Abstract
Recently, the notion of a reinfection threshold in epidemiological models
of only partial immunity has been debated in the literature.
We present a rigorous analysis of a model of reinfection which
shows a clear threshold behaviour at the parameter point where the
reinfection threshold was originally described.
Furthermore, we demonstrate that this threshold is the mean field
version of a transition in corresponding spatial models of
immunization. The reinfection threshold corresponds to the transition
between annular growth of an epidemics spreading
into a susceptible area leaving recovered behind and compact growth
of a susceptible-infected-susceptible region growing into a susceptible area.
This transition between annular growth and compact growth was described
in the physics literature
long before the reinfection threshold debate broke out in the theoretical
biology literature. A pair approximation solution is given for the spatial
epidemic model and its phase transition lines.
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