Figure 2.

Accuracy of the fitted SEIR models and estimated seasonality. (a) Comparison of in-sample model predictions and observations for each city. Expected cases are one-week-ahead predictions from the fitted models. The dashed line shows 1 : 1. Coefficients of determination (R²) were calculated as the reduction in the sum-of-squared errors from model predictions relative to a null model of the mean number of cases (Material and methods). (b) The estimated seasonality of the basic reproductive ratio (${\mathcal{R}}_0$) for each city. ${\mathcal{R}}_0$ was approximated as: ηβ_t/((η + ν)(γ + ν)), where 1/η is the incubation period, 1/γ is the infectious period, β_t is the time-specific rate of transmission, and ν is the death rate. Only β_t is estimated by our model. We set $1/\eta =8\hspace{0.17em}\mathrm{days}$, $1/\gamma =5\hspace{0.17em}\mathrm{days}$, and ν = 0.05 for calculating ${\mathcal{R}}_0$ as shown in this figure. The white line is ${\mathcal{R}}_0$ calculated using the MLE parameters; shaded regions are the bootstrapped 95% confidence intervals. The dashed horizontal lines show the common range of measles ${\mathcal{R}}_0\hspace{0.17em}:\hspace{0.17em}12\hspace{0.17em}\mathrm{to}\hspace{0.17em}18$.