An endemic model with variable re-infection rate and applications to influenza

Horst Thieme, Jinling Yang

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


An epidemic model is considered, where immunity is not absolute, but individuals that have recovered from the disease can be re-infected at a rate which depends on the time that has passed since their recovery (recovery age). Such a model, e.g., can account for the genetic drift in the influenza virus. In the special case that the model has no vital dynamics, there is no obvious disease-free equilibrium and so the model lacks the usual interplay between the basic replacement ratio being > 1 and the disease-free equilibrium being unstable. In fact, this relatively simple model which combines ordinary differential equations with a transport equation shares with general structured population models the feature that the appropriate state space of the solution semiflow is a space of measures, here on the compactified right real half line, with the weak* topology. The disease-free equilibrium, in terms of recovered individuals, is then represented as a Dirac measure concentrated at infinity. Still it is difficult to linearize about it. This makes the concept of persistence very important, for one can show the following: if the basic replacement ratio is > 1, the disease is uniformly strongly persistent, i.e., the number of infectives is ultimately bounded away from 0 with the bound not depending on the initial data. We also derive various conditions for the local and global stability of the endemic equilibrium in terms of the re-infection rate. For instance, the endemic equilibrium is likely to be locally asymptotically stable if the re-infection rate is a highly sub-homogeneous function of recovery age. Conversely, if the re-infection rate is a step function which is zero at small recovery age, the endemic equilibrium can be unstable.

Original languageEnglish (US)
Pages (from-to)207-235
Number of pages29
JournalMathematical Biosciences
Issue number1-2
StatePublished - 2002


  • Dynamical systems
  • Genetic drift
  • Influenza
  • Integral equation
  • Local and global stability
  • Persistence
  • Re-infection
  • State space of measures
  • Transport equation
  • Weak* topology

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Biochemistry, Genetics and Molecular Biology(all)
  • Immunology and Microbiology(all)
  • Agricultural and Biological Sciences(all)
  • Applied Mathematics


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