The Sweataches
It is believed to have been a bison disease, possibly related to the cattle disease brucellosis, in its origins. Specifically, the marsh bison breed of the Lower Mississippi and Masaguay, as the disease when transmitted to humans became a sub-tropical and tropical disease. The disease can spread in any environment through contact with infected animals or humans or their byproduct. The disease is most dangerous, however, in subtropical and tropical regions as it can be spread by insects feeding on both infected animals and humans. The disease best thrived in warm, wet environments and was most common in the Caribbean before the coming of Europeans (possibly due to the Masaguayan expansion and trade networks).
The disease colonises the uterus, placenta and regional lymph nodes of it's host. The time between infection and display of symptoms is generally about two weeks to two months, which allows the disease to be quite unknowingly spread. Symptoms include inconstant fevers, sweating (accompanied by a unique smell akin to wet hay), weakness, migraines, depression and muscular and bodily pain. Contact with infected animals or humans can spread the disease, as can exposure to unpasteurised milk. However, the disease spreads most rapidly through the mosquito, much like malaria. This mutation of the disease will be what makes it such a tenacious killer. Even more that the direct deaths, the effects on fertility will be what makes this disease such a dire foe.
Areas of minimal effect, with deaths between 0% and 5%, will be most of Europe, Russia, central Asia and northern Asia. This will see some dislocation and the odd severe outbreak but in general it won't be catastrophic. A greater effect will be seen in subtropical regions, with perhaps 5% to 15% of the population succumbing. This will occur around the Mediterranean (particularly the Iberian penisula), North Africa, the Middle East (particularly around Mesopotamia), South Africa, the interior of southern China as well as southern Japan. The areas where the disease will find the most fertile ground will be similar areas that are afflicted by malaria. Most of sub-Saharan Africa, India, south east Asia and the southern Chinese coast will be particularly affected. With no resistance, the disease is going to kill a much larger percentage of the population. With the Sweataches spreading through Asia and Africa, and smallpox and other diseases decimating the population of the New World, the global human population will fall significantly in the 16th century.
The Spanish will be largely responsible for bringing the disease to Europe and West Africa, causing epidemics there in the 1510's (in the latter case causing a severe population slump, severely limiting the slave trade. But the main vector of the disease will be Portugeuse expansion in Asia. The first regions hit will be India and the Sultanate of Malacca, then southern China in the 1520's and southern Japan in the 1540's. The Asian trade networks, already reeling from Portugeuse aggression, will be in even more chaos ITTL than in OTL. The Portugeuse themselves will die in large numbers in the Tropics as well, just as susceptible to the Sweataches in that environment.
In general its going to mean that for Portugal, the expansion into Asia will be far less profitable and far more costly than in OTL. Iberian population slumping in this time is going to reduce the rate in which the Spanish and Portugeuse expand into the New World. With the Treaty of Tordesillas no longer relevant (due to the temporary breakup of Spain), the Portugeuse feel free to claim the coast of northern [South America]. As Spain finds gold in its possessions to the north, the Portugeuse become greedy for that easy cash to supplement the increasingly difficult Asian trade. Portugeuse explorers will stumble onto the Chimu, and then the Potosi mine, in the middle of this century. As that vast wealth becomes Portugal's main priority, it will become wealthy and begin to neglect its Asian possessions. The Spanish, wealthy from New World gold but less so than in OTL, will move to expand its Asian interests instead.
The Sultanates of the East Indies will be severely destabilized, delaying for a time the Islamisation of the region as the Sultanates are unable to project their power. The Sultanate of Malacca will collapse more completely than OTL, becoming very difficult for the Portugeuse to administer, however the Sultanates of Johor and Aceh will be similarly affected. As the Portugeuse will be suffering the disease effects as well it may become zero-sum, but as trade dries up the Portugeuse may lose their appetite for it. If Johor weathers the storm, the collapse of Mallacan and Acehnese competition and a Portugeuse withdrawl could see the area come under Johor's control. The Thai Ayutthaya kingdom in the north may try to make a play for the area, however it will also be suffering its own epidemics.
Southern China, particularly the coastline, will be hard-hit by the epidemic. This will limit the ability of the Ming to project their power into south east Asia, so they will be unable to vassalise Vietnam for the time being. With trade disrupted and the population dying in great numbers, the state's naval power is likely to suffer. However, the Japanese pirates infesting the area will also suffer, as southern Japan will host an outbreak of its own. The OTL expansion of Japanese wokou pirates is likely to be reduced ITTL, due to the population slump in both southern Japan and the reduction of southern Chinese overseas trade. When state authority reasserts itself, it should be able to control things better than OTL. This is one part of Asia where the Portugeuse presence is likely to continue, exporting [South American] silver across the Pacific to China.
India will be hit quite hard by the Sweataches, particularly the southern Vijayanagara empire. It is likely to be more vulnerable to assault by the Deccan Sultanates, which may see a greater spread of Islamisation compared to OTL. The empire may split up into bickering feudal kingdoms, becoming more vulnerable to northern aggression and European expansion. The Sultanates themselves will suffer the epidemic, which may cause some of the sultanates to fall. If one state maintains cohesion while others collapse, it could cannibalise the others and then turn southward against the fracturing Vijayanagara. In general, however, the entire subcontinent will suffer political, social and economic disorder, which may instead mean a greater expansion of the Mughals under Babur.
Africa will be hit hard, with the population of much of West Africa slumping. Compared to OTL (which saw a population boom due to the introduction of corn) there are going to be a lot less West Africans, and the slave trade from that region will be much smaller in scope than OTL. Meanwhile, the East African coast will be highly affected, particularly Mombasa and Kilwa. The Portugeuse will probably have an easier time establishing control over this area than in OTL as local power structures collapse. In general, I see the Portugeuse finding control over Indian Ocean trade to be easier, but less profitable, than OTL, and much of the Portugeuse trade network may end up being taken over by Spain or another power.
The Ottoman Empire, though it won't suffer the worst of the Sweataches, will be affected. Greece, Anatolia and the Middle East will be greatly affected. Ottoman expansion into the Red Sea and competition with Portugal may not take place. Mesopotamia will be the largest epicentre, which may weaken the Ottomans in the region enough that the Safavids will retake the area (though they, too, will be somewhat affected by the disease). On the other hand, the weakened Safavids might become subject to an invasion by the alliance of the Uzbek khan Shaybani and the Mughal Babur. That may mean that the Mughals, consolidating control over the southern Persian coast as well as most of India, may be the ones to compete with the Portugeuse with assistance from Ottoman and Venetian advisors.
As for the Mediterranean, there will be severe outbreaks in Iberia, Italy and the North African coast. This will likely assist in the French attempts to dominate Italy in this period, though their local Borgias allies may have a rough time of it. The Ottomans expanding into southeastern Europe are going to be weaker than OTL, and progress slower and with less success. With the Ottoman threat reduced, Spain and Portugal both distracted and weakened, a major issue for Europe will be the conflict between the Holy Roman Empire and France (which controls Italy and the Papacy). In this TL, the political circumstances will mean that the pressures that manifested as the Reformation in OTL may instead manifest as Schism. But more on that later.