ELSEVIER Four SEI Endemic Models with Periodicity and Separatrices LINDA Q. GAO Department of Mathematics, Noah Central College, Napen~ille, Illinois JAIME MENA-LORCA lnstituto de Mathematicas, Universidad Catolica de Valparaiso, Valparaiso, Chile AND HERBERT W. HETHCOTE Department of Mathematics, University of lowa, Iowa City, Iowa ABSTRACT Periodic solutions have been found for some infectious disease models of the SI and SEI types. Here four SEI models with either disease-reduced or uniform reproduction are examined to determine the model features that do and do not lead to periodic solutions. The two SEI models with the simple mass action incidence /3XY can have periodic solutions for some parameter values, but the two SEI models with the standard mass action incidence AXY/N do not have periodic solutions. For some intermediate values of A in the SEI model with incidence AXY/N and uniform reproduction, the interior equilibrium is a saddle whose stable manifold separates the attractive regions for the disease-free equilibrium and the susceptible- free equilibrium. 1. INTRODUCTION Identifying the mechanisms that can lead to periodic solutions in epidemiological models is important so that all possibilities can be considered when periodicity is observed in epidemiological data. It is not surprising that seasonal (periodic) contract rates can cause periodic solutions in models for the spread of infectious diseases, but periodic solutions can also occur in models without seasonal contact rates. In an SIRS model, susceptibles become infectious, then removed with tempo- rary immunity, and then susceptible again when the disease-acquired immunity wears off. Delays in the removed class R in an SIRS model can cause periodic solutions for some parameter values [22, 23]. Some epidemiological models with nonlinear incidence have been shown to Dedicated to the memory of Stavros Busenberg. MATHEMATICAL BIOSCIENCES 128:157-184 (1995) © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010 0025-5564/95/$9.50 SSDI 0025-5564(94)00071-7