International audienceWe consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by Volz (2008)

We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families.

Hepatitis C virus (HCV) seroprevalence remains high in people who inject drug (PWID) populations, often above 60%. Highly effective direct-acting antiviral (DAA) regimens (90% efficacy) are becoming available for HCV treatment. This therapeutic revolution raises the possibility of eliminating HCV from this population. However, for this, an effective cascade of care is required. In the context of the available DAA therapies, we used a dynamic individual-based model including a model of the PWID social network to simulate the impact of improved testing, linkage to care, and adherence to treatment, and of modified treatment recommendation on the transmission and on the morbidity of HCV in PWID in France. Under the current incidence and cascade of care, with treatment initiated at fibrosis stage F2, HCV prevalence decreased from 42.8% to 24.9% (95% confidence interval: 24.8-24.9) after 10 years. Changing treatment initiation criteria to treat from F0 was the only intervention leading to a substantial additional decrease in prevalence, which fell to 11.6% (95% CI: 11.6-11.7) at 10 years. Combining this change with improved testing, linkage to care, and adherence to treatment decreased HCV prevalence to 7.0% (95% CI: 7.0-7.1) at 10 years and avoided 15% (95% CI: 14-17) and 29% (95% CI: 28-30) of cirrhosis complications over 10 and 40 years, respectively. Conclusions: Major decreases in prevalent HCV infections occur only when treatment is initiated at early stages of fibrosis, suggesting that systematic treatment in PWID, where incidence remains high, would be beneficial. However, elimination within the 10 next years will be difficult to achieve using treatment alone, even with a highly improved cascade of care.