We provide an in silico study of stochastic infection extinction from a pharmacokinetical viewpoint. Our work considers a non-specific antiviral drug that increases the virus clearance rate, and we investigate the effect of this drug on early virus extinction. Infection extinction data is generated by a hybrid multiscale framework that applies both partial differential equations and discrete mathematical approaches. The central result of our work is the observation, analysis and explanation of a linear relationship between the virus clearance rate and the probability of early infection extinction. The derivation behind this simple relationship is given by merging different mathematical toolboxes.