The data at the beginning of the epidemic counts are not likely to be very accurate. See some of the discussion on later posts. There are some basic numerical reasons also for this: there are only 2 or 3 significant digits of information in the early numbers, where as the later numbers have higher precision. But the main reason is likely that the early data is also probably not accurate, especially in Hubei where the laboratories probably were quickly overwhelmed. They stopped using laboratory to count cases somewhere in the middle, and the information at WHO suggests that the death counts were only for lab confirmed cases. Then later they added the other death counts in a lump, causing the jump in the accumulated data. What you should try is using the recent data which contains more information and is likely to be more accurate. Also try looking at sequential fits as data accumulates.
nlmd = NonlinearModelFit[#,
logistic, {{k, 0.3}, {L, 3000}, {x0, 20}}, x] & /@
Table[Take[deathCases, i], {i, 5, Length@deathCases}];
Then look at the sequential parameters with ListPlot to see how the parameters stabilize with time. It is a good idea to give NonlinearModelFit some reasonable starting parameters.
