# Explanation

The doubling time is an important variable in assessing the spread of infection. As the name suggests, it indicates the period of time during which the number of infected persons doubles. In the upper graph we have shown three different ways to calculate the doubling time while the width of the time window is varied (see x-axis in the upper graph: Data used to fit until). Two of the calculation are derived from the models used (exponential and logistic). Another doubling time is calculated directly from the empirical (the observed) data. In the next paragraphs, the individual calculations are explained further.

Logistic Model: We use data from the beginning of the recording up to one day $$t_1$$ (this day lies between today and 11 days in the past). We fit the logistic model with the data and obtain the modelled numbers of infected persons. The number of infected persons on the day $$t_1$$ corresponds to the value $$y_1$$ in the left graph. From there, we determine the time interval to the halved case number $$y_0 = \frac{y_1}{2}$$ at time $$t_0$$ and define it as doubling time.

Empirical Data: The doubling time from the empirical data is calculated very similarly. Since the data are only recorded daily, we use an interpolation (smoothing) between two adjacent data points as shown in the graph on the right. Then, the case numbers at the considered date $$t_1$$ are determined and the time window (=doubling time) back to the time $$t_0$$ when the case numbers were halved is calculated ( $$t_1 - t_0$$ ).

Exponential Model: In the case of the exponential model, the doubling time can be calculated directly from the estimated model parameters. Here the equation under the question "For which time $$t$$ does the initial value of infected person $$a$$ double?" has to be solved. To do this, let us take a closer look at the exponential function: $$f(t) = a\,\exp(b\,t)$$ The number of infected persons at the beginning of the modeling period is defined by the parameter $$a$$, because: $$f(t = 0) = a\,\exp(0) = a.$$ The question formulated at the beginning can be expressed in a mathematical equation: $$a\,\exp(b\cdot t) \stackrel{!}{=} 2\cdot a.$$ This leads to the doubling time: $$t_{\mathrm{double}} = \frac{\ln(2)}{b}.$$ The parameter $$b$$ is called growth factor. The greater the growth factor, the shorter the doubling time.

Data is updated automatically as reported by Johns Hopkins University. Last update: 10/18/20.