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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 $$({\mathcal R}_d)$$, the basic epidemiological reproductive number $$({\mathcal R}_0)$$, the ratio $$(\nu)$$ between the average life spans of susceptible and infective class, and the relative fecundity of infectives $$(\theta)$$, 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:
 92D30 Epidemiology 37N25 Dynamical systems in biology 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37G35 Dynamical aspects of attractors and their bifurcations 34D23 Global stability of solutions to ordinary differential equations
##### Keywords:
rescaling; phase portraits