  We examine here the effects of recurrent vaccination and waning immunity on
the establishment of an endemic equilibrium in a population. An
individual-based model that incorporates memory effects for transmission rate
during infection and subsequent immunity is introduced, considering
stochasticity at the individual level. By letting the population size going to
infinity, we derive a set of equations describing the large scale behavior of
the epidemic. The analysis of the model's equilibria reveals a criterion for
the existence of an endemic equilibrium, which depends on the rate of immunity
loss and the distribution of time between booster doses. The outcome of a
vaccination policy in this context is influenced by the efficiency of the
vaccine in blocking transmissions and the distribution pattern of booster doses
within the population. Strategies with evenly spaced booster shots at the
individual level prove to be more effective in preventing disease spread
compared to irregularly spaced boosters, as longer intervals without
vaccination increase susceptibility and facilitate more efficient disease
transmission. We provide an expression for the critical fraction of the
population required to adhere to the vaccination policy in order to eradicate
the disease, that resembles a well-known threshold for preventing an outbreak
with an imperfect vaccine. We also investigate the consequences of unequal
vaccine access in a population and prove that, under reasonable assumptions,
fair vaccine allocation is the optimal strategy to prevent endemicity.
