A system of two differential equations is used to model the transmission dynamics of human immunodeficiency virus between 'persons who inject drugs' (PWIDs) and their syringes. Our vector-borne disease model hinges on a metaphorical urn from which PWIDs draw syringes at random which may or may not be infected and may or may not result in one of the two agents becoming infected. The model's parameters are estimated with data mostly from the city of Omsk in Western Siberia. A linear trend in PWID prevalence in Omsk could only be fitted by considering a time-dependent version of the model captured through a secular decrease in the probability that PWIDs decide to share a syringe. A global sensitivity analysis is performed with 14 parameters considered random variables in order to assess their impact on average numbers infected over a 50-year projection. With obvious intervention implications the drug injection rate and the probability of syringe-cleansing are the only parameters whose coefficients of correlations with numbers of infected PWIDs and infected syringes have an absolute value close to or larger than 0.40.
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