Threshold behaviour of a SIR epidemic model with age structure and immigration
Andrea Franceschetti1, Andrea Pugliese
1Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (TN), Italy. francesc@science.unitn.it
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.
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