Journal of Applied Mathematics and Computing

, Volume 35, Issue 1–2, pp 229–250

# A delayed SIR epidemic model with saturation incidence and a constant infectious period

Article

## Abstract

In this paper, an SIR epidemic model with saturation incidence and a time delay describing a constant infectious period is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. When the basic reproduction number is greater than unity, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained to estimate the eventual lower bound of the fraction of infectious individuals. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global attractiveness of the endemic equilibrium. Numerical simulations are carried out to illustrate the main results.

### Keywords

SIR epidemic model Saturation incidence Infectious period Time delay Permanence Stability

### Mathematics Subject Classification (2000)

34K20 34K60 92D30

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