# American Institute of Mathematical Sciences

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2008, 5(2): 389-402. doi: 10.3934/mbe.2008.5.389

## SEIR epidemiological model with varying infectivity and infinite delay

 1 Analysis and Stochastics Research Group, Hungarian Academy of Sciences, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1., Hungary 2 Center for Disease Modeling & Dept. of Mathematics and Statistics, York University, Toronto 4700 Keele str., M3J 1P3, ON, Canada

Received  August 2007 Revised  January 2008 Published  March 2008

A new SEIR model with distributed infinite delay is derived when the infectivity depends on the age of infection. The basic reproduction number R0, which is a threshold quantity for the stability of equilibria, is calculated. If $R_0$ < 1, then the disease-free equilibrium is globally asymptotically stable and this is the only equilibrium. On the contrary, if $R_0$ > 1, then an endemic equilibrium appears which is locally asymptotically stable. Applying a perma- nence theorem for infinite dimensional systems, we obtain that the disease is always present when $R_0$ > 1.
Citation: Gergely Röst, Jianhong Wu. SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2008, 5 (2) : 389-402. doi: 10.3934/mbe.2008.5.389
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