### Abstract

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 language | English (US) |
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Pages (from-to) | 207-235 |

Number of pages | 29 |

Journal | Mathematical Biosciences |

Volume | 180 |

DOIs | |

Publication status | Published - Nov 2002 |

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### Keywords

- 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

- Agricultural and Biological Sciences(all)
- Ecology, Evolution, Behavior and Systematics