Journal of Dynamics and Differential Equations

, Volume 20, Issue 3, pp 699–717

# Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments

Article

## Abstract

The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.

## Keywords

Compartmental models Reproduction ratio Periodicity Threshold dynamics

## Mathematics Subject Classification (2000)

34D20 37B55 92D30

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## Copyright information

© Springer Science+Business Media, LLC 2008

## Authors and Affiliations

1. 1.Department of MathematicsSouthwest UniversityChongqingPeople’s Republic of China
2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada