and death, the “Confirmed” cases must be “lagged” by \(\sim1\) day. In the best case, the time from the end of incubation period (in average \(\sim5-6\) days) to death is an average \(\sim14\) days. In this case, the average Confirmed infection to Death period is \(\sim14\) days), the “Confirmed” cases must be correctly “lagged” by \(\sim14\) days. For the assumed time periods see (RKI 2020c), (RKI 2020b), for Case Fatality Rate and Incubation Period in general see (Wikipedia contributors 2020a), (Wikipedia contributors 2020c).

The simple calculation with unlagged cumulative confirmed cases divided by cumulative deaths results in a significant underestimation of the CFR in health systems with early disease detection. If the number of cumulative cases is already large compared to the number of active cases (~ cases from the past two weeks), the “lagged” rsp. “unlagged” values converge.

The Infection Fatality Rate (IFR) is the fatality rate of all infection, that means detected confirmed cases and undetected cases (asymptomatic and not tested group). This lethality is assumed to be country independent and only rough estimates exist (RKI: bottom of existing estimates \(\sim0.56\%\)).

Case Fatality Rate (in %) - for CFR_mean_daily (mean of past 7 days) and CFR_total assuming a mean time lag of 12 days between Confirmed => Death

Case Fatality Rate (in %) - selected countries

Confirmed and Deaths - cumulative and mean daily Cases, overall and per 100,000 Inhabitants (absolute and relative data)

Population numbers are taken from (UNO 2020). In order to compensate for daily fluctuations, the mean number of cases for the past seven days (Mean_Daily_*) is used instead of the daily cases.

Confirmed and Deaths - cumulative and mean daily Cases, overall and per 100,000 Inhabitants (absolute and relative data)

Population numbers are taken from (UNO 2020). In order to compensate for daily fluctuations, the mean number of cases for the past seven days (Mean_Daily_*) is used instead of the daily cases.

World

Germany Rolling Mean of daily Cases

Rolling Mean of daily Cases

The 7-days Rolling Mean/Moving Average of the Daily Confirmed and Death Cases smooths out the short-term weekly fluctuations (weekend).

The daily confirmed cases are related to the left y-axes, the daily death cases are related to the right y-axes. This clearly outlines the 12 days delay relation between daily confirmed and death cases and also the roughly the factor of ~1/25 (~4%).

Note: Age group dependent 7-Day Incidence plots for Germany** are largely provided weekly.

The age group specific incidence data RKI - COVID-19-Fälle nach Altersgruppe und Meldewoche - is provided weekly, every Tuesday evening.

Germany Calculated Reproduction Number

Calculated Reproduction Number

The calculation of the reproduction number \(R(t)\) uses a R function provided by (Thomas Hotz 2020b) on GitHub.

The (effective) reproduction number \(R(t)\) at day \(t\), i.e. the average number of people someone infected at time \(t\) would infect if conditions remained the same.

For further German federal states figures (based on the data provided by Robert Koch Institut) see (Thomas Hotz 2020a) and for worldwide figures (based on the data provided by Johns Hopkins University) see (Thomas Hotz 2020c).

For the calculation the assumption of 7-days reporting delay (confirmed is reported 7-days after ‘real’ infection) is unchanged and the same modelled infectivity profile w is used. The lower and upper confidence interval lines provide the (approximate, pointwise) 95% confidence interval (only based on statistical numbers, possible changes in e.g. counting measures can not be considered).

This is the reason why (Thomas Hotz 2020b) “do not compute an average over a sliding window of seven days so the viewer immediately recognizes the size of such artefacts, warning her to be overly confident in the results. In fact, these artefacts are much larger than the statistical uncertainty due to the stochastic nature of the epidemic which is reflected in the confidence intervals.”

Nevertheless, here the calculation is based on the 7-days rolling mean and therefore the figure smooths over the the weekly rhythm.

Exponential Growth Cumulated Cases

China and South Korea slowed down exponential growth significantly at an early stage. Their lines on log10 scale have had no longer a significant slope.

In early phases countries have a more or less unchecked exponential growth. If countermeasures are effective, reduced exponential growth is reflected in a reduced slope of the cumulative cases again.

Virus Spread - Cumulative Cases (absolute on log10 / relative on linear scale)

Exponential Growth Daily Cases

In early phases countries have a more or less unchecked exponential growth resulting in a significant curve slope for the daily cases.

Successful countermeasures are reflected in a reduced exponential growth, the slope of the curve decreases.

In the steady state, the slope disappears and if the daily cases decrease, the slope becomes negative.

Virus Spread - Daily Cases (absolute on log10 / relative on linear scale)

Estimation spread of the Coronavirus with Linear Regression of log data

Exponential Growth and Doubling Time \(T\)

Exponential growth over time can be fitted by linear regression if the logarithms of the case numbers is taken. Generally, exponential growth corresponds to linearly growth over time for the log (to any base) data (see also Wikipedia contributors 2020b).

The semi-logorithmic plot with base-10 log scale for the Y axis shows functions following an exponential law \(y(t) = y_0 * a^{t/\tau}\) as straight lines. The time constant \(\tau\) describes the time required for y to increase by one factor of \(a\).

If e.g. the confirmed or death cases are growing in \(t-days\) by a factor of \(10\) the doubling time \(T \widehat{=} \tau\) can be calculated with \(a \widehat{=} 2\) by

\(T[days] = \frac {t[days] * log_{10}(2)} {log_{10}(y(t))-log_{10}(y_0)}\)

\(log_{10}(y(t))-log_{10}(y_0) = = log_{10}(y(t))/y_0) = log_{10}(10*y_0/y_0) = 1\)

\(T[days] = t[days] * log_{10}(2) \approx t[days] * 0.30\).

For Spain, Italy, Germany we have had a doubling time of only \(T \approx 9days * 0.3 \approx 2.7 days\) at the beginning of the pandemic!!.

The doubling time \(T\) and the Forecast is calculated for following selected countries: Austria, France, Germany, Italy, India, Japan, Spain, EU, United Kingdom, United States of America and World in total (see Forecast / Doubling Time).

Germany - Trend with Forecast on a linear scale

14-day Forecast plot of expected Cases

The plot shows the daily cases forecast increase in case of unchecked exponential growth. The dark shaded regions show the 80% rsp. 95% prediction intervals. These prediction intervals are displaying the uncertainty in forecasts based on the linear regression of the logarithmic data over the past 14 days.

Comparison Exponential Growth (Daily rolling mean cases on log10 scale)

Germany - Cumulative cases with ~linear slope on a log10 scale

Comparison Exponential vs. Linear Growth

The charts compare the different forecasts for an exponential rsp. linear growth model. Due to the large fluctuations of the daily cases regression of three weeks is required. Otherwise the prediction levels are much too big.

The dark shaded regions are indicating the \(80\%\) rsp. \(95\%\) prediction intervals. These prediction intervals are displaying the “pure” statistical uncertainty in forecasts based on the regression models.

For doubling periods in the order of period of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty, see (RKI 2020b), we no longer have exponential growth. The “old” infected cases are at the end of the doubling period no longer infectious (active). This results in a constant infection rate with basic reproduction number \(R_t \sim 1\) or even \(<1\).

Note: for case numbers of German federal states see (RKI 2020a).

Comparison Exponential and Linear Growth

Forecast of Daily and Cumulative Cases and check of Accuracy

The forecast accuracy is checked by using the forecast method for the past three weeks before the past week (training data). Subsequent forecasting of the past week enables comparison with the real data of these days (test data).

The comparison is also an early indicator if the exponential growth is declining. However, possible changes in underreporting (in particular the proportion confirmed / actually infected) requires careful interpretation.

For doubling periods of the total cumulative cases in the order of infectivity (RKI assumption: \(\sim9-10\) days, with great uncertainty, (see RKI 2020b), we have no
exponential growth for the total cumulative cases. Since the “old” infected cases are no longer infectious after these periods and we then have a constant infection rate with basic reproduction number \(R_t \sim 1\).

Instead, we have “only” linear growth of the cumulative Confirmed Cases and the Daily Confirmed Cases remain more or less constant if \(R_t \sim 1\).

However, the basic reproduction number \(R_0 (\approx 3.3 - 3.8)\) (RKI 2020d) is a product of the average number of contacts of an infectious person per day, the probability of transmission upon contacts and the average number of days infected people are infectious. With the current uncertainty of the average duration of the infectivity duration, \(R_0\) can therefore be estimated from the doubling time only to a very limited extent. See also (CMMID 2020).

Germany - Forecast Mean Daily Cases based on regression of past 14 days

Germany - Forecast accuracy check of expected Mean Daily Cases for the past 7 days

Forecasting with lagged Predictors

The number of confirmed cases is an early predictor of the number of deaths. The number of today’s deaths is already determined by the infections about by \(\sim19\) days ago or respectively by the confirmed cases about by \(\sim12\) days ago see Bar Chart - CFR.

The country-specific case fatality rate (CFR, proportion of deaths from confirmed cases) and changes over time can be an indicator of different

• test rates
• infection penetration into vulnerable groups
• average age of the population
• quality/capacity of the health system.

Overall a rough conclusion on the country specific underreporting rate (lack of diagnostic confirmation; proportion of all infected to confirmed cases) is feasible if the infection fatality rate (IFR, confiremd cases plus all asymptomatic and undiagnosed infections) is assumed to be country independent and the IFR is known (bottom of existing estimates \(\sim0.56\%\), assumption by RKI see (RKI 2020b).

In this case an estimation of the CFR of \(0.06\) \((6\%)\) indicates an underreporting by a by a factor of \(\sim10\). A CFR of \(0.20\) \((20\%)\) indicates an underreporting by a by a factor of \(\sim30\). This corresponds to RKI assumption of a underreporting by a factor of \(11-20\) (RKI 2020c). Unfortunately, the IFR or lethality is still far too imprecise for concrete conlusions.

In the model paper RKI assumes for the

• Incubation period \(\sim5-6\) days - Day of infection day until symptoms are upcoming)
• Hospitalisation \(+4\) days - Admission to the hospital (if needed) after Incubation Period)
• Average period to death \(+11\) - if the patient dies, it takes an average of \(11\) days after admission to the hospital

Depending on the country-specific test frequency (late or early tests), the

*lag_days - time from receipt of the confirmed test result to death, Confirmed to Death, is about \(11-13\) days.

Note: these methods are also used for example for advertising campaigns. The campaign impact on sales will be some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months (see Hyndman and Athanasopoulos 2020).

Daily Confirmed and Death Cases

Case Fatality Rate mean daily (CFR in %, past 4 months)

Daily Deaths Forecast depending on lagged Daily Confirmed Cases (ARIMA model)

Exampla Germany - White Noise of Forecast Residuals

Forecast residuals
indicate quality of
Arima model fit: