We consider a model for mortality rates that includes both the long and short term effects of switching from an initial to a second state, for example, when patients receive an initial treatment and then switch to a second treatment. We include transient effects associated with the switch in the model through the use of time-dependent covariates. One can choose the form of the time-dependent covariate to correspond with a variety of possible transition patterns. We use an exponential decay model to compare the survival experience of transplant versus dialysis treatment of end stage renal disease (ESRD) patients from the Michigan Kidney Registry (MKR). This model involves a hazard function that has an initial effect in mortality at the time of transplant, expected to be higher, followed by a smooth exponential decay to a long term effect, expected to be lower than the risk for those remaining on dialysis. Cox and Oakes used this model to analyse the Stanford Heart Transplant data. The model implicitly suggests there is a time at which the hazard curves (and survival curves) for the treatment groups cross. Those crossing times are useful in advising patients who have the option of receiving a transplant. We describe methods for obtaining estimates of the crossing times and their associated variances, and then apply them in analysing the MKR data.