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

Modelling, Analysis, Observability and Identifiability of Epidemic Dynamics with Reinfections