We estimate the number of times each syringe is used for non-shared injections according to the following mathematical argument which is also based on syringe distribution and behaviour data. If P syringes are distributed each year and a proportion ω of all syringes are not used, then P(1 - ω) syringes are used. If a syringe is used δp times before disposal for personal (unshared) injections, then the number of syringes used for individual injecting episodes among non-sharing IDUs is nN(1 - s)/δp. Similarly, the total number of syringes used for individual injecting among all sharing IDUs is n(1 - q)sN/δp and the total number of syringes used in sharing events is nqsN/δs.

Therefore, P(1 - ω) = nN(1 - s)/δp + n(1 - q)sN/δp + nqsN/δs = nN[δs-sq(δs - δp)]/δpδs defines a relationship between the total number of syringes distributed and the use of syringes. Changes in the number of syringes distributed are likely to change any, or all, of the following factors: the proportion of syringes that remain unused (ω), the proportion of injections that are shared (q), or the average number of times each syringe is used (in shared (δS) or individual (non-shared) injections (δp)). Substituting the known values and solving for δp leads to an average of 2.1 (1.2, 2.9) uses of non-shared syringe