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where $f$ denotes a cubic spline function in countries with large year-to-year fluctuations in $N$, or else, the identity function. The incidence series are completed using assumptions about improvements in CFR over time in countries with evidence of improvements in TB control performance such as an increased detection coverage over time or improved treatment outcomes, ensuring that the following inequality holds:   \begin{align*}  0 \leq \lvert \left|  \frac{dI}{dt} \rvert \leq \lvert \right|  \frac{dM}{dt} \rvert \end{align*}  where $I$ denotes incidence and $M$ denotes mortality.