Mathematical Biosciences and Engineering (MBE)

A Simple Epidemic Model with Surprising Dynamics

Pages: 133 - 152, Volume 2, Issue 1, January 2005      doi:10.3934/mbe.2005.2.133

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F. Berezovskaya - Department of Mathematics, Howard University, Washington D.C., 20059, United States (email)
G. Karev - Oak Ridge Institute for Science and Education (ORISE) 8600 Rockville Pike, Bldg. 38A, Rm. 5N511N, Bethesda, MD 20894, United States (email)
Baojun Song - Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, United States (email)
Carlos Castillo-Chavez - Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, United States (email)

Abstract: 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.

Keywords:  Epidemic model, dynamical system, bifurcation analysis, global stability.
Mathematics Subject Classification:  92D30, 34C37, 37G35.

Received: July 2004;      Accepted: August 2004;      Available Online: November 2004.