HIV infections in Africa fell by almost 40 percent over the past 15 years [pdf]

The posting title misrepresents the contents.

What the document actually says is that the HIV infection rate among the studied group (age 15-49) dropped 16%, not 40%. See page 39 of the document (page 59 of the PDF).

The AIDS-related death rate dropped 40%, but that's a different criterion -- one made possible by better modern medical care, especially antiretroviral drugs.