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By comparing mathematical models with the data obtained from patients being treated with antiviral drugs, it has been possible to determine many quantitative features of these infections. The most dramatic finding has been that even though AIDS and hepatitis C are diseases that occur on a timescale of one or more decades, there are very rapid dynamical processes that occur on timescales of hours to days, as well as slower processes that occur on timescales of weeks to months. We show how dynamical modeling and parameter estimation techniques have uncovered these important features of HIV and HCV infection and subsequently impacted the way in which patients are treated with potent antiviral drugs. 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