The statistics of epidemic transitions
Fig 2
Critical slowing down is illustrated in the potential function of the linearized SIR model.
Disease prevalence (I/N) is represented by the horizontal position of a ball sliding through viscous fluid in a well having a height determined by the potential function. Both the depth of the well and the viscosity of the fluid in the equivalent physical system are affected by vaccine coverage. The well is shallowest near the immunization threshold, which illustrates the slowing down of the dynamics as the critical point (νc ≈ 0.941; green dashed line) is approached. Oscillatory dynamics occur at another immunization level (ν ≈ 0.939) corresponding to the system becoming underdamped (pink region). Model parameters: b = 2 × 105 y−1, μ = 0.02 y−1, γ = 365/22 y−1, η = 2 × 10−5 y−1, R0 = 17. To write the potential function in terms of prevalence, we scaled the deviations of the linearized system by the square root of the equilibrium population size (i.e., scaled by ). Critical slowing down is seen in this figure in the relative magnitude of the displacement of the ball with respect to the distance from the critical level of immunization.