Table 1 Model comparison.

The maximum likelihood estimates (95% CI) are presented for the stochastic variant of the models, estimated using the MIF algorithm. The best AIC value is indicated in boldface. SE, standard error.

QuantityNo-loss modelLeaky modelWaning modelWaning+Leaky model
log L−3726.9 (SE: 0.4)−3664.9 (SE: 0.9)−3594.5 (SE: 0.6)−3598.5* (SE: 1.8)
AIC7474735672157217*
ΔAIC25914102
Rp2.4 (1.8, 2.7)1.6 (1.3, 2.2)1.8 (1.5, 2.0)1.8 (1.6, 2.0)
R013.6 (7.5, 23.0)12.6 (9.0, 19.4)10.1 (6.5, 17.2)9.1 (5.3, 16.2)
Vaccine impact0.85 (0.70, 0.95)0.90 (0.81, 0.95)0.85 (0.75, 0.93)0.83 (0.70, 0.92)

*Because the two models are nested, the likelihood of the full model should be higher or equal to that of the waning model. The small difference indicates that the MLE of the leakiness for the full model is 0. Consequently, the AIC was calculated with the likelihood of the waning model.