# American Institute of Mathematical Sciences

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2005, 2(1): 133-152. doi: 10.3934/mbe.2005.2.133

## A Simple Epidemic Model with Surprising Dynamics

 1 Department of Mathematics, Howard University, Washington D.C., 20059, United States 2 Oak Ridge Institute for Science and Education (ORISE) 8600 Rockville Pike, Bldg. 38A, Rm. 5N511N, Bethesda, MD 20894, United States 3 Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043 4 Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287

Received  July 2004 Revised  August 2004 Published  November 2004

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 ($\nu$) 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.
Citation: F. Berezovskaya, G. Karev, Baojun Song, Carlos Castillo-Chavez. A Simple Epidemic Model with Surprising Dynamics. Mathematical Biosciences & Engineering, 2005, 2 (1) : 133-152. doi: 10.3934/mbe.2005.2.133
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