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Threshold behaviour of a SIR epidemic model with age structure and immigration. (English) Zbl 1141.92037

Summary: 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.

MSC:

92D30 Epidemiology
45G10 Other nonlinear integral equations
47N60 Applications of operator theory in chemistry and life sciences
35F25 Initial value problems for nonlinear first-order PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
46N60 Applications of functional analysis in biology and other sciences
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