Threshold behaviour of a SIR epidemic model with age structure and immigration

J Math Biol. 2008 Jul;57(1):1-27. doi: 10.1007/s00285-007-0143-1. Epub 2007 Nov 6.

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.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Age Factors
  • Communicable Diseases / epidemiology*
  • Disease Outbreaks* / statistics & numerical data
  • Disease Transmission, Infectious
  • Emigration and Immigration*
  • Fertility*
  • Humans
  • Italy
  • Mathematics
  • Models, Biological
  • Population Growth*
  • Reference Values
  • Time Factors