## Results: 8

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Surfaces representing all the eigenvalues of the next generation matrix for

*R*_{0}= 1.6. All the eigenvalues of the matrix but two are zero (plotted in yellow). The ones that are non-zero are plotted above in red and blue. The green plot corresponds to the plane*z*= 1. Here, the blue surface corresponds to the effective reproduction number with vaccination,*R*(_{f}*f*_{1},*f*_{2}).6.

The shaded regions indicate the

**intermediate regions**for a specific*R*_{0}, (where*R*_{0}∈ [1.2, 1.8]) that is, the region where the threshold for a mixture of vaccine-induced immunity and natural immunity should lie. The level of combined immunity (vaccine-induced and naturally-acquired immunity) at the end of winter 2010, calculated from on [23] and [27], is given by the dotted square*S*.7.

Sensitivity analysis for the recovery rates for children and adults considered. Each recovery rate was independently taken from a uniform distribution of length 0.2 centered around the values used. Only those pairs for which the

*R*_{0}fell in one of the following intervals were used: 1.15 <*R*_{0}< 1.25, or 1.35 <*R*_{0}< 1.45, or 1.55 <*R*_{0}< 1.65 or 1.75 <*R*_{0}< 1.85. The results are very robust with respect to these parameters.8.

Sensitivity analysis for the contact rates taken from a uniform distribution of length 0.2, centered at the value of the

*c*used. The contact rates were taken independently but only the triplets that gave 1.15 <_{ij}*R*_{0}< 1.25, or 1.35 <*R*_{0}< 1.45, or 1.55 <*R*_{0}< 1.65 or 1.75 <*R*_{0}< 1.85 and where*c*_{11}>*c*_{12}and*c*_{11}>*c*_{22}were considered. The results are generally robust under this sensitivity. As long as*R*_{0}< 1.6, all the runs lie below the dotted rectangle, indicating that a recurrent epidemic wave is very unlikely. For*R*_{0}= 1.6, most of the curves intersect the rectangle, suggesting that a following wave is plausible. For*R*_{0}= 1.8 most of the curves overlap with the rectangle or lie above the rectangle, indicating that a third wave would be very likely. The black curve indicates the critical vaccination curve used in the analysis.