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]]}]}
   ];

So far the epidemic has not ended anywhere. This is a problem of population and resources to support the population. Humans are the substrate for the virus population to propagate. When a person becomes infected, if he can be isolated from the rest of the population then the virus cannot spread. Unfortunately there is a period of time (incubation period) that a person is infected, where he is not symptomatic, and can spread the virus, so standard epidemiologic technique is to identify the contacts and try to isolate them as well. If quarantine measures succeed then the curve will flatten. If the measures don't succeed then potentially a large population is at risk. So far the numbers from China, if they can be trusted, seem to be showing deceleration of growth, and over the past week the fit to the curve from the Chinese data continue to show the deceleration and limiting population in the 80,000 to 90,000 range. Note that the logistic model always has positive growth rate until the limit is reached at infinity. In real life the cases are in integers, so if the last infected person is isolated and does not spread the infection, it stops. Unfortunately there are other possibilities that some people or animals may become carriers, in which case a lower level of infection may continue indefinitely. Hopefully most people who survive will also develop immunity so that they won't be re-infected.

The model could also be wrong, but so far all epidemics have either ended or continue at low levels that do not threaten the whole population.

Why do you assume a logistic growth curve that flattens out? None of the data points presented are in the area where the estimated curves flatten out.

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