Table 1 Schematic overview of the 3-step Monte Carlo simulation model to estimate ever IDU prevalences ^Ɨ

STEP 1:

Imputation by Chained Equations: missing risk factor information

Missing risk factor information was imputed using Imputation by Chained Equations. The imputation model contains the variables: injecting drug use, sex, nationality, year at registration and age at registration. The imputation results in one complete dataset X _k , containing original and imputed values.

STEP 2:

Stochastic Mortality Modeling: lacking follow-up of the HIV ⁺ /AIDS ^– cases

For a complete dataset k, the number of registered HIV-cases for whom injecting drug use was the most probable route of transmission and who were alive at time t is calculated as ${\widehat{N}}_x^k\left(t\right)={\displaystyle \sum _{i=1}^{n_t}\left(I_i^k\left|X_{3i}^k=1\right.\right)}\left(t\right)\mathit{with}{I}_i^k=\left\{\begin{array}{l}0,T_{di}<t\mathit{or}{T}_{li}<t\hfill \\ 1,T_{ai}\ge t\mathit{or}{T}_{di}\ge t\mathit{or}{T}_{li}\ge t\hfill \\ S_i~\mathit{bern}\left({\left(1-p_d\right)}^{r_i}\right),\mathit{otherwise},\hfill \end{array}\right.$

where I _i indicates the ‘vital’ status with I _i  = 1 if person i is still alive and living in Belgium and I _i  = 0 otherwise, where r _i is the number of years since HIV registration or r _i  = t − t _hi and where p _d is the annual non-AIDS mortality rate among seropositive IDUs with p _d  ~ betapert*(0.58%, 1.08%, 1.58%).

STEP 3:

Benchmark-multiplier method: population size estimation

The number of ever-injecting drug users being alive at time t is given by ${\widehat{N}}_y^k\left(t\right)={\widehat{p}}_{\mathit{HIV}}^{-1}{\widehat{N}}_x^k\left(t\right)-n^{-1}{\widehat{N}}_x^k\left(t\right)\left(1-{\widehat{p}}_{\mathit{HIV}}\right){\widehat{p}}_{\mathit{HIV}}^{-1},$ with ${\widehat{N}}_x^k\left(t\right)$ obtained from step 2, ${\widehat{p}}_{\mathit{HIV}}~\mathit{beta}\left(21,620\right)$ and n = 639.