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A simple epidemic model with surprising dynamics. (English) Zbl 1061.92052

Summary: A simple model incorporating demographic and epidemiological processes is explored. Four re-parameterized quantities, the basic demographic reproductive number ( d ), the basic epidemiological reproductive number ( 0 ), the ratio (ν) between the average life spans of susceptible and infective class, and the relative fecundity of infectives (θ), are utilized in the qualitative analysis.

Mathematically, non-analytic vector fields are handled by blow-up transformations to carry out a complete and global dynamical analysis. A family of homoclinics is found, suggesting that a disease outbreak would be ignited by a tiny number of infectious individuais.

MSC:
92D30Epidemiology
37N25Dynamical systems in biology
34C37Homoclinic and heteroclinic solutions of ODE
37G35Attractors and their bifurcations
34D23Global stability of ODE