The effect of seasonal forcing on oscillations in influenza incidence. We simulate disease dynamics in a population of 500,000 people both with (blue) and without (red) demographic stochasticity. varies sinusoidally between 9.6 and 10.4. The black curve shows the (nearly invisible) oscillations in incidence that these variations in would cause if disease dynamics responded instantaneously to changes in transmission, without resonance or transient fluctuations. (a) Weak resonance. The intrinsic oscillatory period is T ≈ 0.59 years (duration of infectiousness 0.02 yr, duration of immunity 4 yr). (b) Strong resonance. T ≈ 0.94 yr (duration of infectiousness 0.025 yr, duration of immunity 8 yr). When parameters are drawn at random from the ranges given for , D, and L in the text, we find strong oscillations due to resonance (winter peak/summer trough >5) for 21% of parameter sets (Fig. 2).

Magnitude of observed oscillations (ratio of peak-to-trough incidence) in the stochastic forced SIRS epidemic model plotted against the approximated period of endogenous oscillations in the SIRS model , for 2,000 sets of parameters randomly chosen from the ranges given in the text (shown are the 1,560 trials where the disease persisted for at least 20 years after being started from the deterministic equilibrium). Underlying variation in transmission rate is ± 4% (i.e., β₁ = 0.04 in Eq. 3). Strong resonance occurs when the approximate endogenous period is near 1 year.