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}_{3\phantom{\rule{0.2em}{0ex}}i}^{k}=1\right.\right)}\left(t\right)\phantom{\rule{0.6em}{0ex}}\mathit{with}\phantom{\rule{0.6em}{0ex}}{I}_{i}^{k}=\left\{\begin{array}{l}0,\phantom{\rule{0.5em}{0ex}}{T}_{d\phantom{\rule{0.2em}{0ex}}i}<t\phantom{\rule{0.4em}{0ex}}\mathit{or}\phantom{\rule{0.3em}{0ex}}{T}_{l\phantom{\rule{0.2em}{0ex}}i}<t\phantom{\rule{4.9em}{0ex}}\hfill \\ 1,\phantom{\rule{0.9em}{0ex}}{T}_{a\phantom{\rule{0.2em}{0ex}}i}\phantom{\rule{0.25em}{0ex}}\ge \phantom{\rule{0.25em}{0ex}}t\phantom{\rule{0.4em}{0ex}}\mathit{or}\phantom{\rule{0.2em}{0ex}}{T}_{d\phantom{\rule{0.2em}{0ex}}i}\phantom{\rule{0.25em}{0ex}}\ge \phantom{\rule{0.25em}{0ex}}t\phantom{\rule{0.4em}{0ex}}\mathit{or}\phantom{\rule{0.3em}{0ex}}{T}_{l\phantom{\rule{0.2em}{0ex}}i}\phantom{\rule{0.25em}{0ex}}\ge \phantom{\rule{0.25em}{0ex}}t\phantom{\rule{0.2em}{0ex}}\hfill \\ {S}_{i}~\mathit{bern}\phantom{\rule{0.25em}{0ex}}\left({\left(1-{p}_{d}\right)}^{{r}_{i}}\right),\phantom{\rule{0.4em}{0ex}}\mathit{otherwise},\phantom{\rule{0.7em}{0ex}}\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. |