Alternative graphical displays for the monitoring of epidemic outbreaks, with application to COVID-19 mortality

Abstract

Background: Classic epidemic curves - counts of daily events or cumulative events over time -emphasise temporal changes in the growth or size of epidemic outbreaks. Like any graph, these curves have limitations: they are impractical for comparisons of large and small outbreaks or of asynchronous outbreaks, and they do not display the relative growth rate of the epidemic. Our aim was to propose two additional graphical displays for the monitoring of epidemic outbreaks that overcome these limitations.

Methods: The first graph shows the growth of the epidemic as a function of its size; specifically, the logarithm of new cases on a given day, N(t), is plotted against the logarithm of cumulative cases C(t). Logarithm transformations facilitate comparisons of outbreaks of different sizes, and the lack of a time scale overcomes the need to establish a starting time for each outbreak. Notably, on this graph, exponential growth corresponds to a straight line with a slope equal to one. The second graph represents the logarithm of the relative rate of growth of the epidemic over time; specifically, log₁₀(N(t)/C(t-1)) is plotted against time (t) since the 25th event. We applied these methods to daily death counts attributed to COVID-19 in selected countries, reported up to June 5, 2020.

Results: In most countries, the log(N) over log(C) plots showed initially a near-linear increase in COVID-19 deaths, followed by a sharp downturn. They enabled comparisons of small and large outbreaks (e.g., Switzerland vs UK), and identified outbreaks that were still growing at near-exponential rates (e.g., Brazil or India). The plots of log₁₀(N(t)/C(t-1)) over time showed a near-linear decrease (on a log scale) of the relative growth rate of most COVID-19 epidemics, and identified countries in which this decrease failed to set in in the early weeks (e.g., USA) or abated late in the outbreak (e.g., Portugal or Russia).

Conclusions: The plot of log(N) over log(C) displays simultaneously the growth and size of an epidemic, and allows easy identification of exponential growth. The plot of the logarithm of the relative growth rate over time highlights an essential parameter of epidemic outbreaks.

Keywords: COVID-19; Epidemic curve; Growth rate.

Fig. 1

Representations of the accrual of deaths from the SIR model: daily deaths over time (upper panel), log of daily deaths versus log of total deaths (middle panel; the dotted line shows the slope expected with exponential growth), and log of relative growth rate over time (lower panel)

Fig. 2

Logarithm of daily number of deaths attributed to COVID-19 versus logarithm of cumulative number of deaths in 11 European countries, as of June 5, 2020 (from top right down): United Kingdom (teal), Italy (light pink), France (mustard), Spain (orange), Belgium (light blue), Germany (dark green), Russia (navy blue), Netherlands (purple), Sweden (dark pink), Switzerland (red), Portugal (light green). Smoothed lines were obtained by non-parametric regression. Dotted line is the identity function, parallel to exponential growth

Fig. 3

Logarithm of daily number of deaths attributed to COVID-19 versus logarithm of cumulative number of deaths in 11 non-European countries, as of June 5, 2020 (from top right down): USA (navy blue), Brazil (light blue), Mexico (light pink), India (mustard), Canada (red), Iran (teal), Peru (purple), Turkey (light green), China (dark green), Egypt (orange), South Africa (dark pink). Smoothed lines were obtained by non-parametric regression. Dotted line is the identity function, parallel to exponential growth

Fig. 4

Logarithm of relative growth rate of deaths attributed to COVID-19 over time in 11 European countries, as of June 5, 2020 (from top right down): Russia (navy blue), Italy (light pink), Sweden (dark pink), Portugal (light green), United Kingdom (teal), France (mustard), Spain (orange), Germany (dark green), Netherlands (purple), Belgium (light blue), Switzerland (red). Smoothed lines were obtained by non-parametric regression

Fig. 5

Logarithm of relative growth rate of deaths attributed to COVID-19 over time in 11 non-European countries, as of June 5, 2020 (from top right down): Brazil (light blue), South Africa (dark pink), Mexico (light pink), India (mustard), Peru (purple), Egypt (orange), USA (navy blue), Canada (red), Turkey (light green), Iran (teal), China (dark green). Smoothed lines were obtained by non-parametric regression