Each country's response will be different, but I expect the data from democracies should be more transparent. The interesting thing about cumulative data is that to some extent it will be self-correcting. Cases discovered with a lag will eventually be added, so as the epidemic proceeds the early data points can be discarded to get a more reliable fit. Here is a fit based on the differential equation that may help with your tracking:
Clear[logisticDEFit];
logisticDEFit[data_, graphic_: True] :=
Module[{dataFn, dataDE, dataDEPlot, deModel, deNLM},
dataFn = Interpolation[data];
dataDE = {dataFn[#], dataFn'[#]} & /@ data[[All, 1]];
dataDEPlot =
ListPlot[dataDE, PlotStyle -> Darker@Red, GridLines -> Automatic,
PlotRange -> All];
deModel = k x (1 - x/L);
deNLM = NonlinearModelFit[dataDE, deModel, {k, L}, x];
If[graphic,
Print@Show[
Plot[deNLM[x], {x, 0, L /. deNLM@"BestFitParameters"},
GridLines -> Automatic, Frame -> True,
FrameLabel -> {"Cases", "LogisticFunction'[t]"},
PlotRange -> All,
PlotLabel -> "Logistic Differential Equation"], dataDEPlot]];
Flatten@{deNLM@"BestFitParameters",
FindRoot[
dataFn[t0] == L/2 /. deNLM["BestFitParameters"], {t0,
data[[-1]][[1]]}]}
];