Sick residents do not die, but their state of health has a negative impact on the overall health of the district.
If you do not treat the residents, you can quickly bring a situation in which the health of the district falls below serious level. That's just a step away from the critical state of the district.
If the condition of a district falls below critical, the district will collapse. The people will die and enemies will appear in the region, which have been friendly so far.
It is therefore worth treating the sick on an ongoing basis and ensuring that the health condition of the district is maintained at a high level. Probably you are planning to play a "bad character" - then you can aim to exterminate the inhabitants as much as you can. You will then be able to gain access to many closed chests and private goods of various characters.
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