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\subsubsection {Results from national TB prevalence surveys.} Incidence was estimated using prevalence survey results in 19 countries that accounted for 46\% of the estimated global number of incident cases in 2014. Two approaches were used to derive incidence from prevalence.  In a first approach, incidence is estimated using measurements from national surveys of the prevalence of TB disease combined with estimates of the duration of disease. Incidence is estimated as the prevalence of TB divided by the average duration of disease assuming that the rate of change of prevalence with respect to time is negligible: let $N$ denote the size of a closed population with the number of birth and deaths the same for a period $\Delta t > 0$, let $C$ be the number of prevalent TB cases, $P$ the prevalence rate so that $P = C/N$. $C$ is small relative to $N$ and $(1-P) \approx P$. 1$.  Let $m$ denote the rate of exit from the pool of prevalent cases through mortality, spontaneous self-cure or cure from treatment, and $I$ the rate new cases are added to the pool. At equilibrium during the time period $\Delta t$ and further assuming exponentially distributed durations $d$ such that $d = m^{-1}$, \begin{align*}  I(N - C)\Delta t = m C \Delta t