Covid-19 Case Fitting Curves

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 ($n_{i}=16.48$).

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.

Figure 2 shows the case curves for the complete Covid-19 case data from China. The whole data seems to have a subseries described by the curve \(Z_{21}\) (the violet curve), obtained using the first 21 days, which indicates a (local) flattening of the cases over two weeks (around 53,700 cases). In spite of this curve $Z_{21}$ being a false lead in searching for the case flattening, since we know the whole history, it has all features of a typical (complete) case curve. In fact, a close analysis in this subseries shows an inflection point around $N=14$. Also, while the nearest case curves $Z_{N\leq 21}$ approach $Z_{21}$ in an oscillatory way, all case curves $Z_{N\geq 23}$ approach monotonically the global case curve $Z_{70}$ (the black curve in Figure 2), obtained using the first 70 days in the case data from China, which indicates a flattening around 81,000 cases. Therefore, the (black) curve $Z_{70}$ shown in Figure is the lower limit for all curves $Z_{N\geq 22}$ and has its inflection point around $N=21$. All curves $Z_{N\geq 29}$ converge monotonically and rapidly to $Z_{70}$, giving a 10 days prediction for the (global) flattening.  Best parameters for these two curves are shown in Table Par-China.

We have learned that the case curves $Z_{N}$ obtained after the inflection point converge monotonically, in an increasing speed, to the global case curve. Of course, mass testing is presumed. Figure rms-China shows the root mean square (rms) deviations of each case curve $Z_{N\geq 22}$, confirming the monotonic convergence.

Some questions. What did happen at day 22? Was it an increase in the total tests? Or is this jump in the number of cases from day 21 to the day 22 (see the solid green circles in Figure Cases-China) a typical feature of the global case curve around its inflection point (where the rate of change is maximum)?

South Korea has a case data with two series: (a) day 1 to 30 , \(1\leq N\leq 30\), fitted curve $Z_{a}$ (green line in Figure 3), and (b) day 31 and after ($N\geq 31$), fitted curve $Z_{b}$ (blue line in Figure 3). These two series are quite different from the Chinese two subseries, where there is a global fitting curve (see Figure 2). As we can see in Figure 3, the thin curve ($Z_{79}$) does not fit properly the whole data. A second wave of Covid-19 cases is in progress in South Korea (curve $Z_{b}$ in Figure 3). Fortunately, this second wave is flattening. The inlet in Figure 3) shows the root mean square (rms) deviations for the last $N$ days from the curve with the full data (upper grey dots) and the curve $Z_{b}$ from the second wave (lower blue dots).

An very important lesson can be learned here: a second wave of infections can come quickly.

New Zealand seems to have reached a global flattening, as we can see in Figure 4. Its case data has one jump at day 31 to 32 and its root mean square deviations are getting smaller (see inlet in Figure 4). There are repeated data resembling jumps at days 19-22 and 25-28, but probably they are not.

There is a growing oscillation in the case data around the last day fitted curve which can leads to a false flattening, like the case data from South Korea. This growing oscillation can be also the indicative of a second wave in course.

The case data from Spain seems to have three harmonics (or waves). The last wave $Z_{c}$ (blue line) seems to be reaching its flattening. The root mean square (rms) deviations in the inlet were calculated by the fitted curve \(Z_{71}\) using the whole case data.

The case data from Italy seems to have three harmonics (or waves). The last wave $Z_{c}$ (blue line) is still growing. The root mean square (rms) deviations in the inlet were calculated by the fitted curve \(Z_{78}\) using the whole case data.

The case data from Iran seems to have four harmonics (or waves). The last wave $Z_{d}$ (blue line) is still growing. The root mean square (rms) deviations in the inlet were calculated by the fitted curve \(Z_{76}\) using the whole case data.

The case data from Australia seems to have four harmonics (or waves). The last wave can keep growing. The root mean square (rms) deviations in the inlet were calculated by the fitted curve \(Z_{78}\) using the whole case data.

South Korea has a case data with two series: (a) day 1 to 30 (\(1\leq N\leq 30\)), fitted curve $Z_{a}$ (green line in Figure Cases-South Korea), and (b) day 31 and after ($N\geq 31$), fitted curve $Z_{b}$ (blue line in Figure Cases-South Korea). These two series are quite different from the Chinese two subseries, where there is a global fitting curve (see Figure Cases-China). As we can see in Figure Cases-South Korea, the thin red curve ($Z_{73}$) does not fit properly the whole data. A second wave of Covid-19 cases is in progress in South Korea (blue line $Z_{b}$ in Figure Cases-South Korea). Fortunately, this second wave is flattening. An important lesson can be learned here: a second wave of case infections can come very fast.

Parameters \(\{A,a,b,C\}\) appearing in the model function were obtained inside Maple and Mathematica by a (local) non-linear fitting minimizing the least-squares error (both worksheet and notebook can be found here and here). An update pdf version can be found here. All case data points are weighted equally. Case data from every country are available at Worldometer (www.worldometers.info),  Coronavirus (www.worldometers.info/coronavirus).

[1] S. Maltezos, Parametrization Model Motivated from Physical Processes for Studying the Spread of COVID-19 Epidemic. arXiv:2004.05992v2 [physics.soc-ph] (2020).

[2] P. Girardi et al., Robust inference for nonlinear regression models from the Tsallis score: application to Covid-19 contagion in Italy. arXiv:2004.03187v2 [stat.AP] (2020).

[3] M. A. V. Arias, Using generalized logistics regression to forecast population infected by Covid-19. arXiv:2004.02406v1 [q-bio.PE] (2020).

[4] J. Kumar and K. P. S. S. Hembram, Epidemiological study of novel coronavirus (COVID-19). arXiv:2003.11376v1 [q-bio.PE] (2020).

\(

\begin{thebibliography}{10}
\bibitem{Maltezos}
  S. Maltezos,
  \newblock{Parametrization Model Motivated from Physical Processes
    for Studying the Spread of COVID-19 Epidemic}.
  \newblock \href{https://arxiv.org/abs/2004.05992}{arXiv:2004.05992v2
    [physics.soc-ph]} (2020).
 
\bibitem{Girardi}
  P. Girardi \textit{et al.},
  \newblock{Robust inference for nonlinear regression models from the
    Tsallis score: application to Covid-19 contagion in Italy}.
  \newblock \href{https://arxiv.org/abs/2004.03187}{arXiv:2004.03187v2
    [stat.AP]} (2020).

\bibitem{Arias}
  M. A. V. Arias,
  \newblock{Using generalized logistics regression to forecast
    population infected by Covid-19}.
  \newblock \href{https://arxiv.org/abs/2004.02406}{arXiv:2004.02406v1
    [q-bio.PE]} (2020).
 
\bibitem{Kumar}
  J. Kumar and K. P. S. S. Hembram,
  \newblock{Epidemiological study of novel coronavirus (COVID-19)}.
  \newblock \href{https://arxiv.org/abs/2003.11376}{arXiv:2003.11376v1
    [q-bio.PE]} (2020).
\end{thebibliography}
\)