The basic reproduction number of discrete SIR and SEIS models with periodic parameters

Hui Cao; Yicang Zhou

doi:10.3934/dcdsb.2013.18.37

Article Contents

2013, Volume 18, Issue 1: 37-56. Doi: 10.3934/dcdsb.2013.18.37

This issue Previous Article Second order corrector in the homogenization of a conductive-radiative heat transfer problem Next Article Weak KAM theory for nonregular commuting Hamiltonians

The basic reproduction number of discrete SIR and SEIS models with periodic parameters

Hui Cao^1, and
Yicang Zhou^2,

1.
School of Science, Shaanxi University of Science & Technology, Xi'an, 710021
2.
Department of Mathematics, Xi'an Jiaotong University, Xi'an, 710049

Received: April 2011

Revised: May 2012

Published: January 2013

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Abstract

Seasonal fluctuations have been observed in many infectious diseases. Discrete epidemic models with periodic epidemiological parameters are formulated and studied to take into account seasonal variations of infectious diseases. The definition and the formula of the basic reproduction number $R_0$ are given by following the framework in [1,2,3,4,5]. Threshold results for a general model are obtained which show that the magnitude of $R_0$ determines whether the disease will go extinct (when $R_0<1$) or not (when $R_0>1$) in the population. Applications of these general results to discrete periodic SIR and SEIS models are demonstrated. The disease persistence and the existence of the positive periodic solution are established. Numerical explorations of the model properties are also presented via several examples including the calculations of the basic reproduction number, conditions for the disease extinction or persistence, and the existence of periodic solutions as well as its stability.

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Mathematics Subject Classification: Primary: 39A10; Secondary: 92D30.

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