A simple model for complex dynamical transitions in epidemics

Dramatic changes in patterns of epidemics have been observed throughout this century. For childhood infectious diseases such as measles, the major transitions are between regular cycles and irregular, possibly chaotic epidemics, and from regionally synchronized oscillations to complex, spatially incoherent epidemics. A simple model can explain both kinds of transitions as the consequences of changes in birth and vaccination rates. Measles is a natural ecological system that exhibits different dynamical transitions at different times and places, yet all of these transitions can be predicted as bifurcations of a single nonlinear model. Apart from their public health importance, epidemics of childhood infections have pro-vided valuable insights into theories of pop-ulation dynamics (1–3). Before mass vacci-nation began in the 1960s, epidemics of mea-sles exhibited both regular (annual, biennial