Pre-exposure prophylaxis (PrEP) is increasingly used to prevent the transmission of H.I.V. in at-risk populations. However, PrEP users may discontinue use of the medicine due to side effects, lower perceived risk, or other reasons. The usage metrics of 594 individuals was tracked over 350 days using the Wisepill electronic monitoring system. We model the PrEP drug adherence level using an absorbing Markov chain with a unique absorbing state. The transition matrix T obtained from the Wisepill data will have a trivial eigenvector (eigendistribution) associated with the first (i.e., largest) eigenvalue 1. The 2nd eigenvalue(s) then become important in determining the asymptotic behavior of the Markov chain, dictating how fast the Markov chain decays to the absorbing state. Under a fairly general assumption, we prove that the second positive eigenvalue is unique and the corresponding eigenvector will have nonnegative entries with exceptions at absorbing states. In addition, we define the asymptotic half life of the absorbing Markov chain directly from the 2nd eigenvalue. We then determine the 2nd eigenvalue of T and the asymptotic half life of the Markov chain, which turns out to be very close to the real half life of the Markov chain. Finally, we interpret the 2nd eigenvector as the relative probability distribution of X