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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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