Covid-19 Case Fitting Curves
Last update: 29/08/2020
1. Model Function
Based on data from Covid-19 cases from China, a universal fitting curve is presented. Only four parameters are required to be determined by a given Covid-19 case over a number of days. Under the special condition of reliable mass testing, this curve can indicate the flattening of a given case after its inflection point is reached.
The hyperbolic tangent function,
\begin{equation}
\label{eq:tgh}
Z_{N}(n)=A\tanh(an-b)+C,
\end{equation}
has two desirable features of any Covid-19 case data over the number of days $n$: it is flat at its endpoints and has one inflection point (at $n_{i}=b/a$. Parameter $A$ (or $C=Z(n_{i})$) gives the magnitude of the accumulated cases over $N$ days (amplitude is $2A$). Its steepness is tuned by the slope $a$ of the straight line $an-b$.
Figure 1 shows a typical case for the model function, in which \(A=4086.9\) is the magnitude and $0.170n-2.807$ (dashed line) is its steepness. The green dot is the inflection point ($n_{i}=16.48$).
its steepness (dashed line $0.170n-2.807$).
Parameters $\{A,a,b,C\}$ appearing in the model function were obtained inside Maple and Mathematica by a (local) non-linear fitting minimizing the root mean square (rms) deviation (both worksheet and notebook can be found here and here. All case data points are weighted equally. Case data from every country are available at Worldometer.
Root mean square (rms) deviations shown by inlets in figures belong to the fitted curve $Z_{N}$, where $N$ is the number of days in a given case data. Lower the rms, better the fitting. Residual is the difference between reported and predicted cases, using the last day fitted curve.
Nonetheless, it is important to say that despite the (weak) evidences presented below, any prediction here is not to be taken for granted. Also, this is a simple exercise of reconversion, since I am not an expert in the Statistics Science. See Refs.[1]-[4] for more accurate fitting schemes.