A time since onset of injection model for hepatitis C spread amongst injecting drug users.

Studies of hepatitis C virus (HCV) infection amongst injecting drug users (IDUs) have suggested that this population can be separated into two risk groups (naive and experienced) with different injecting risk behaviours. Understanding the differences between these two groups and how they interact could lead to a better allocation of prevention measures designed to reduce the burden of HCV in this population. In this paper we develop a deterministic, compartmental mathematical model for the spread of HCV in an IDU population that has been separated into two groups (naive and experienced) by time since onset of injection. We will first describe the model. After deriving the system of governing equations, we will examine the basic reproductive number R₀, the existence and uniqueness of equilibrium solutions and the global stability of the disease free equilibrium (DFE) solution. The model behaviour is determined by the basic reproductive number, with R₀ = 1 a critical threshold for endemic HCV prevalence. We will show that when R₀ ≤ 1, and HCV is initially present in the population, the system will tend towards the globally asymptotically stable DFE where HCV has been eliminated from the population. We also show that when R₀ > 1 there exists a unique non-zero equilibrium solution. Then we estimate the value of R₀ from epidemiological data for Glasgow and verify our theoretical results using simulations with realistic parameter values. The numerical results suggest that if R₀ > 1 and the disease is initially present then the system will tend to the unique endemic equilibrium.