title: Threshold behaviour of a SIR epidemic model with age structure and immigration.
abstract: We consider a SIR age-structured model with immigration of infectives in all epidemiological compartments; the population is assumed to be in demographic equilibrium between below-replacement fertility and immigration; the spread of the infection occurs through a general age-dependent kernel. We analyse the equations for steady states; because of immigration of infectives a steady state with a positive density of infectives always exists; however, a quasi-threshold theorem is proved, in the sense that, below the threshold, the density of infectives is close to 0, while it is away from 0, above the threshold; furthermore, conditions that guarantee uniqueness of steady states are obtained. Finally, we present some numerical examples, inspired by the Italian demographic situation, that illustrate the threshold-like behaviour, and other features of the stationary solutions and of the transient.
authors.affiliation: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (TN), Italy. francesc@science.unitn.it
authors.forename: Andrea
authors.initials: A
authors.lastname: Franceschetti
authors.forename: Andrea
authors.initials: A
authors.lastname: Pugliese
mesh_terms: D000367:Age Factors; D003141:Communicable Diseases / Q000453:epidemiology*; D004196:Disease Outbreaks / Q000706:statistics & numerical data; D018562:Disease Transmission, Infectious; D004641:Emigration and Immigration; D005298:Fertility; D006801:Humans; D007558:Italy; D008433:Mathematics; D008954:Models, Biological; D011158:Population Growth; D012016:Reference Values; D013997:Time Factors