Figure 1.

Bifurcation delay is the time between the tipping point at R₀ = 1 (intersection of grey horizontal line and blue vertical line) and the beginning of an actual disease epidemic (orange vertical line, definition in text). This simulated time series is a stochastic realization of the hinge SIR model, where the transmission rate (β) begins to increase at a set time (here, t = 5 years) at rate α. This example contains multiple ‘sparks’ after the transition to supercriticality that do not lead to epidemics, demonstrating that bifurcation delay is not simply the waiting time for a spark to occur after R₀ > 1. Other parameters for this simulation: rate of increase in transmission, α = 0.001 yr⁻¹, initial population size (N₀) = 1000, recovery rate, γ = 365/14 yr⁻¹ (corresponding to a 14 day infectious period), spontaneous infection rate, ξ = 0.00067 yr⁻¹, birth and death rate, μ = 1/60 yr⁻¹. (Online version in colour.)