We compare the aftershock decay rate in natural data with predictions from a stochastic analytical model based on a Markov process with stationary transition rates. These transition rates vary according to the magnitude of a scalar representing the state of stress and defined as the overload. Thus, the aftershock decay rate in the model is a sum of independent exponential decay functions with different characteristic times. From different shapes of the overload distribution and different expressions of the transition rates, we discuss the magnitude of the exponent of the power law aftershock decay rate and the time interval over which we can expect to observe this regime. Before and after this time interval, we show that the decay is linear and exponential, respectively. From our analytical solutions, we deduce a model of aftershock decay rate in which a power law scaling exponent and two characteristic rates emerge. One rate is a short-term linear decrease before the onset of the power law decay to account for a finite number of events at zero time, and the other one can be interpreted as an inverse correlation time, after which aftershocks no longer occur. Then, we interpret the empirical modified Omori law (MOL) and its parameters in the framework of our theoretical model. We suggest a technique to systematically estimate and interpret the temporal limits of the power law aftershock decay rate in real sequences. We approximate these temporal limits from data available from several well-known aftershock sequences and show from an Akaike Information Criteria (AIC) that, in almost all cases examined here, our model fits better the aftershock decay rate than the MOL despite a quantitative penalty for the extra parameter required. From this work, we conclude that the time delay before the onset of the power law decay may be related to the recurrence time of an earthquake. Finally, we suggest that the power law decay rates extend over longer times according to the concentration of the deformation along dominant major faults.