All linear DNA molecules face special problems in replicating their 5' ends, as DNA polymerases add nucleotides only to pre-existing strands with free 3'-OH groups. Parvoviruses, a group of small animal viruses with a linear single-stranded DNA genome, cope with this problem by having palindromic terminal sequences that can fold back on themselves to form hairpin structures essential in priming DNA replication. The 3' terminal sequence that initiates replication becomes reversed in orientation during the process, and if the palindrome is imperfect, two different, reverse-complementary terminal sequences are generated. The relative abundances of the terminal sequence orientations at each end of the DNA molecules can be measured and give information about the replication process. From such clues, we developed a "kinetic hairpin transfer model" based on differential rates of hairpin formation and inversion processes depending on the conformations of the 3' termini. Numerical studies showed that this simple idea can account for the diverse pattern of DNA distributions observed in the family Parvoviridae. In this paper, we simplify the model to a set of coupled linear first-order ordinary differential equations in order to delineate its essential properties by Perron-Frobenius theory. Secondly, we examine our assumption of linear kinetics by modeling enzyme catalysis of the component steps of the hairpin transfer process. We show that the rate-determining step of the process is the binding of initiation complex to the self-priming hairpin structures. Furthermore, we find that if the replication machinery is saturated by DNA substrate late in an infection, the differential equations become non-linear but the steady-state DNA distribution is still given by the solution of our original linear equations.