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Four SEI endemic models with periodicity and separatrices. (English) Zbl 0834.92021
Summary: 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 $$\beta XY$$ can have periodic solutions for some parameter values, but the two SEI models with the standard mass action incidence $$\lambda XY/N$$ do not have periodic solutions. For some intermediate values of $$\lambda$$ in the SEI model with incidence $$\lambda XY/N$$ 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.

##### MSC:
 92D30 Epidemiology 34C25 Periodic solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
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