### Abstract

A new age-structured model, which incorporates the use of treatment, is designed and qualitatively analysed. The model is, first of all, shown to be properly-posed mathematically by formulating it as an abstract Cauchy problem. For the case where the contact rate is separable (i.e., β(a,b)= β_{1}(a) β_{2}(b)), it is shown that the disease-free equilibrium of the model is locally- and globally-asymptotically stable whenever a certain epidemiological threshold, denoted by R0s, is less than unity. Furthermore, the model has a unique endemic equilibrium when the threshold exceeds unity (this equilibrium is shown to be locally-asymptotically stable if another condition holds). For the case where the natural death and contact rates are constant (i.e., independent of age), the unique endemic equilibrium of the resulting model is shown, using Lyapunov function theory and LaSalle's Invariance Principle, to be globally-asymptotically stable when it exists. Furthermore, for this reduced version of the model (with constant natural death and contact rates), it is shown that the use of treatment could offer positive or negative population-level impact, depending on the size of the parameter associated with the reduction of infectiousness of treated individuals.

Original language | English (US) |
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Pages (from-to) | 10627-10642 |

Number of pages | 16 |

Journal | Applied Mathematics and Computation |

Volume | 219 |

Issue number | 22 |

DOIs | |

State | Published - May 13 2013 |

Externally published | Yes |

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### Keywords

- Abstract Cauchy problem
- Age-structure
- C-semigroup
- Equilibria
- Stability

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Mathematics and Computation*,

*219*(22), 10627-10642. https://doi.org/10.1016/j.amc.2013.03.126