## An epidemic in a dynamic population with importation of infectives

#### Frank Ball (University of Nottingham)

We consider a large uniformly mixing dynamic population, which has
constant birth rate and exponentially distributed lifetimes, with mean pop-
ulation size $$n$$. A Markovian SIR (susceptible → infective → recovered)
infectious disease, having importation of infectives, taking place in this pop-
ulation is analysed. The asymptotic behaviour of the model is determined
as $$n \to \infty$$, keeping the basic reproduction number R 0 as well as the impor-
tation rate of infectives fixed, but assuming that the quotient of the average
infectious period and the average lifetime tends to 0 faster than $$1/ \log n$$. We
show that, as $$n → ∞$$, the behaviour of the 3-dimensional process describing
the evolution of the fraction of the population that are susceptible, infec-
tive and recovered, is encapsulated in a 1-dimensional regenerative process
$$S = \{S(t); t ≥ 0\}$$ describing the limiting fraction of the population that are
susceptible. The process $$S$$ grows deterministically, except at one random
time point per regenerative cycle, where it jumps down by a size that is com-
pletely determined by the waiting time since the start of the regenerative
cycle. Properties of the process $$S$$, including the jump size and stationary
distributions, are determined.