A simple epidemic model with surprising dynamics.

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