The Markovian approach, which assumes constant transmission rates and thus
leads to exponentially distributed inter-infection times, is dominant in
epidemic modeling. However, this assumption is unrealistic as an individual's
infectiousness depends on its viral load and varies over time. In this paper,
we present a SIRVS epidemic model incorporating non-Markovian infection
processes. The model can be easily adapted to accurately capture the generation
time distributions of emerging infectious diseases, which is essential for
accurate epidemic prediction. We observe noticeable variations in the transient
behavior under different infectiousness profiles and the same basic
reproduction number R0. The theoretical analyses show that only R0 and the mean
immunity period of the vaccinated individuals have an impact on the critical
vaccination rate needed to achieve herd immunity. A vaccination level at the
critical vaccination rate can ensure a relatively low incidence among the
population in case of future epidemics, regardless of the infectiousness
profiles