Table 1.
Posterior model parameter ranges used in the full model, obtained through the incarceration sub-model calibration
| Parameter | Symbol | Prior distribution | Posterior parameter range | Source and Comments |
|---|---|---|---|---|
| Death rate (per year) 1 | μ1 | Sampled from a Poisson distribution with mean (10), with sampled values divided by 1000. | 0.006 – 0.014 | (25) |
| Average duration injecting (years) 1 | μ2 | Uniform on (5,20) | 5.1 – 17.7 | (26) |
| Factor increase in mortality rate for 2 weeks following prison release | μ* | Lognormal with parameters (2.0053,0.1393) truncated to 95% confidence interval [5.7,9.9] | 6.3–8.2 | (20) |
| Percentage of prison population that are current PWID | P | Normal with parameters (0.19, .006) truncated to 95% CI (0.18 – 0.21) | 18.7–21.0% | Scottish prison survey(9) |
| Current PWID population size | N | N/A | N/A | Parameter sets are rejected if model population size not within 11500–18600(24) |
| Percentage of PWID initiating injecting when 3 | Dirichlet distribution with parameters (10,1,1,1,1) | - | Obtained through model fitting. | |
| Never incarcerated | p1 | 72.2–92.5% | ||
| Incarcerated for first time | p2 | 1.6%–12.0% | ||
| Community, incarcerated once | p3 | 1.4–10.3% | ||
| Incarcerated for second or more time | p4 | 3.2–13.7% | ||
| Community, incarcerated twice or more | p5 | 0.2–8.3% | ||
| Incarceration rates per year | γ | Obtained through model fitting | ||
| Recent PWID (<5yrs injecting) | Uniform on (0,0.25) | 0.12–0.17 | ||
| Non-recent PWID (>5yrs injecting) | Uniform on (0,0.25) | 0.03–0.06 | ||
| Re-incarceration rates per year | δ | Obtained through model fitting | ||
| Recent PWID | Uniform on (0,1) | 0.63–0.88 | ||
| Non-recent PWID | Uniform on (0,1) | 0.08–0.17 | ||
| Release rate per year | τ | Normal with parameters (0.48, .019) truncated to 95% CI (0.44 – 0.52) | 0.47–0.51 | Scottish prison survey2. Corresponds to an average 5.7–6.1 months spent in prison per incarceration. |
The PWID leaving rate, μ, is given by: μ1 + 1/μ2
We used the weighted average time between date of incarceration and earliest date of liberty for current PWID, i.e. weighted by the reciprocal of these times to allow for the likely oversampling of prisoners with long sentences.
In the final model which does not stratify incarceration history into incarcerated once and twice or more, p2 and p4 are combined to give the proportion of PWID initiating injecting in prison. Whilst, p3 and p5 are combined to give the proportion of PWID initiating injecting in the community having been incarcerated – a random proportion of which have been recently released.