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In infectious disease epidemiology, it is useful to know how quickly genetic markers of pathogenic agents evolve while inside hosts. We propose a modular framework with which these genotype change rates can be estimated. The estimation scheme requires a model of the underlying process of genetic change, a detection scheme that filters this process into observable quantities, and a monitoring scheme that describes the timing of observations. We study a linear "birth-shift-death" model for change in transposable element genotypes, obtaining maximum-likelihood estimators for various detection and monitoring schemes. The method is applied to serial genotypes of the transposon IS6110 in Mycobacterium tuberculosis. The estimated birth rate of 0.0161 (events per copy of the transposon per year) and death rate of 0.0108 are both significantly larger than the estimated shift rate of 0.0018. The sum of these estimates, which corresponds to a "half-life" of 2.4 years for a typical strain that has 10 copies of the element, substantially exceeds a previous estimate of 0.0135 total changes per copy per year. We consider experimental design issues that enable the precision of estimates to be improved. We also discuss extensions to other markers and implications for molecular epidemiology.