Chimpanzees have been used extensively as a model system for laboratory research on infectious diseases. Ironically, we know next to nothing about disease dynamics in wild chimpanzee populations. Here, we analyze long-term demographic and behavioral data from two habituated chimpanzee communities in Taï National Park, Côte d'Ivoire, where previous work has shown respiratory pathogens to be an important source of infant mortality. In this paper we trace the effect of social connectivity on infant mortality dynamics. We focus on social play which, as the primary context of contact between young chimpanzees, may serve as a key venue for pathogen transmission. Infant abundance and mortality rates at Taï cycled regularly and in a way that was not well explained in terms of environmental forcing. Rather, infant mortality cycles appeared to self-organize in response to the ontogeny of social play. Each cycle started when the death of multiple infants in an outbreak synchronized the reproductive cycles of their mothers. A pulse of births predictably arrived about twelve months later, with social connectivity increasing over the following two years as the large birth cohort approached the peak of social play. The high social connectivity at this play peak then appeared to facilitate further outbreaks. Our results provide the first evidence that social play has a strong role in determining chimpanzee disease transmission risk and the first record of chimpanzee disease cycles similar to those seen in human children. They also lend more support to the view that infectious diseases are a major threat to the survival of remaining chimpanzee populations.