a, The trajectories of I for three different combinations of latency (1/σ days) and loss of immunity (1/ω years) for R0 = 3, 1/γ = 14 days. TE is the inter-epidemic interval (years), I(∞) is the value at the endemic equilibrium and the gray trajectory is the prediction from the closed epidemic SIR model1. b, The I versus S phase planes of the 1/ω = 1 year epidemic waves depicted in a, showing the effect of doubling 1/σ from 7 to 14 days. c, The effect of doubling 1/ω from 1 to 2 years. Points along the spirals indicate the one-year mark after the start of the first epidemic. Start and progression of spirals are indicated by a black arrow (from bottom right and counterclockwise).