  We consider in this paper a general SEIRS model describing the dynamics of an
infectious disease including latency, waning immunity and infection-induced
mortality. We derive an infinite system of differential equations that provides
an image of the same infection process, but counting also the reinfections.
Existence and uniqueness of the corresponding Cauchy problem is established in
a suitable space of sequence valued functions, and the asymptotic behavior of
the solutions is characterized, according to the value of the basic
reproduction number. This allows to determine several mean numbers of
reinfections related to the population at endemic equilibrium. We then show how
using jointly measurement of the number of infected individuals and of the
number of primo-infected provides observability and identifiability to a simple
SIS model for which none of these two measures is sufficient to ensure on its
own the same properties.
