An epidemic in a dynamic population with importation of infectives

Frank Ball (University of Nottingham)

Frank Adams 2,

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.

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