Well… Great Zimbabwe left no written records and there are no contemporary sources for how they lived. So, while an anthropologist might be able to answer that, historians have no primary or secondary resources to rely upon.
Now, in the Mali Empire, which largely was inland, there likely were devastating epidemics in 14th and 15th centuries. This likely was the bubonic plague. And while not a tropical disease, it shows epidemic diseases ravaged west Africa as much as anywhere.
While technically post colonization, the people’s of the valley of Mexico suffered greatly at the hands of cocoliztli. It’s unclear whether this was either native to North America, or a European disease, but it obliterated what was left of the population after small pox had ravaged the region. That’s said, there’s been theories that it not only was native to North America, but also responsible for the collapse of the Classical Maya.
In India, Ayurvedic medicine, and the focus on preventing illness and hygiene may helped mitigate the effects of tropical disease. One of the primary sources, the Sushruta Samhita, has substantial portions on hygiene. Now this is more my speculation, but the focus this probably allowed ancient Indian culture to handle tropical diseases.
Disease ravaged all cultures, regardless of location. Cities, particularly prior to the 20th centuries, were often dirty and dangerous. The high density of people, poor sanitary conditions, and lack of medical knowledge meant that no place was really safe from disease. Tropical diseases were really just another thing that might kill you.
Bhishagratna, Kaviraj KL (1907). An English Translation of the Sushruta Samhita in Three Volumes, (Volume 1, Archived by University of Toronto). Calcutta.
Gallagher, Daphne E., and Stephen A. Dueppen. “Recognizing Plague Epidemics in the Archaeological Record of West A...” Afriques. Débats, Méthodes Et Terrains D'histoire, Institut Des Mondes Africains (IMAF, 24 Dec. 2018, journals.openedition.org/afriques/2198?lang=en.
Rodolfo Acuna-Soto, David W. Stahle, Matthew D. Therrell, Sergio Gomez Chavez, Malcolm K. Cleaveland, Drought, epidemic disease, and the fall of classic period cultures in Mesoamerica (AD 750–950). Hemorrhagic fevers as a cause of massive population loss, Medical Hypotheses, Volume 65, Issue 2, 2005, Pages 405-409, ISSN 0306-9877,
I’ll actually format these citations at a later point in time.