Modeling the spread of hepatitis C virus amongst people who inject drugs

This article discusses two models, with two different needle assumptions for the transmission of hepatitis C virus (HCV) between people who inject drugs (PWIDs) who share needles and syringes. Our analysis demonstrates that the basic reproduction number R-0 determines how the model behaves. R-0 = 1 is a crucial threshold parameter which divides two qualitatively different scenarios. It has been shown that if R-0 <= 1 only a disease-free equilibrium point exists. On the other hand if R-0 > 1 a unique endemic equilibrium point exists as well as the disease-free one. The disease-free equilibrium point is globally asymptotically stable if R-0 <= 1 otherwise unstable. We look at an approximation to this model by using the fact that the timescale on which the individual PWIDs inject with needles or syringes is much faster than the timescale of epidemiological change. Also, we showed that if R-0 > 1 the endemic equilibrium point is locally asymptotically stable for our approximation to this model. Additionally, we perform some simulations with realistic parameter values which confirm that if R-0 > 1 then the solutions to this model tend to this endemic equilibrium value giving a steady-state nonzero HCV prevalence. With realistic parameter values for both assumptions we find that R-0 = 2.9987 and the total endemic equilibrium fraction of infectious PWIDs is 0.5348 and the total equilibrium fraction of antibody positive PWIDs is 0.65. The total fraction of infectious needles is 0.275.